The speed of life in sharks and rays: methods,patterns, and data-poor applications

by

Sebastián A. Pardo

B.Mar.St. (Hons.), University of Queensland, 2006

Thesis Submitted in Partial Ful�llment of theRequirements for the Degree of

Doctor of Philosophy

in theDepartment of Biological Sciences

Faculty of Science

© Sebastián A. Pardo 2017SIMON FRASER UNIVERSITY

Spring 2017

All rights reserved.However, in accordance with the Copyright Act of Canada, this work may be reproduced

without authorization under the conditions for “Fair Dealing.” Therefore, limitedreproduction of this work for the purposes of private study, research, education, satire,parody, criticism, review and news reporting is likely to be in accordance with the law,

particularly if cited appropriately.

Approval

Name:

Degree:

Title:

Examining Committee:

Sebastián A. Pardo

Doctor of Philosophy

The speed of life in sharks and rays: methods, patterns, and data-poor applications

Chair: Michael A. SilvermanAssociate Professor

Nicholas K. DulvySenior SupervisorProfessor

Andrew B. CooperSupervisorAdjunct ProfessorSchool of Resource and EnvironmentalManagement

John D. ReynoldsSupervisorProfessor

Isabelle M. CôtéInternal ExaminerProfessor

William W. L. CheungExternal ExaminerAssociate ProfessorInstitute for the Oceans and FisheriesThe University of British Columbia

Date Defended: 11 April 2017

ii

Abstract

Since the theory of evolution by natural selection was �rst postulated, biologists have noted thatlife histories evolve following broad patterns across all organisms. Understanding the mecha-nisms causing these relationships is the central focus of life history theory; these insights can alsobe used to better estimate the biology and extinction risk of data-poor species. Sharks, rays, andchimaeras (class Chondrichthyes) are an ideal taxon to explore these relationships as they haveevolved a broad range of life history strategies. In this thesis, I focus on two key time-related lifehistory parameters that are often used as a measure of productivity: growth coe�cient k , which isestimated from the von Bertalan�y growth function (VBGF), and maximum intrinsic rate of pop-ulation increase rmax , estimated by simplifying the Euler-Lotka equation. I begin by clarifyingtwo methodological problems regarding the estimation of growth and productivity. I �rst showthat �xing the y-intercept in the VBGF, a common approach in chondrichthyan age and growthstudies, often causes considerable bias in growth coe�cient estimates, and recommend using thethree-parameter VBGF instead. I then point out an important omission in a method commonlyused for estimating rmax in chondrichthyans and clarify the correct way to estimate it. Next Iexplore the e�ect of uncertainty on the estimation of rmax and show that species with low an-nual reproductive outputs are bound to have very low productivities, thus focus should be placedinto accurately estimating litter sizes, breeding intervals, and the variability of these traits. Asan example of how these insights can be applied, I better estimate growth and productivity for adata sparse species of conservation concern, the Spinetail Devil Ray (Mobula japanica), and showit has a much lower somatic growth rate than previously thought and one of the lowest produc-tivities among chondrichthyans. Finally, I show that productivity in chondrichthyans varies withtemperature as well as depth, and that the scaling of this relationship changes with temperatureaccording to the expectation from Bergman’s rule. My thesis demonstrates that simple insightsfrom life history theory can further our knowledge on the broad patterns that shape the evolutionof life histories we see today, which can be used to inform management of data-poor species.

Keywords: Elasmobranchs; demography; extinction risk; comparative biology; evolutionary biol-ogy; marine ecology

iii

Dedication

This thesis is dedicated to my mother and my father.

iv

“We have also here an acting cause to account for thatbalance so often observed in nature,—a de�ciency in oneset of organs always being compensated by an increaseddevelopment of some others—powerful wingsaccompanying weak feet, or great velocity making up forthe absence of defensive weapons; for it has been shownthat all varieties in which an unbalanced de�ciencyoccurred could not long continue their existence.”

— Alfred R. Wallace, 1858

v

Acknowledgements

I would �rst like to acknowledge that almost all of the work for this thesis was conducted on thetraditional territories of the Musqueam, Squamish, Tsleil-Waututh, and Kwikwetlem peoples.

I am very grateful for having the opportunity to pursue this academic endeavour at SimonFraser University. Nick Dulvy, thank you very much for taking me on as a student (without havingeven met me!) and mentoring me during all these years. I really appreciated your disposition topush me forward as well as give me space when I needed it. Andy Cooper, thank you for allthe stats-related insights and suggestions, my thesis is much more robust thanks to you. JohnReynolds, thank you for the many discussions on life history theory and extinction risk, and forshowing me the local avifauna since day one. I’d also like to thank Isabelle Côtéfor agreeing toexamine both my proposal and my thesis, and William Cheung for being my external examiner.

I’d like to thank the funding sources that supported me over the years: J. Abbott / M. FretwellGraduate Fellowship in Fisheries Biology, President’s PhD Scholarship, as well as Graduate Fel-lowships. I think Leonardo DiCaprio might have chipped in as well...

I am also very grateful to the Biology Department sta� who constantly support studentsthrough the many trials and tribulations of grad school. Marlene Nguyen, Barb Sherman, AmeliaShu, Debbie Sandher, thank you for helping me navigate the stormy waters of grad school.

Over my time in the Biology Department and at the Earth to Ocean Research Group I’ve be-come friends with so many wonderful people who have enriched my life while in Vancouver. Thereare too many of you to remember: Amanda Kissel, Chris Mull, Elly Knight, Philina English, RachelWalls, Jenn and Joel “El Douche-o” Harding, Aleks Maljković, Dan Greenberg, Mikaela Davis, SeanAnderson, Morgan Hocking, Jenny Bigman, Rowan Trebilco, Ryan Cloutier, Noel Swain, MarineOut, I’m sure I’m missing a few! All these years in grad school have been worth it just yourfriendships. I am also grateful to many friends from outside the department, university, and evencountry: Mike Boyd, Jim Palmer, Lauren Ham, Tim Ennis, Leah Ballin, Pete “Swamp Donkey”Kyne, Mark Carras, and too many friends from Chile to list here! Thank you all for your friend-ship through these years. I am also grateful to Bev and Rob Knight, who have the best Vancouverparents I could have hoped for.

I am ine�ably grateful to Michelle, my partner in crime, for all the experiences we’ve sharedand the adventures we’ve had. Your love has constantly helped me grow as a person and nurtured

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me when I’ve needed it, and I truly think I might have not �nished this thesis without your unwa-vering love and support. While our paths may be diverging in the future, I’ll always cherish thatcozy, messy, and loving cuddle puddle you carved out in my heart. I love you.

Lastly, I’d like to thank my family, and especially my parents, for their never-ending supportand continuing encouragement to pursue my curiosity. Me hacen falta las palabras para agrade-cerles por el interminable afecto y apoyo que me han dado para perseguir mis intereses, inclusocuando esto signi�có estar tan lejos de ustedes. Esta tesis se la dedico a ustedes.

vii

Table of Contents

Approval ii

Abstract iii

Dedication iv

Acknowledgements vi

Table of Contents viii

List of Tables xii

List of Figures xiii

1 Introduction 11.1 Life history theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Chondrichthyan life histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Somatic growth and productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Conservation applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Metabolic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Avoiding �shy growth curves 82.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Comparison of observed size at birth and estimated L0 . . . . . . . . . . . 112.3.2 Data creation through simulation . . . . . . . . . . . . . . . . . . . . . . . 122.3.3 How does the 2-p VBGF perform with sparse data? . . . . . . . . . . . . . 142.3.4 Does performance of the 2-p VBGF vary across a range of life histories? . 14

2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Comparison of observed size at birth and estimated L0 . . . . . . . . . . . 14

viii

2.4.2 Fixing L0 often results in biased growth estimates . . . . . . . . . . . . . . 162.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.1 Comparison of observed size at birth and estimated L0 . . . . . . . . . . . 182.5.2 The future of the two-parameter von Bertalan�y growth model . . . . . . 192.5.3 Accuracy versus parsimony in model �tting . . . . . . . . . . . . . . . . . 192.5.4 Consequences for �sheries assessment and marine conservation . . . . . . 20

2.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.7 Supplementary materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7.1 R code for �tting 3-p and 2-p VBGFs using the nls function . . . . . . . . 222.7.2 R code for assessing the e�ect of slight variations in growth coe�cient k

estimates on a simple yield-per-recruit model . . . . . . . . . . . . . . . . 222.7.3 Supplementary Tables and Figures . . . . . . . . . . . . . . . . . . . . . . 23

3 Incorporating survival to maturity in chondrichthyan rmax 263.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Original derivation of rmax . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Accounting for survival to maturity . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.6 Supplementary materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6.1 Detailed derivation of rmax . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6.2 Understanding the confusion around α̃ . . . . . . . . . . . . . . . . . . . . 373.6.3 Assumptions of M = 1/ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.6.4 Supplementary Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Estimating rmax in the face of uncertainty 424.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3.2 Annual reproductive output . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3.3 Age at maturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.4 Instantaneous natural mortality . . . . . . . . . . . . . . . . . . . . . . . . 474.3.5 Estimating rmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.6 Separating uncertainty in input parameters . . . . . . . . . . . . . . . . . 484.3.7 Model sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.8 Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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4.4.1 Estimation of rmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.4.2 Comparing uncertainty in input parameters . . . . . . . . . . . . . . . . . 494.4.3 Model sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.7 Supplementary materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7.1 Alternate scenario with a more uncertain M . . . . . . . . . . . . . . . . . 574.7.2 Supplementary Figures and Tables . . . . . . . . . . . . . . . . . . . . . . 61

5 Growth, productivity, and relative extinction risk of a devil ray 635.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3.1 Part 1: Re-estimating VBGPs for the Spinetail Devil Ray . . . . . . . . . . 655.3.2 Part 2. Estimating total mortality using the catch curve . . . . . . . . . . . 675.3.3 Part 3. Estimating M. japanica maximum population growth rate . . . . . 685.3.4 Part 4. Comparison of Mobula rmax among chondrichthyans . . . . . . . . 69

5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4.1 Part 1: Re-�tting the growth curve for Mobula japanica . . . . . . . . . . . 695.4.2 Part 2. Estimating total mortality using the catch curve . . . . . . . . . . . 705.4.3 Part 3. Maximum population growth rate rmax of the Spinetail Devil Ray . 725.4.4 Part 4. Comparing Mobula rmax to other chondrichthyans . . . . . . . . . 72

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Correlates of productivity and its mass scaling in sharks and rays 796.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3.2 Metabolic scaling expectations . . . . . . . . . . . . . . . . . . . . . . . . 826.3.3 Does depth play a role in the metabolic scaling relationship of rmax? . . . 836.3.4 Metabolic scaling of growth coe�cient k . . . . . . . . . . . . . . . . . . . 85

6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.4.1 Metabolic scaling of rmax . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.4.2 Metabolic scaling of growth coe�cient k . . . . . . . . . . . . . . . . . . . 85

6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.7 Supplementary materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.7.1 Supplementary Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

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7 General discussion 977.1 Main �ndings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 977.2 Signi�cance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 987.3 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 997.4 Concluding thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 101

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List of Tables

Table 2.1 VBGPs used in bias estimaton . . . . . . . . . . . . . . . . . . . . . . . . . . 24Table 2.2 Data and sources for size at birth and L0 values . . . . . . . . . . . . . . . . 25

Table 4.1 Values and sources of life history parameters used to estimate rmax for theten populations studied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Table 4.2 Coe�cients of variation of models with varying levels of uncertainty . . . 52Table 4.4 Coe�cients of variation of models with varying levels of uncertainty in the

scenario with uncertain M numerator . . . . . . . . . . . . . . . . . . . . . 58

Table 5.1 Priors used in the three di�erent Bayesian von Bertalan�y growth models. 67Table 5.2 Von Bertalan�y growth parameter estimates for models with di�ering priors 70

Table 6.1 Productivity models tested in our analysis . . . . . . . . . . . . . . . . . . . 84Table 6.2 Comparison of loд(rmax ) models using AICc, log likelihood, and adjusted R2 89Table 6.3 Coe�cient estimates for all models of loд(rmax ) . . . . . . . . . . . . . . . . 90Table 6.4 Comparison of loд(rmax ) models using AICc, log likelihood, and adjusted R2 93Table 6.5 Coe�cient estimates for all models of loд(k) . . . . . . . . . . . . . . . . . 94Table 6.6 Di�erences ∆AICc across models using 20 di�erent phylogenetic trees . . . 96

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List of Figures

Figure 2.1 Flow diagram of simulation model . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.2 Relationship between L0 and observed size at birth . . . . . . . . . . . . 15Figure 2.3 Biased growth estimates from the two-parameter von Bertalan�y growth

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Figure 3.1 Histogram of percent di�erence between updated rmax values (this study)and previous ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Figure 3.2 Percent di�erence in updated vs previous rmax estimates across life histories 32Figure 3.3 Percent di�erence in updated vs previous rmax estimates with relation to

the αmat/αmax ratio and b . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.4 Comparison of updated rmax estimates with the previous estimates as out-

lined by García et al. (2008) . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 3.5 Comparison between updated rmax and previous method using di�erent

M estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 3.6 Proportional di�erence between updated and previous rmax estimates con-

trasted with life histories . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 4.1 Flow chart of model for estimating uncertainty in rmax . . . . . . . . . . 46Figure 4.2 Estimated rmax values grouped by uncertainty type . . . . . . . . . . . . 50Figure 4.3 Relationship between coe�cient of variation in rmax estimates and life

histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 4.4 Comparison of potential �shing limits based on rmax estimates with and

without accounting for uncertainty . . . . . . . . . . . . . . . . . . . . . 51Figure 4.5 Contour plots of rmax values across life history parameters . . . . . . . . 54Figure 4.6 Estimated rmax values grouped by uncertainty types in the scenario with

uncertain M numerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Figure 4.7 Estimated rmax values for all uncertainty types and associated life history

parameter uncertainty in the scenario with uncertain M numerator . . . 60Figure 4.8 Estimated rmax values grouped by species . . . . . . . . . . . . . . . . . 61Figure 4.9 Estimated rmax values for all uncertainty types and associated uncertainty

in each life history parameter . . . . . . . . . . . . . . . . . . . . . . . . 62

Figure 5.1 Prior and posterior distributions of VBGP for M. japanica . . . . . . . . . 70

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Figure 5.2 Length-at-age data for M. japanica showing multiple model �ts . . . . . 71Figure 5.3 Total mortality (Z ) estimates for M. japanica . . . . . . . . . . . . . . . . 73Figure 5.4 Comparison of maximum intrinsic rate of population increase (rmax ) for

96 elasmobranch species . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 5.5 Comparison of rmax estimates and life history parameters for 96 elasmo-

branch species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 6.1 Phylogeny, body mass, productivity, temperature, and depth in sharks,rays, and chimaeras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 6.2 Coe�cient plots for the best loд(rmax ) models . . . . . . . . . . . . . . . 87Figure 6.3 Relationship between maximum weight and rmax highlighting tempera-

ture and median depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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Chapter 1

Introduction

As a result of natural selection, individuals tend to pass on life history traits that maximize their�tness (Darwin and Wallace 1858). Patterns emerge when comparing these life histories acrosspopulations and species, indicating that the space in which these traits evolve is limited (Stearns1989; 1992). The �eld of life history theory focuses on exploring these relationships to understandthe action of natural selection, and is the main thread that weaves this thesis together.

1.1 Life history theory

Life history traits correlate with each other due to constraints and trade-o�s, evolving followingpredictable patterns (Law 1979; Charnov 1993; Ro� 2002). Constraints can be physical, allometric,physiological, demographic, and genetic conditions that limit the potential life histories of a group(Ro� 1986; 2002; Wootton 1992). Within those constraints, trade-o�s arise when a trait change thatincreases �tness is accompanied with a detrimental change in another trait, thus constraining thepotential parameter space in which these life history can evolve (Stearns 1989; Ro� 2002).

When compared numerically with each other, life history traits often yield ratios that are rel-atively constant (Beverton and Holt 1959; Charnov 1993). For example, Beverton and Holt (1959)recognised there is a positive relationship between natural mortality M and growth coe�cientk among species of a given taxon, the ratio of which was seen to be relatively constant acrossspecies. Thus, gaining knowledge of this relationship enables us to have a rough idea of the natu-ral mortality of a species by just by knowing its growth coe�cient, especially when comparisonsare made within constrained taxonomic groups (Frisk et al. 2001). While these relationships havetraditionally been considered to be relatively invariant, at least within taxa (Charnov 1993), recentevidence suggests they are not always constant and can vary in predictable ways (Prince et al.2014). Thus, exploring this additional axis of variation can improve the inferences made basedon these relationships, which is best accomplished by doing comparative studies encompassing adiverse range of life history strategies.

1

1.2 Chondrichthyan life histories

Sharks, rays, and chimaeras, also known as cartilaginous �shes or chondrichthyans (class Chon-drichthyes) are the oldest extant lineage of jawed vertebrates that originated around 420 millionyears ago (Stein et al. 2017). There are over 1200 known chondrichthyan species, and most (∼1100)are in the order Elasmobranchii that includes both sharks and rays, while the remaining speciesare the chimaeras, members of the order Holocephali. Chondrichthyans are often described asbeing slow growing, late maturing, and having low fecundities yet this only glosses over the truediversity of their life history strategies. Chondrichthyans encompass considerable variability insize, growth, productivity, and reproductive strategies (Compagno 1990), and therefore are idealgroup to study life history relationships. They also inhabit the world’s oceans covering a widerange of temperatures and habitats. For example, they range in size from the Pygmy Shark (Eu-protomicrus bispinatus) at 10 cm to the world’s largest �sh, the Whale Shark (Rhincodon typus)at 20 m. There is also considerable variation in their “speed of life”. For example, age at matu-rity can vary from 2 years in most urolophid and urobatid stingrays (families Urolophidae andUrobatidae, respectively) to over a century in squaliform sharks (Last and Stevens 2009). Theyalso include the longest-living vertebrate known, the Greenland Shark (Somniosus microcephalus)which is estimated to have a lifespan of around 390 years (Nielsen et al. 2016). Chondrichthyansalso encompass a range of reproductive strategies, from egg laying to placental live bearing, and amuch larger range of litter sizes (2–200) than mammals, birds, and many reptiles. Large variabilityin life histories can even be found within a single family; for example the requiem sharks (familyCarcharhinidae) include species ranging over two-orders of magnitude in maximum adult weightfrom 7 kg (Australian Sharpnose Shark, Rhizoprionodon taylori) to over 800 kg (Tiger Shark, Ga-leocerdo cuvier). These contrasts in chondrichthyan life histories makes them an ideal taxon forthe comparative study of life history relationships. Leonard Compagno said it best in his 1990monograph on chondrichthyan diversity:

“It is time to nail a list of ‘chondrichthyian heresies’ on the cathedral door of marinebiology. As with dinosaurs, recent research on cartilaginous �shes has yielded a verydi�erent biological and evolutionary picture (e.g. Northcutt 1977a, Moss 1984, Randall1984, Stevens 1987) from the old mythos of sharks and other cartilaginous �shes beingsimple, stupid, clumsy, vicious, primitive, harmful, asocial, undiverse and unimpor-tant animals. It is time also to combat the teleost and tetrapod chauvinism shownby researchers and the general public, and to consider chondrichthyians as somethingother than a minor and morbid sideshow of the anthropocentric circus of more diverse,useful bony �shes, ‘lovable’ marine mammals and aesthetically pleasing marine birds.”

While exploring the relationships between any life history traits would be a rewarding en-deavour, in this thesis I will focus on two time-related traits: somatic growth rate, and maximumintrinsic rate of population increase. I focus on these two traits as they are often used as a measure

2

of productivity among species (e.g. Musick 1999). Unlike teleosts �shes which often have a pelagiclarval stage, chondrichthyans produce fully developed young which allows for the estimation ofproductivity with limited life history data (Myers and Mertz 1998; Pardo et al. 2016). Furthermore,recent research in tunas has shown that time-related traits, hereafter referred to as the speed oflife, are more related to population declines than those related with size or reproductive output(Juan-Jordá et al. 2015).

1.3 Somatic growth and productivity

Somatic growth, hereafter referred to as growth, is one of the most important measurable lifehistory parameters for individuals and species (Austin et al. 2011; Einum et al. 2012; Paine et al.2012) and is fundamental to understanding life histories, demography, ecosystem dynamics and�sheries sustainability (Beddington and Kirkwood 2005; Frisk et al. 2005). Across species, growthcorrelates with a number of life history traits including natural mortality rate (Pauly 1980; Charnovet al. 2013), lifespan (Hoenig 1983), and reproductive allocation (Lester et al. 2004; Charnov 2008).Importantly, these traits also in�uence the response of species to exploitation (Jennings et al. 1998;Frisk et al. 2005). Natural mortality is notoriously di�cult to estimate for marine �shes (see Pope1975, for a common example of its guesstimation), particularly for poorly studied species. It is oftenindirectly estimated by using known relationships between natural mortality and biological traitssuch as age at maturity, somatic growth rate, and longevity (see Kenchington 2014; Then et al.2014). The intrinsic link between life histories and demography allows the use of life history traitsto quantify extinction risk (Musick 1999; Frisk et al. 2001; Dulvy et al. 2004). Identifying whichlife history traits relate to resilience to a range of selective pressures (such as �shing) is crucialfor averting over-exploitation or extinction, particularly of data poor species (Reynolds 2003).These relationships between growth, mortality and ultimately productivity, ensure that somaticgrowth has a central role in life histories (Charnov 1993), population dynamics (Jennings et al.1998) and risk assessment (Hoenig 1983; Charnov et al. 2013; Jennings et al. 1998). It is thereforeimperative to be able to estimate growth correctly accounting for any sources of uncertainty whichcan potentially impact the estimation of biological reference points (Tsai et al. 2011). Once theestimation of growth and rmax under uncertainty is understood, we can investigate how variationin growth rates might arise, and in turn in�uence extinction risk. These emergent patterns on howpopulations evolve provide insights into the evolutionary mechanisms that result in the myriad oflife history strategies we see in living beings today.

The most widely used method of describing growth in sharks and rays is the von Bertalan�ygrowth function, or VBGF (von Bertalan�y 1938; 1957). A key factor that enables chondrichthyansto exploit a whole gamut of habitat and trophic levels is that, like most �shes, they grow contin-uously and asymptotically throughout their lives; this growth is well-described by the von Berta-lan�y model (Beverton and Holt 1959; Cailliet et al. 2006). Von Bertalan�y hypothesised that netgrowth, i.e. the change in mass over time resulting from the di�erence between anabolism and

3

catabolism, is approximately a one-third power function of size describing the net e�ect of bothmetabolic processes. One of its key parameters, the von Bertalan�y growth coe�cient (k), is therate at which growth approximates theoretical asymptotic size (L∞) such that it takes ln2/k unitsof time to grow halfway toward L∞ at any given point (Fabens 1965).

While the variant of the VBGF most often used for chondrichthyans estimates three parame-ters (growth coe�cient k , asymptotic size L∞, and size-at-age zero L0), another variant has beenincreasingly used (e.g. Neer et al. 2005; Braccini et al. 2007; Pierce and Bennett 2010), which �xesthe size-at-age zero parameter to a known value, the empirical size at birth. This two-parametervariant presents the opportunity to save one degree of freedom in the model �tting process, butimplicitly assumes that the L0 parameter (better described as the theoretical average length whenage is zero) is identical to, and can be replaced by, an empirical estimate of size at birth. Whilethere are speci�c situations where �xing model parameters may improve growth estimates, suchas the case of �edgling growth (Tjørve and Tjørve 2010; Austin et al. 2011), the consequences of�xing parameters on model performance in the von Bertalan�y growth function have only rarelybeen examined. In Chapter 2 I use simulation modelling to compare, in terms of bias and un-certainty, the estimation of the growth coe�cient (k) between the two- and three-parameter vonBertalan�y growth functions in data-rich as well as data-sparse scenarios, and determine whetherthese di�erences vary across a range of carcharhiniform shark life histories.

Another fundamental measure of the “speed of life”, and more speci�cally of productivity, is theintrinsic rate of population increase better known as r (in year-1), which describes the abundancetrajectory of a population that arises from birth and death rates. The intrinsic rate of increase canvary from values near zero when a population is at carrying capacity (and sometimes negativevalues when above carrying capacity or due to overexploitation) to its maximum value, denotedas rmax .

This metric is also known as the maximum intrinsic rate of population increase and re�ectsthe productivity of depleted populations where density-dependent regulation is absent (Myerset al. 1997; Myers and Mertz 1998). When population trajectories are lacking, rmax is useful forevaluating a species’ relative risk of overexploitation (e.g. Dulvy et al. 2014b) as it is equivalent tothe �shing mortality that will drive a species to extinction, Fext (Dulvy et al. 2004; Gedamke et al.2007; Cortés et al. 2014). Given the paucity of life history data in chondrichthyans, rmax is oftenestimated from an unstructured derivation of the Euler-Lotka equation, which in its simplest formonly required information on age at maturity, annual reproductive output, and natural mortality(Myers et al. 1997; Hutchings et al. 2012; Pardo et al. 2016; Cortés 2016).

Surprisingly, survival to maturity has not been incorporated into calculations of rmax for chon-drichthyans. As most of these species lack stock-recruitment relationships, survival to maturityat low population sizes has been assumed to be very high and hence set to one because they havehigh investment per o�spring (García et al. 2008; Hutchings et al. 2012; Dulvy et al. 2014b). Inother words, species with one or hundreds of o�spring annually were assumed to have the samesurvival through the juvenile life stage. However, juvenile survival is likely to vary among chon-

4

drichthyans even in the absence of density-dependence as they have a wide variety of reproductivemodes (ranging from egg-laying to placental live-bearing) including some of the longest gestationperiods in the animal kingdom (Branstetter 1990). Sensitivity analyses of age- and stage-structuredmodels show that juvenile survival is a key determinant of population growth (λ), especially forspecies with low rmax (Cortés 2002; Frisk et al. 2005; Kindsvater et al. 2016). In Chapter 3, I clarifya commonly-made mistake regarding the estimation of rmax for chondrichthyans and propose aupdated method for its estimation.

1.4 Uncertainty

Recently, there has been increasing awareness of the importance of considering multiple sourcesof uncertainty in demographic parameter estimation, such as rmax (Simpfendorfer et al. 2011). Inaddition, demographic modelling frameworks quantify the degree of caution that should be ex-ercised for their sustainable management and can have major implications for the conservationof species (Caswell et al. 1998; Cortés 2002; Cortés et al. 2014). The following sources of uncer-tainty are among those that can be easily propagated in a modelling framework: measurementerror (or trait error) stems from uncertainty in the empirical estimation of a life history parameter(Harwood and Stokes 2003; Quiroz et al. 2010), and coe�cient error is derived from the uncer-tainty in the values of the coe�cients of a model (e.g., uncertainty around the intercept of a linearmodel) (Quiroz et al. 2010). In Chapter 4, I explore how including uncertainty in an unstructureddemographic model (described in Chapter 3), stemming from measurement error, a�ects uncer-tainty in estimates of the maximum intrinsic rate of population increase rmax , as well as assess thesensitivity of these estimates to uncertainty in each input parameter.

1.5 Conservation applications

Sharks, rays, and chimaeras face a conservation problem. Due to increasing demand of their prod-ucts and the subsequent increase in �shing e�ort, 17.4% of all chondrichthyans are threatened,facing an elevated risk of extinction (Dulvy et al. 2014a). Importantly, chondrichthyans have thehighest proportion of species with unknown threat status among vertebrates (Ho�mann et al.2010), with almost half (46.8%) of all known species classi�ed as Data De�cient by the InternationalUnion for the Conservation of Nature (IUCN). In other words, there is not enough information onthe risk of extinction for nearly half of the world’s sharks, rays, and chimaeras to assess their con-servation status. This poses a question: how can we measure risk of extinction for species that aredata-poor? With the increasing push to assess the conservation status of all species, life historytheory provides a framework with which to estimate life history traits and productivity of speciesthat are poorly studied.

Understanding the sustainability and extinction risk of data-sparse species is a pressing prob-lem for policy-makers and managers. This challenge can be compounded by economic, social and

5

environmental actions. One group of chondrichthyans that are facing considerable threats are themobulid rays (family Mobulidae). This group includes two species of charismatic and relativelywell-studied manta rays (Manta spp.), which support a circumtropical dive tourism industry withan estimated worth of $73 million USD per year (O’Malley et al. 2013). The Mobulidae also includesnine described species of devil rays (Mobula spp.). Devil rays are increasingly threatened by targetand incidental capture in a wide range of �sheries, from small-scale artisanal to industrial trawland purse seine �sheries targeting pelagic �shes (Zeeberg et al. 2006; Croll et al. 2016). The meatis sold or consumed domestically, and the gill plates have been exported, mostly to China, to beconsumed as a health tonic (Couturier et al. 2012). Furthermore, small-scale subsistence and arti-sanal �sheries, mainly for meat, have operated throughout the world for decades (Couturier et al.2012). Even if devil rays are handled carefully and released, their post-release mortality might besigni�cant (Francis and Jones 2016). Currently, we do not know whether current �shing pressureand international trade demand for devil rays are signi�cant enough to cause population declinesand increase extinction risk. The degree to which devil ray populations can withstand currentpatterns and levels of �shing mortality depends on their intrinsic productivity, which determinestheir capacity to replace individuals removed by �shing. In Chapter 5, I apply the insights pro-vided in the previous chapters on growth and rmax to a species of special conservation concern:the Spinetail Devil Ray (Mobula japanica), with the objective to inform the decision-making pro-cess for its proposed listing under the Convention on International Trade in Endangered Species(CITES).

1.6 Metabolic theory

Life history relationships can also be examined using a metabolic theory of ecology framework.The metabolic theory of ecology postulates that smaller organisms, and those at higher temper-atures, have higher mass-speci�c metabolic requirements than larger, colder ones as a functionof the fractal arrangement of their vascular networks (West et al. 1997; Brown et al. 2004). Thistheory also predicts that scaling relationships in multiples of 1⁄4 should be seen in biological ratessuch as rmax , and is supported by empirical data (Slobodkin 1980; Savage et al. 2004). Over recentyears it has become more apparent that scaling coe�cients are not �xed, but vary as a result ofother factors aside size and temperature (Clarke 2006; Glazier 2005; 2010). For example, Sibly andBrown (2007) suggested that in mammals there is another axis of variation which is driven bylifestyle as a result of evolved functional traits aside from allometric variation due to body size(Sibly and Brown 2007, note that temperature is not an axis of variation in endotherms).

In marine ecosystems, exploring variation in productivity across depth presents a unique op-portunity to investigate this lifestyle axis across an energetic gradient. There are multiple ways inwhich energy availability decreases with increasing depth: temperature, light, and consequentlyprimary productivity decrease below the photic zone (Gage and Tyler 1991; Jahnke 1996); metabolicrequirements in species decrease with increasing depth as the distance at which predators and

6

prey interact is reduced as light levels drop (Childress et al. 1990; Seibel and Drazen 2007); animalbiomass decreases with increasing depth (Rex et al. 2006); even the unique physiology of chon-drichthyans can increase the energetic cost of living in the deep, and consequently reduce theenergy available for production (Treberg and Speers-Roesch 2016). All these potential energeticgradients are likely to impact the productivity of chondrichthyans across a depth gradient, evenafter temperature is accounted for.

Productivity among chondrichthyans is known to decrease with increasing size (Hutchingset al. 2012; Dulvy et al. 2014b) and depths (García et al. 2008; Simpfendorfer and Kyne 2009).However, there is some evidence that the relationship between productivity and body size breaksdown in the deep sea as even small deep-water chondrichthyans have very low productivities(Rigby and Simpfendorfer 2015). From a metabolic theory perspective, this suggests there arestronger constraints on the mass scaling of production rates occurring at greater depths than thoseimposed from metabolism alone. In Chapter 6, I use a metabolic theory framework to tease apartthe roles of body mass, temperature, and depth on the productivity of chondrichthyans, and toassess whether the mass scaling relationship varies with depth or temperature.

1.7 Contributions

This introduction and Chapter 7 (General discussion) are written in the �rst-person singular. Chap-ters 2–6 are written in the �rst-person plural since they are derived from published manuscripts(Chapters 2, 3 and 5) or from manuscripts written for submission to scienti�c journals with co-authors (Chapters 4 and 6). While I wrote the code, analysed the data, and wrote the �rst draftsof the text for all chapters, they all bene�ted from discussions, editing, and comments from theco-authors listed at the beginning of each chapter and colleagues mentioned in each chapter’sacknowledgements.

7

Chapter 2

Avoiding �shy growth curves1

2.1 Abstract

1. Somatic growth is a fundamental property of living organisms, and is of particular impor-tance for species with indeterminate growth that can change in size continuously through-out their life. For example, in �shes, an individual can increase in size by 2-6 orders ofmagnitude during its lifetime, resulting in changes in production, consumption, and func-tion at the ecosystem scale. Among species, growth rates are traded o� against other lifehistory parameters, hence an accurate description of growth is essential to understandingthe comparative demography, productivity, �sheries yield, and extinction risk of populationsand species.

2. The growth trajectory of indeterminate growing species is usually modelled using a threeparameter logarithmic function, the von Bertalan�y growth function (VBGF), to describe thetotal length of the average individual at any given age. Recently, however, a two-parameterform has gained popularity. The third y-intercept parameter (L0) of the VBGF has beeninterpreted as being biologically equivalent and �xed as the size at birth, rather than beingestimated in the model-�tting process.

3. We tested the equivalence assumption that L0 is the same or similar to size at birth by com-paring direct estimates of size at birth available from the literature and compared them withestimates of L0 from published data from elasmobranchs, and found that even though thereis an overlap of values, there is a high degree of variability between them.

4. We calculate the level of bias in the growth coe�cient (k) of the VBGF by comparison be-tween the two- and three-parameter estimation methods. We show that slight deviations in�xed L0 can cause considerable bias in growth estimates in the two-parameter VBGF whileproviding no bene�t even when L0 matches the true value. We show that the e�ect of this

1A version of this chapter appears as Pardo, S. A., Cooper, A. B., Dulvy, N. K. 2013. Avoiding �shy growth curves.Methods in Ecology and Evolution 4(4): 353-360. http://dx.doi.org/10.1111/2041-210x.12020.

8

biased growth estimate has profound consequences on the assessment of �sheries stock sta-tus.

5. We strongly recommend the use of the three-parameter VBGF and discourage the use ofthe two-parameter VBGF because the latter results in substantially biased growth estimateseven with slight variations in the value of the �xed L0 parameter.

2.2 Introduction

Growth is one of the most important measurable life history parameters for individuals and species(Austin et al. 2011; Einum et al. 2012; Paine et al. 2012). Recent comparative and analytical workhas shown that understanding growth is fundamental to understanding life histories, demogra-phy, ecosystem dynamics and �sheries sustainability (Beddington and Kirkwood 2005; Frisk et al.2005). Across species, growth correlates with a number of life history traits including natural mor-tality rate (Pauly 1980; Charnov et al. 2013), lifespan (Hoenig 1983), and reproductive allocation(Lester et al. 2004; Charnov 2008); traits that also in�uence the response of species to exploitation(Jennings et al. 1998; Frisk et al. 2005).

A widely used method of describing growth, currently utilized in at least one hundred pub-lished articles per year, is the von Bertalan�y growth function, or VBGF (von Bertalan�y 1938;1957). This model has been used to describe the change in body size over time of fossil and modernand species across a wide range of taxa, including mammals (English et al. 2012), birds (Tjørve andTjørve 2010), reptiles (including dinosaurs) (Lehman and Woodward 2008), amphibians (Arntzen2000), but it is most extensively applied across the most speciose vertebrate taxon—the �shes(Chen et al. 1992; Frisk et al. 2001). Most �sheries stock assessment models rely on von Bertalan�ygrowth models to convert between population numbers and biomass.

Von Bertalan�y hypothesised that net growth, i.e., the change in mass over time resulting fromthe di�erence between anabolism and catabolism, is approximately a one-third power function ofsize describing the net e�ect of both metabolic processes. By integrating and converting to a lengthformulation (assuming weight is proportional to the third power of length) von Bertalan�y de�nedgrowth in length as:

Lt = L∞ − (L∞ − L0)e−kt (2.1)

where Lt is length-at-age t (age in years, length in cm), L∞ is the asymptotic size (in cm), kis the growth coe�cient (in years-1), and L0 is the length-at- age zero (in cm) (Fig. 1a). Whileasymptotic size (L∞) is the maximum theoretical size that a species will tend towards to but neveractually reach, the growth coe�cient (k) is the rate at which growth approximates this asymptotesuch that it takes ln2/k units of time to grow halfway towards L∞ at any given point (Fabens1965). The third parameter used in the von Bertalan�y growth equation is the size-at-age zero (L0)which equates to the y-intercept. Note that two key parameters often lie well beyond the data (the

9

smallest theoretical size L0 and the largest asymptotic theoretical size L∞). Von Bertalan�y growthmodels are �tted to empirical length-at-age data (in �shes, age is usually estimated from tree ring-like growth checks in the otoliths, vertebrae, or spines), with age on the x-axis and length on they-axis, and models are �t using non-linear sum-of-squares �tting methods (Appendix I). Someteleost age and growth studies also �x the intercept to zero (McGarvey and Fowler 2002; Tayloret al. 2005; Gwinn et al. 2010), but the two-parameter von Bertalan�y growth function is mostwidely applied to elasmobranchs as they tend to have a large size at hatch or at birth.

Elasmobranchs, like most �shes, grow continuously and asymptotically throughout their livesand their growth is well-described by the von Bertalan�y model (Beverton and Holt 1959; Cail-liet et al. 2006). In a recent review of elasmobranch age and growth studies, Cailliet et al. (2006)recommended the use of the von Bertalan�y growth function based on the L0 parameter. Thisformulation then allows �xing L0 to a known value, the empirical size at birth, and presents theopportunity to save one degree of freedom in the model �tting process. The key assumption is thatthe L0 parameter (better described as the theoretical average length when age is zero) is identicalto, and can be replaced by, an empirical estimate of size at birth. As a consequence, the two-parameter von Bertalan�y growth function only requires the estimation of the remaining growthparameters L∞, and k from the available length-at-age data. The use of this two-parameter vonBertalan�y growth function has proliferated in recent years (Neer et al. 2005; Braccini et al. 2007;Pierce and Bennett 2010). While there are speci�c situations where �xing model parameters mayimprove growth estimates, such as the case of �edgling growth (Tjørve and Tjørve 2010; Austinet al. 2011), the consequences of �xing parameters on model performance in the von Bertalan�ygrowth function have only rarely been examined. A recent comparison of growth models showedthat and even though the two-parameter von Bertalan�y growth model was overall the most parsi-monious model (i.e., best ranked using Akaike Information Criteria, or AIC), it appears to performbetter only in data-sparse simulations compared to the three-parameter variant which performsbest in data-rich settings (Thorson and Simpfendorfer 2009).

In addition to �tting two- and three-parameter von Bertalan�y growth models there is anemerging practice of �tting multiple models (both von Bertalan�y models as well as others, suchas Gompertz and logistic), comparing them using Akaike Information Criteria (AIC), and reportingparameter estimates of all candidate models or a single set of estimates from multi-model aver-aging (Katsanevakis 2006; Katsanevakis and Maravelias 2008; Thorson and Simpfendorfer 2009).This approach addresses the question of which model is most parsimonious with the availabledata, trading o� model complexity with goodness-of-�t. Unfortunately, a model can be the mostparsimonious while still incorrectly describing the underlying growth trajectory. In this study wetest performance not by the parsimony approach of AIC, but instead by determining whether thetwo- or the three-parameter von Bertalan�y growth model provides parameter estimates that areclosest to the true (simulated) values.

The speci�c aims of this study are: (1) to test the equivalence assumption that L0 is the sameor similar to empirically estimated size at birth, (2) to compare, in terms of bias and uncertainty,

10

the estimation of the growth coe�cient (k) between the two- and three-parameter estimationmethods of the von Bertalan�y growth function in data-rich as well as data-sparse scenarios, and(3) to determine whether these di�erences vary across a range of life histories. We show that L0is not equivalent to size at birth, and that assuming so results in severely biased growth estimates,which in turn adversely biases understanding of �sheries stock status. This case study provides ageneral caution against �xing parameters in order to save one degree of freedom, especially whenthe underlying parameters covary.

2.3 Methods

First, we assessed the relevance of our analyses by comparing literature estimates of empiricalsize at birth with estimates of L0 from published data from elasmobranchs. Second, to evaluate theperformance of the two-parameter von Bertalan�y growth model when L0 is uncertain, we sim-ulated a length-at-age dataset for a hypothetical ground shark (Carcharhinus sp.) from which wesubsampled points with replacement (bootstrap), and then �tted the three growth models to eachbootstrap sample and calculated a range of k values. For the two- parameter models we systemati-cally varied L0 as a proportion of the true value of L0, i.e., used to create the simulated length-at-agedata. Third, we generalise our �nding across a wider range of life histories by running our analysesusing covarying combinations of simulated k , L∞, and L0 values.

2.3.1 Comparison of observed size at birth and estimated L0

Published empirical size at birth estimates were compared with L0 estimates available for 30 elas-mobranch species from 12 families. Bias was de�ned as the ratio of size at birth and L0 (i.e. size atbirth/L0), with high values > 1 indicating empirical size at birth is greater than the statistical pa-rameter and vice versa. This analysis is not a direct measure of the actual bias in published growthstudies, but rather the potential bias that might arise if the available species-level empirical size atbirth estimates were substituted for the L0 parameter.

Estimates of von Bertalan�y growth parameters were obtained from a database search in ISIWeb of Science using the following search terms (elasmo*, shark, skate, ray and chimaera in com-bination with age, growth, demography and age determination), combined with manual searchesof references cited in these papers. We retained only those parameter estimates from wild caughtspecimens (not aquarium studies), and only where asymptotic size L∞, growth coe�cient k , andeither theoretical age-at-length zero t0 or length-at-age zero L0 were estimated. Where possible,we used growth parameters for both sexes combined, and used females estimates if sexes wereseparated. Parameter estimates were retained for equations computed from both observed andback-calculated data. If t0 was estimated instead of L0 we calculated the latter using the followingequation:

L0 = L∞(1 − ekt0) (2.2)

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where k , L∞, and t0 are the von Bertalan�y growth parameters. Estimates of size at birth wereobtained from a life history database compiled by NKD, containing all records of elasmobranch sizeat birth in elasmobranchs published in the literature up to the year 2008, and is based primarilyon the Food and Agriculture Organisation Fisheries Synopses (Compagno 1984a;b).

2.3.2 Data creation through simulation

We simulated a length-at-age dataset for a hypothetical ground shark (which closely resemblesthe life history of the spinner shark Carcharhinus brevipinna) from which we bootstrapped points.The simulated dataset consisted of 300 length-at-age estimates for each of 20 age classes, totalling6000 datapoints.

First, we created a von Bertalan�y growth curve for our hypothetical elasmobranch specieswith growth coe�cient k = 0.1, asymptotic size L∞ = 200 cm, L0 = 59.06 cm (equivalent to t0 =-3.5), and Tmax = 19 (Figs 2.1a,b; Carlson and Baremore 2005).

Second, to simulate uncertainty in size-at-age values, for each of the 20 age classes we drew 300random draws from a log-normal distribution with bias correction centered on the mean (Fig. 2.1)(Hilborn and Mangel 1997):

Ltε = Lt × eN(µ,σ )+ σ 2

2 (2.3)

where Ltε is the distribution of lengths at age t with error included, Lt is length-at-age t fromthe original model, and N(µ,σ ) is a normal distribution with a mean of 0 (µ) and a standard devi-ation (σ 2) of 0.1.

Third, from this simulated dataset, we drew 250 samples (with replacement) of 150 pointseach, using a negative exponential probability distribution (Fig. 2.1d). This distribution was usedto approximate the observed distribution of size frequencies of length-at-age data as expected fora population where mortality is constant (there are always more juveniles than adults).

For each subsample, three variants of the von Bertalan�y growth function were �tted by non-linear least squares using the nls function (Appendix 1) in R version 2.14.2 (R Development CoreTeam 2012): (i) a three-parameter VBGF (ii) the modi�ed two-parameter VBGF where averagelength-at-age zero (L0) is �xed or set to the empirical size at birth, and (iii) a version of the two-parameter VBGF where the L0 is �xed iteratively from a normal distribution of possible sizes atbirth based on the methodology outlined by Neer et al. (2005). In the three-parameter model allthree von Bertalan�y growth parameters (k , L∞, and L0) are estimated in the equation; however,in both two-parameter variants of the VBGF the L0 is �xed or drawn iteratively from a knownnormal distribution in each subsample. In the case of the two-parameter growth models, the valueof the L0 parameter used in both two-parameter von Bertalan�y growth function were systemati-cally �xed as a proportion of real L0 (over the range from 0.7 to 1.3) to assess the e�ect of �xing L0

[hereafter referred to as assumed L0 (Fig. 2.1e,f)]. Values were calculated at 0.01 intervals between0.85 and 1.15 and at 0.05 intervals at the remainder of assumed L0 values. Both two-parameter

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Set VBG parameters(L∞, k, L0)

Add lognormal variability (x300) at each age

class

Subsample 150 points

drawn from negative

exponential distribution

x250

Fit 2-parameter

VBGF by fixing assumed L0

Extract k (coef &s.d.) and

calculate 10,000 k values from

normal distribution

Calculate quantiles of k for each biased L0

(2.5, 25, 50, 75, and 97.5 percentiles)

x30

Extract k (coef & s.d.) and

calculate 10,000 k values from

normal distribution

Generate points for each age

class (0 to 19)

Set assumed L0 as a proportion of real L0 (0.7 to 1.3

in 30 steps)

Fit 3-parameter

VBGF

(a)

(d)

(c)

(b)

(e) (f)

(g)

Figure 2.1: Flow diagram of simulation model created for assessing the e�ect of �xing the L0 pa-rameter on the estimation of growth coe�cient k in the von Bertalan�y growth function.

13

formulations of the von Bertalan�y growth model produced very similar growth estimates, hereafter we only compare the version where L0 is replaced with a single empirical estimate of L0.

In order to assess the bias of the models, 20,000 random draws of the estimated growth coe�-cients k were taken from normal distributions, which were then compared with the real estimateof k (Fig. 2.1g). Median values, 50%, and 95% quantiles were calculated from the pooled coe�cientsin each variant of the von Bertalan�y growth function.

2.3.3 How does the two-parameter VBGF perform with sparse data?

We measured performance under two common data scenarios and an extreme one: (1) a data-richscenario, in which a total of 150 length-at-age points are subsampled across all age classes; (2) adata-sparse where only 20 length-at-age points are subsampled across all age classes (hereafterreferred to as “thinned scenario”); and (3) a second data-sparse scenario in which 20 length-at-agepoints are subsampled excluding the youngest three age classes (ages 0-2; hereafter referred to as“thinned/no-juveniles scenario”) (Fig. 2.1d).

2.3.4 Does performance of the two-parameter VBGF vary across a range of life his-tories?

Our initial performance analysis focused on a species with a relatively slow life history (i.e., slowgrowth rates and large size; k = 0.1 and L∞ = 200 cm, respectively). Rather than comparing twospecies with contrasting life histories (Thorson and Simpfendorfer 2009) we expanded our simu-lation to cover a broader range of life histories for species with growth rates ranging from k = 0.09(slow) to 0.54 year-1 (fast) and asymptotic sizes ranging for L∞ = 70 to 225 cm (see Table S1). Todo this we need to understand the degree to which the von Bertalan�y growth parameters covary.Therefore, we modeled the covariance between von Bertalan�y growth parameters by �tting alinear model to the log- transformed coe�cients (k , L∞, and L0) of ten carcharhinid shark pop-ulations (Carlson and Baremore 2003; 2005; Carlson et al. 2003; 2006; 2007; Lombardi-Carlsonet al. 2003; Neer et al. 2005; Piercy et al. 2007; 2010). We used these �tted models to provide acontinuous range of k , L∞, and L0 values (see Table S1). We calculated bias in the estimation ofgrowth coe�cient k for this range of life histories as previously; by systematically �xing assumedL0 as a proportion of real L0 (over the range from 0.7 to 1.3). Given that we were not estimatinguncertainty, we only drew 1000 k values from each growth model �tted.

2.4 Results

2.4.1 Comparison of observed size at birth and estimated L0

The ratio between empirical size at birth of elasmobranchs and estimatedL0 from published growthcurves ranged between 0.5 and 4.11, but with all ranges falling between 0.7 and 1.3 (Fig. 2.2, seeTable S2). While the two are correlated, only just over half of species (18 out of 30) had overlapping

14

0 20 40 60 80 1000

20

40

60

80

100(a)

L 0 parameter (cm)

Siz

e a

t bir

th (

cm

)

0.0 0.1 0.2 0.3 0.4 0.5

0.5

1.0

1.5

2.0

(b)

Growth coefficient k

Ratio b

etw

een s

ize a

t bir

th a

nd L

0

Figure 2.2: The L0 parameter of the von Bertalan�y growth equation is not the same as the ob-served size at birth. (a) Discrepancy between observed sizes at birth and estimated L0 parameterfor 30 elasmobranch species for 41 studies published in the literature. Box width and height rep-resent range in size at birth and variation in L0 amongst published studies on the same species,respectively. Diagonal line indicates 1:1 relationship. (b) Discrepancy between size at birth and L0does not vary systematically across life histories, as indexed by growth coe�cient k . The ratio forDeania calcea lies o� the plot and is not included in the graph.

15

size at birth values and L0 estimates (Fig. 2.2a). There were no clear patterns or biases in eitherparameter, and there is no apparent correlation between growth coe�cient k and the ratio of sizeat birth to L0 estimates (Fig. 2.2b).

2.4.2 Fixing L0 often results in biased growth estimates

The growth coe�cient (k) estimated using the two-parameter model was increasingly biased asa result of the discrepancy between the actual and assumed L0. This bias increased at a rate ofapproximately a 2.7% change in growth rate (k) for every per cent bias in assumed L0. The growthcoe�cients of both models were signi�cantly di�erent (i.e. the 95% quantiles of growth coe�cientestimates stopped overlapping) from the true value of k after an approximately 12% discrepancybetween the assumed and real L0. Fixing L0 with a smaller than true size at birth resulted in anoverestimated growth rate, and vice versa. Uncertainty in the growth estimate was comparablewhen both two and three-parameter growth models were �t to a large number of data pointsspanning the complete lifespan of the species (as described by the width of 95% con�dence interval;Fig. 2.3a).

Uncertainty in the estimate of the growth coe�cient (k) was greater for both models whenfewer length-at-age data were available. More importantly, the bias of the two-parameter modelincreased slightly from 2.7% for every percent change in assumed L0 in the data-rich scenario(Fig. 2.3a) to approximately 3% in the thinned scenario (Fig. 2.3b). In the thinned/no-juvenilesscenario, uncertainty in the estimation of k was high for both two- and three-parameter modelsyet it was much higher in the three-parameter von Bertalan�y growth function (Fig. 2.3c). Therate at which k deviates from its true value as L0 is systematically biased was also slightly reducedin this scenario, with approximately 2% change in k for every per cent change in assumed L0.

The degree of bias in growth estimates was greatest for species with slow growth (k ≈ 0.1year-1); however, bias was reduced when no data were available in the youngest age classes (notethe increased space between isopleths in Fig. 2.3f compared with Figs 2.3d,e). In the data-richscenario, any given discrepancy in L0 bias was almost twice as high for slow growing species(k ≈ 0.1 year-1) compared to faster growing species (k ≈ 0.5 year-1; Fig. 2.3d). Bias in growth rateestimation decreased considerably as growth coe�cient k approached 0.25 year-1, whereupon itlevelled o� and remained fairly constant for higher growth rates. The bias in k estimates acrossa range of life histories in the thinned scenario was similar to the bias in the data-rich scenario,albeit at slightly higher rate (Fig. 2.3e). As in the previous scenarios, the thinned/no-juvenilesscenario resulted in higher uncertainty for slower-growing species than for faster-growing ones.Nonetheless, in fast growing species there was a positive bias in the estimation of k even whenthere was no bias in the assumed L0 (Fig. 2.3f).

The range of observed discrepancies between empirical sizes at birth and L0 (0.62–3.63, seeTable S2) is larger than the range of variability in assumed L0 values used for the simulation modelsin this study (0.7–1.3, Figs 2.3d,e,f). Thus, there is scope for even larger biasing of growth estimates

16

0.7

0.8

0.9

1.0

1.1

1.2

1.3 3-parameter VBGF2-parameter VBGF

(a)

0.7

0.8

0.9

1.0

1.1

1.2

1.3 (b)

0.00 0.05 0.10 0.15 0.20 0.250.7

0.8

0.9

1.0

1.1

1.2

1.3 (c)

-50 -40 -30

-20

-10

0

10

20

304050

(d)

-50 -40 -30

-20

-10

0

10

20

304050

(e)

0.1 0.2 0.3 0.4 0.5

-20-40 -30

-10

0

1020304050

(f)

Estimated growth coefficient k Real growth coefficient k

Ratio

bet

ween

ass

umed

L0

and

true

L0

Figure 2.3: Biased growth estimates from the two-parameter von Bertalan�y growth model for (a,d) data-rich scenario with 150 lengths at each age across all age classes; (b, e) data-poor scenariowith 20 lengths at each age across all age classes; and (c, f) no juvenile data-poor situation with 20lengths at each age but none from the youngest three (0-2 years) age classes. The left-hand panels(a–c) show the bias in estimated growth coe�cient k across a range of assumed L0 for a blacknoseshark Carcharhinus acronotus life history (L∞ = 200 cm, k = 0.1 year-1, L0 = 59.06 cm). Dark linesare the median estimate of k , with 50% (darker shading) and 95% (lighter shading) quantiles for thetwo- (red) and three-parameter (grey) model. The right-hand column (d–f) shows the median biasin estimated k (as a percentage di�erence from real k) across a fuller range of life histories (x-axis)for varying bias in L0 (y-axis). Grey dots represent the mean ratios of size at birth to L0 publishedin the literature (see Table 1 and Fig 2.2b). The dotted lines represent the k value at which (a), (b),and (c) were respectively computed (k = 0.1 year-1). All lines are lowess-smoothed.

17

than those explored in this analysis if those extremely dissimilar values of size at birth were usedto �x L0 in the von Bertalan�y growth model.

2.5 Discussion

Fixing model parameters may seem like an appealing approach, particularly when faced with fewdata and one degree of freedom can be saved in the estimation of parameters. However, for the caseof the von Bertalan�y growth model, we show how �xing one parameter results in a substantialrisk of estimating a biased growth parameter, which far outweighs any bene�ts of this approach.The modi�ed two-parameter von Bertalan�y growth model does not reduce bias in growth esti-mates, at least in a scenario with robust length-at-age data, and in fact can potentially increasebias if the value of L0 is even marginally di�erent from the underlying “true” value. Furthermore,in all scenarios we explored except the thinned/no juveniles scenario, there was no added bene�tof �tting a two-parameter von Bertalan�y growth model to the data.

The �ndings in this study are also applicable to bony �shes (teleosts) where the t0 parameter-ization of the von Bertalan�y is used, with some studies advocating for �xing the t0 parameter tozero in particular cases (McGarvey and Fowler 2002; Taylor et al. 2005; Gwinn et al. 2010). Giventhat the t0 and the L0 parameterizations are mathematically equivalent (Cailliet et al. 2006), �xingit to a speci�ed value will result in the same mathematical constraints and hence similar e�ectson model performance as those highlighted in our study.

2.5.1 Comparison of observed size at birth and estimated L0

The L0 parameter from the von Bertalan�y growth model is not equivalent to the size at birthobserved empirically. This can be due to a least three factors. Firstly, in their �rst year individualsspan a range of sizes, as a result of di�ering birth size and foraging competency in the face of pre-dation risk at a time when the gain in weight per gram of individual is greatest. Secondly, whileage should be a continuous variable, it is usually binned in yearly groups due to the practical con-straints of age determination. Thirdly, size at birth usually represents the minimum size at which aspecies is born, and is derived from comparisons between neonates with umbilical scars and fullydeveloped embryos. The suggestion by Cailliet et al. (2006) to use all known values of size at birthfor a species is not likely to reduce bias in the estimation of growth, as suggested by the similarperformance of both two parameter models tested. We do support their suggestion of comparingthe estimated L0 parameter from the traditional three-parameter von Bertalan�y growth modelwith published accounts of size at birth to evaluate how reasonably the model �ts the data, in thesame way the estimated L∞ is compared with maximum published size of a given species to assessgoodness of �t. As pointed out by Knight (1968) the problem lies when parameters from the vonBertalan�y growth model are regarded “as a fact of nature rather than as a mathematical artifactof the data analysis”.

18

2.5.2 The future of the two-parameter von Bertalan�y growth model

So when might we consider using a two-parameter von Bertalan�y growth model instead ofthe three-parameter variant? Under typical sampling strategies (shown in Figs 2.3a,b), the two-parameter von Bertalan�y growth model provides little reduction in uncertainty and risks biasingthe estimates of the growth coe�cient k . When data are sparse and the sample is mainly com-posed of adults (i.e., with few juveniles and sub-adults available) the L0 parameter will be poorlyestimated and, due to the correlation in parameters, will result in a poorly �t model with broadcon�dence intervals. While bias may be reduced, uncertainty is not vastly improved by using thetwo-parameter von Bertalan�y growth model. In such a data-poor situation with predominantlyadult samples, �xing the L0 parameter might be justi�ed. However, we warn that even if L0 isestimated correctly, a sample size of 20 length-at-age points with no juveniles (ages 0 to 3) resultsin an estimate of k which has con�dence intervals that encompass a ± 75% di�erence from the truevalue (Fig. 2.3c, 95% con�dence intervals in red). If such a growth curve is �t, it must be recog-nised that the uncertainty in growth coe�cient estimates must be accounted for and propagatedin any stock assessments. A novel approach to improving estimation of growth in data-sparsesituations with the aid of a known, empirical size at birth, but without using a two-parameter vonBertalan�y growth function, has been outlined by Smart et al. (2013) based on back-calculationtechniques and has been shown to perform well from an information-theoretic perspective. Test-ing the performance of this approach under uncertain L0 parameter values should be the focus offurther investigation.

2.5.3 Accuracy versus parsimony in model �tting

The AIC approach, commonly used in age and growth studies, balances model complexity (num-ber of parameters) and goodness of �t (the likelihood), but does not necessarily provide the mostunbiased parameter estimates. Previous work has shown that the two-parameter von Bertalan�ygrowth model was selected most often by AIC (partially because it estimates one less param-eter) while at the same time providing the least accuracy in parameter recovery (Thorson andSimpfendorfer 2009). In their study the �xed L0 in the two-parameter models equalled true L0

(i.e., the value of L0 used to create simulated data points). This scenario is rarely encountered:size at birth is not the same as L0 (Fig. 2.1). As we show here, once uncertainty in size at birth isaccounted for, the performance of two-parameter von Bertalan�y model is more biased than orig-inally suspected (Thorson and Simpfendorfer 2009). While AIC can be useful for discriminatingbetween models it does not necessarily provide unbiased parameter estimates, which is the key�nding of our study.

19

2.5.4 Consequences for �sheries assessment and marine conservation

“It ain’t what you don’t know that gets you into trouble.It’s what you know for sure that just ain’t so.”

— Mark Twain

These �ndings have important implications for �sheries management and the conservation ofmarine resources. The growth coe�cient k is a key parameter for the estimation of biomass and assuch any stock assessment which calculates spawning stock biomass can potentially be profoundlya�ected by inaccurate estimates of growth. The impact on stock assessments can be explored usinga simple yield-per-recruit model (See Appendix 2). An 8% overestimate of L0 results in a k estimateof 0.08 year-1 (Fig. 2.3a), which is 20% lower than true k . This in turn leads to a 20.1% reduction inthe estimate of yield-per-recruit biomass at F0.1 when compared with calculations based on trueL0. Whereas an 8% underestimate in L0 causes an overestimation of k of 20% (0.12 year-1), resultingin a F0.1 yield-per-recruit estimate that is 20.8% higher than if true L0 was used in the estimationof k.

Many life history invariant relationships are also derived from von Bertalan�y growth param-eters (Charnov 1993; 2008; Charnov et al. 2013). In turn, population assessment models, partic-ularly those for data poor species, make routine use of invariants (Dulvy et al. 2004; Le Quesneand Jennings 2012; Pardo et al. 2012). For example, a natural mortality parameter is commonlycalculated in stock assessments using von Bertalan�y growth parameters as proxies (Pauly 1980;Charnov et al. 2013). Hence using biased estimates of natural mortality derived from biased esti-mates of growth can lead to erroneous assessments of stock status. Furthermore, given that vonBertalan�y growth parameters are correlated with each other (Pilling et al. 2002), biases in theestimation of growth coe�cient (k) will have impacts on the estimation of asymptotic size (L∞);these parameters covary negatively. Thus, �xing L0 with a smaller than true value will result inan underestimate of asymptotic size as well as an overestimate of k , which can further a�ect theire�ective use as proxies in life history estimation and stock assessments.

In conclusion, we strongly discourage the use of empirical size at birth for �xing the L0 param-eter in the von Bertalan�y growth model. Furthermore, where multiple growth models are �t andmulti-model averaging framework used, two-parameter models that �x the intercept parametershould not be considered as candidate models unless data are sparse for juvenile age classes. Moregenerally, our case study suggests cautions against �xing parameters to save a degree of freedomand lower the AIC score of a model, without an understanding of the biases that may arise.

2.6 Acknowledgements

The authors are grateful to Lucy Harrison and John Carlson for providing data, and we thank theEarth to Ocean Research group for their critical comments, and Elly Knight, María José Juan Jordá,Lindsay Davidson, Chris Mull, Rowan Trebilco, Calen Ryan, and the two anonymous reviewers

20

for their constructive comments on the manuscript. This work was funded by the Natural Scienceand Engineering Research Council of Canada.

21

2.7 Supplementary materials

A version of the following supporting materials is available online athttp://onlinelibrary.wiley.com/doi/10.1111/2041-210x.12020/suppinfo.

2.7.1 R code for �tting three- and two-parameter von Bertalan�y growth functions(VBGF) using the nls function.

#########################################################################################

### #

### Fitting three- and two-parameter von Bertalanffy growth functions (VBGF) using #

### the nls() function. For a more in depth review of model fitting, see: #

### http://www.ncfaculty.net/dogle/fishR/gnrlex/VonBertalanffy/VonBertalanffy.pdf #

### #

### The VBGF equation is in the form of: #

### #

### Length = Linf - ((Linf - L0) * exp(-k*Age)) #

### #

#########################################################################################

# Creating simulated length-at-age points with lognormal error centered around the median

# with growth parameters Linf = 200, k = 0.1, and L0 = 59.06

Age <- rep(0:19, each = 10)

lnorm.error <- exp(rnorm(length(Age), 0, 0.1) + 0.1^2/2)

Length <- (200 - ((200 - 59.06) * exp(-0.1 * Age[]))) * lnorm.error[]

# The von Bertalanffy growth function is defined as:

VBGF <- Length ~ Linf - ((Linf - L0) * exp(-k * Age))

# In order to fit at three-parameter VBFG the parameters do not have to be specified,

# but some starting values for the optimization need to be provided

nls(VBGF, start = list(Linf = 100, k = 0.2, L0 = 20))

# to fit the two-parameter VBGF, the L0 parameter has to be specified as well as starting

# values for the remaining two parameters

L0 <- 59.06

nls(VBGF, start = list(Linf = 100, k = 0.2))

2.7.2 R code for assessing the e�ect of slight variations in growth coe�cient k esti-mates on a simple yield-per-recruit model of Carcharhinus brevipinna.

#########################################################################################

### #

### Assessing the effect of slight variations in growth coefficient (k) estimates #

22

### on a simple yield-per-recruit model of Carcharhinus brevipinna #

### #

#########################################################################################

# Loading package contaning the yield-per-recruit (ypr) function

library(fishmethods)

# Setting von Bertalanffy growth parameters

k <- 0.1 # growth coefficient

Linf <- 200 # asymptotic length

L0 <- 59 # length-at-age zero

# Values for the length-weight conversion (from Carcharhinus brevipinna in FishBase)

a <- 0.0075

b <- 2.97

# Range of ages

age.range <- 0:19

# Selectivity of each age class (to be multiplied with instantaneous fishing mortality)

sel.at.age <- c(rep(0, 2), rep(1, 18))

# Calculating weighs-at-age for three different scenarios:

# A) k = 0.10

weighs01 <- a*(Linf - ((Linf - L0) * exp(-k * age.range)))^b

# B) k = 0.08

weighs008 <- a*(Linf - ((Linf - L0) * exp(-0.08 * age.range)))^b

# C) k = 0.12

weighs012 <- a*(Linf - ((Linf - L0) * exp(-0.12 * age.range)))^b

# Calculating the yield-per-recruit value at F_0.1

# A) k = 0.10

YPR01 <- ypr(age = age.range, wgt = weighs01, partial = sel.at.age, M = 0.2,

plus = FALSE, maxF = 4,incrF = 0.01)$Reference_Points[1, 2]

# B) k = 0.08

YPR008 <- ypr(age = age.range, wgt = weighs008, partial = sel.at.age, M = 0.2,

plus = FALSE, maxF = 4,incrF = 0.01)$Reference_Points[1, 2]

# C) k = 0.12

YPR012 <- ypr(age = age.range, wgt = weighs012, partial = sel.at.age, M = 0.2,

plus = FALSE, maxF = 4,incrF = 0.01)$Reference_Points[1, 2]

# Ratio of YPR between estimates with k = 0.08 and k = 0.10

YPR008/YPR01

# Ratio of YPR between estimates with k = 0.12 and k = 0.10

YPR012/YPR01

2.7.3 Supplementary Tables and Figures

23

Table 2.1: Range of von Bertalan�y growth parameter values (L∞, k , and L0) used for the estimationof bias from �xing L0 through a life history continuum. Life histories were modeled based oncarcharhinid and sphyrnid sharks. Table is ordered from low to high L∞ values. with the tablesplit into two columns.

L∞ k L0

70 0.546 31.4275 0.491 32.6380 0.444 33.8185 0.405 34.9690 0.370 36.0895 0.341 37.16100 0.315 38.23105 0.292 39.27110 0.272 40.29115 0.254 41.29120 0.238 42.26125 0.223 43.22130 0.210 44.17135 0.198 45.09140 0.187 46.01145 0.177 46.90

L∞ k L0

150 0.168 47.79155 0.160 48.66160 0.152 49.51165 0.145 50.36170 0.139 51.19175 0.133 52.02180 0.127 52.83185 0.122 53.63190 0.117 54.43195 0.112 55.21200 0.108 55.98205 0.104 56.75210 0.100 57.51215 0.096 58.26220 0.093 59.00225 0.090 59.73

24

Tabl

e2.2:

Dat

ause

dfo

rcal

cula

ting

empi

rical

disc

repa

ncie

sbet

wee

nsiz

eatb

irth

and

theL

0pa

ram

eter

from

thev

onBe

rtala

n�y

grow

thfu

nctio

n.Sp

ecie

sare

orde

red

taxo

nom

ical

ly.F

L=

fork

leng

th,T

L=

tota

llen

gth,

PCL

=pr

ecau

dall

engt

h,an

dD

W=

disc

wid

th.R

efer

ence

suse

din

thet

able

are

liste

dbe

low.

Ord

erSp

ecie

sM

easu

rem

ent

L 0es

timat

eSi

zeat

birth

Grow

thco

e�ci

entk

Size

atbi

rth/L

0ra

tioRe

fere

nces

used

(min

–max

,cm

)(m

in–m

ax,c

m)

(min

–max

,yea

r-1)

(mea

n)(m

in–m

ax)

Squa

lifor

mes

Deaniacalcea

TL8.3

29–3

40.

083.6

33.5

1–4.

11Cl

arke

etal

.(20

02);

Cox

and

Fran

cis(

1997

)Etmopterusspinax

TL7.2

8.5–1

10.

161.4

91.1

8–1.5

3Ge

nnar

iand

Scac

co(2

007)

;Vac

chia

ndO

rsiR

elin

i(19

79);

Fisc

her

(1987

)Squa

lusacan

thias

TL16

.9–23

.519

–30

0.07

–0.17

1.14

0.81

–1.77

Avsa

r(20

01);

Hen

ders

onet

al.(

2002

);Pa

wso

nan

dEl

lis(2

005)

Lam

nifo

rmes

Carchariastaurus

TL95

–109

.491

–105

0.09

–0.18

0.96

0.83

–1.11

Gold

man

etal

.(20

06);

Bran

stet

ter

and

Mus

ick

(1994

);Gi

lmor

eet

al.(

1983

)Alopias

pelagicus

PCL

94.5

72–9

10.

080.

890.

76–0

.96Li

uet

al.(

1999

)Alopias

superciliosus

PCL

72.1

66.1–

73.7

0.09

1.18

0.92

–1.0

2Ch

enet

al.(

1997

);Li

uet

al.(

1998

)La

mna

ditropis

PCL

62.1–

67.1

60–7

00.

17–0

.181.0

80.

89–1

.13Go

ldm

anet

al.(

2006

);N

agas

awa

(1998

)La

mna

nasus

FL78

.6–9

3.655

–67

s0.

720.

59–0

.85Fr

anci

seta

l.(2

007)

;Fra

ncis

and

Mao

lagá

in(2

000)

;Jen

sen

etal

.(2

002)

;Nat

anso

net

al.(

2002

)Ca

rcha

rhin

iform

esFurgaleusmacki

FL22

.019

.8–24

.30.

370.

960.

9–1.1

1Si

mpf

endo

rfere

tal.

(200

0);S

impf

endo

rfera

ndUn

swor

th(19

98)

Galeorhinus

galeus

TL36

.930

–36

0.09

0.88

0.81

–0.98

Fran

cisa

ndM

ullig

an(19

98);

Lena

nton

etal

.(19

90)

Musteluscanis

TL53

.433

–40

0.29

0.62

0.62

–0.75

Conr

ath

etal

.(20

02);

Conr

ath

and

Mus

ick

(200

2)Mustelusman

azo

TL33

.620

–35

0.11

0.71

0.6–

1.04

Yam

aguc

hiet

al.(

1996

;200

0)Mustelusmustelus

TL39

.339

–41

0.06

0.99

0.99

–1.0

4Go

osen

and

Smal

e(19

97);

Smal

ean

dCo

mpa

gno

(1997

)Carcharhinu

sacronotus

FL37

.9–59

29.2–

34.1

0.18

–0.35

0.89

0.5–

0.9

Bran

stet

ter(

1981

);Ca

rlson

etal

.(19

99);

Drig

gers

etal

.(20

04a;

b)Carcharhinu

sbrevipinna

FL58

.6–7

4.7

51.6

–55.8

0.08

–0.15

0.94

0.69

–0.95

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nan

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er(2

002)

;Car

lson

and

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mor

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005)

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nget

al.(

2005

);Br

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90)

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sfalcifo

rmis

TL71

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70.

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get

al.(

2008

);Br

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r(19

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Carcharhinu

sisodon

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0.24

0.80

0.78

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onet

al.(

2003

);Br

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0.71

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al.(

2004

);Jo

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(1995

);M

cAul

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2006

);Ro

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al.(

2006

);Ta

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71)

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ana

(1998

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80.

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ntan

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dLe

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ompa

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(1984

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0.86

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bone

etal

.(20

08);

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(1987

);N

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130.

630.

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1979

);Sk

omal

and

Nat

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n(2

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0.46

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6–1.3

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rlson

and

Bare

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003)

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rnalewini

TL31

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.246

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0.09

–0.25

1.28

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1.6A

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do-T

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and

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-Men

doza

(200

1);A

nisla

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al.(

2008

);Pi

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07);

Chen

etal

.(19

90);

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al.(

2001

);Ch

enet

al.(

1988

);Br

anst

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r(19

87c)

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and

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ons(

1997

);Pa

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b)Ra

jifor

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Rhinobatos

rhinobatos

TL24

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0.13

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0.99

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.(20

08);

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(1993

)Malacorajasenta

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10.

1–0.

120.

960.

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.09

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and

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9);N

atan

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.(20

07)

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101.0

00.

96–1

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rra-

Pere

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al.(

2008

);Pa

wso

nan

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005)

Uroloph

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aculatus

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0.26

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0.68

0.55

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tean

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tter(

2005

);Ed

war

ds(19

80)

25

Chapter 3

Maximum intrinsic rate of populationincrease in sharks, rays, and chimaeras:the importance of survival to maturity2

3.1 Abstract

The maximum intrinsic rate of population increase rmax is a commonly estimated demographicparameter used in assessments of extinction risk. In teleosts, rmax can be calculated using anestimate of spawners per spawner, but for chondrichthyans, most studies have used annual re-productive output b instead. This is problematic as it e�ectively assumes all juveniles survive tomaturity. Here, we propose an updated rmax equation that uses a simple mortality estimator whichalso accounts for survival to maturity: the reciprocal of average lifespan. For 94 chondrichthyans,we now estimate that rmax values are on average 10% lower than previously published. Our up-dated rmax estimates are lower than previously published for species that mature later relativeto maximum age and those with high annual fecundity. The most extreme discrepancies in rmax

values occur in species with low age at maturity and low annual reproductive output. Our resultsindicate that chondrichthyans that mature relatively later in life, and to a lesser extent those thatare highly fecund, are less resilient to �shing than previously thought.

3.2 Introduction

The rate of increase is a fundamental property of populations that arises from birth and death rates.A commonly used metric for guiding assessments of extinction risk and setting limit referencepoints is the maximum intrinsic rate of population increase rmax ; it re�ects the productivity of

2A version of this chapter appears as Pardo, S. A., Kindsvater, H. K., Reynolds, J. D., Dulvy, N. K. 2016. Maximumintrinsic rate of population increase in sharks, rays, and chimaeras: the importance of survival to maturity. CanadianJournal of Fisheries and Aquatic Sciences, 73(8): 1159-1163. http://dx.doi.org/10.1139/cjfas-2016-0069.

26

depleted populations where density-dependent regulation is absent (Myers et al. 1997; Myers andMertz 1998). When population trajectories are lacking, rmax is useful for evaluating a species’relative risk of overexploitation (Dulvy et al. 2014b) as it is equivalent to the �shing mortality thatwill drive a species to extinction, Fext (Myers and Mertz 1998).

A fundamental parameter in calculating rmax is the product of survival to maturity lαmat andannual fecundity b. Fisheries biologists studying teleost �shes often calculate it based on lifetimespawners per spawner (α̂ ), which is related to the slope near the origin of a stock-recruitmentrelationship (Denney et al. 2002; Dulvy et al. 2004; Hutchings et al. 2012). In other words, thespawners per spawner incorporates juvenile survival and approximates lαmatb.

Surprisingly, survival to maturity has not been incorporated into calculations of rmax for chon-drichthyans (sharks, rays, and chimaeras). As most of these species lack stock-recruitment rela-tionships, survival to maturity at low population sizes has been assumed to be very high and henceset to one because they have high investment per o�spring (García et al. 2008; Hutchings et al.2012; Dulvy et al. 2014b). In other words, species with one or hundreds of o�spring annually wereassumed to have the same survival through the juvenile life stage. However, juvenile survival islikely to vary among chondrichthyans even in the absence of density-dependence as they havea wide variety of reproductive modes (ranging from egg-laying to placental live-bearing) includ-ing some of the longest gestation periods in the animal kingdom (Branstetter 1990). Sensitivityanalyses of age- and stage-structured models show that juvenile survival is a key determinant ofpopulation growth (λ), especially for species with low rmax (Cortés 2002; Frisk et al. 2005; Kinds-vater et al. 2016).

To correct for the assumption that all juveniles survive to maturity, here we show how thecommonly used equation to estimate rmax was derived and then indicate where juvenile survivalis accounted for in the model but has been overlooked. We then introduce a simple updated methodfor estimating rmax that takes into account juvenile survival. Finally, we re-estimate rmax for 94chondrichthyans using our updated equation and the same life history parameters used previously(see supplementary material in García et al. 2008), compare our updated rmax estimated withprevious ones, and discuss which species’ rmax were previously overestimated.

3.3 Methods

3.3.1 Original derivation of rmax

The maximum rate of population increase rmax can be derived from the Euler-Lotka equation indiscrete time:

ω∑t=1

ltbte−r t = 1 (3.1)

27

where t is age, ω is maximum age, lt is the proportion of individuals that survive to age t , btis fecundity at age t , and r is the rate of population increase. This rate changes with populationdensity, but we are concerned with the maximum intrinsic rate of population increase rmax , whichoccurs at very low densities in the absense of density dependence. If we assume that after reachingmaturity at age αmat annual fecundity and annual survival are constant (b and p, respectively), wecan estimate the probability of survival to ages t > αmat as survival to maturity lαmatp

t−αmat ,where lαmat is the proportion of individuals surviving to maturity (Myers et al. 1997).

Annual survival of adults is calculated as p = e−M where M is the species-speci�c instanta-neous natural mortality rate. This allows for survival to maturity lαmat and annual fecundity b tobe removed from the sum and the equation to be rewritten as follows (equation 6 in Myers et al.1997):

lαmatbω∑

t=αmat

pt−αmat e−rmax t = 1 (3.2)

If we solve the summation we obtain the following (see Charnov and Scha�er 1973; Myers et al.1997, and Supplementary materials for a more detailed derivation)

lαmatbe−rmaxαmat

1 − pe−rmax= 1 (3.3)

which we can rearrange as

lαmatb = ermaxαmat − p(ermax )αmat−1 (3.4)

The term outside of the sum lαmatb has been equated to the maximum spawners per spawnerα̃ , thus we can rewrite the equation as

α̃ = ermaxαmat − p(ermax )αmat−1 (3.5)

This is the same equation used by Hutchings et al. (2012) to solve for rmax when estimates ofα̃ are available, and is mathematically equivalent to the equation used by García et al. (2008) inthe case where age of selectivity into the �shery αsel = 1. Equation 3.2 shows that survival tomaturity is only accounted for in lαmat . Calculations of α̃ for chondrichthyans have ignored lαmat ,e�ectively equating it to 1, assuming α̃ = b:

b = ermaxαmat − p(ermax )αmat−1 (3.6)

Hence, the previous equation of rmax for chondrichthyans assumed all individuals surviveduntil maturity. This formulation was used for chondrichthyans by García et al. (2008), Hutchingset al. (2012), and Dulvy et al. (2014b), and is hereafter referred to as the “previous” equation.

The oversight in the previous formulation of rmax is comparable to an erroneous assumptionin �sheries models where steepness — the productivity of the population — is held constant or

28

set to 1 if data from stock-recruitment relationships are not available (reviewed in Mangel et al.2010). Low-fecundity species such as chondrichthyans are assumed to have extremely high juve-nile survival relative to teleost �shes, given that fecundity of sharks and rays is one or two ordersof magnitude lower than most teleosts. However, steepness itself is fundamentally a property ofearly life history traits (Myers et al. 1999; Mangel et al. 2010) and hence should be calculated fromdemographic data or life history relationships.

Furthermore, it is often assumed that density dependence acts mainly upon juvenile survival.When estimating intrinsic rate of population increase, juvenile mortality is assumed to be lowestat very low population sizes, which may have justi�ed its omission from earlier formulations ofthe rmax equation (E.L. Charnov, pers. comm.).

3.3.2 Accounting for survival to maturity

We revise the previous method by incorporating an estimate of juvenile survival that depends onage at maturity and species-speci�c natural mortality. We calculate the proportion of individualssurviving until maturity with the following equation:

lαmat = (e−M )αmat (3.7)

We chose to use a simple estimate of natural mortality M based on average lifespan. Assumingthat the natural mortality rate of a cohort is exponentially distributed, the average mortality rateis the mean of that distribution, which is equivalent to the reciprocal of average lifespan (Dulvyet al. 2004), such that M = 1/ω, where ω is an estimate of average lifespan, in years (See Sup-plementary materials). Since cohort data on average lifespan are di�cult to obtain, we assumeω = (αmax + αmat )/2 — the midpoint between age at maturity and maximum age. We do thisfor three reasons. First, estimates of maximum age are readily available for many chondrichthyanspecies, and they are applicable to most chondrichthyan populations since they have truncatedsize class distributions due to prolonged �shing exposure (Law 2000). Second, chondrichthyanshave low fecundity and large o�spring, which are much more likely to survive to maturity thanspecies with very high fecundity. This means that the average lifespan and the maximum lifespanare likely much closer together for chondrichthyans than for teleosts. Third, some of the commonmethods for estimating M , e.g., Jensen (1996) or Hewitt and Hoenig (2005), result in unrealisticestimates of rmax for many species (i.e., zero or negative, see Fig. 3.5 in Supplementary materials)probably due to natural mortality being overestimated for many chondrichthyan species whenusing estimators based mostly on teleost data. In preliminary analyses we found that when usingthese teleost-based mortality estimators, we could only obtain plausible estimates of rmax for allspecies when ignoring juvenile mortality.

One reason for the overestimation may be that the Hewitt and Hoenig (2005) equation coe�-cients are estimated from data on �sh that have extremely low juvenile survival (mostly teleosts).By contrast, our method assumes that 36.8% of o�spring reach average lifespan (see explanation

29

and Supplementary materials in Hewitt and Hoenig 2005). Put simply, for a species with an aver-age lifespan of ten years, 9.5% of the population must die each year for there to be a 37% chance ofsurviving for ten years. While in teleosts average lifespan is probably less than the age of maturity,for chondrichthyans it is likely greater, which is why we assume it is the mean of age at maturityand maximum observed lifespan. We recalculate rmax for 94 chondrichthyan species examinedin García et al. (2008) and Dulvy et al. (2014b) using our updated method that combines equa-tions 3.4 and 3.7, as well as using the previous method that uses equation 3.6 and Jensen’s (1996) Mestimator. Finally, we compare rmax values from previous and updated methods and explore therelationship between life history parameters and discrepancies in rmax values.

3.4 Results and Discussion

Our updated estimates of maximum intrinsic population growth rates (rmax ) for chondrichthyansare on average 10% lower than previous estimates (Fig. 3.1). For the most fecund species (b >10 female o�spring per year) updated rmax estimates were always 10-20% lower than previousestimates. This means that for species with high fecundity, rmax has been overestimated in thepast (see right side of Fig. 3.2a,b; large circles in Fig. 3.3). In contrast, for less fecund species (b < 5female o�spring per year), discrepancy in rmax between updated and previous estimates variesfrom 30% lower to 80% higher (small circles in Fig. 3.3). Two of the most fecund chondrichthyans,the Big Skate (Raja binoculata) and the Whale Shark (Rhincodon typus), have lower intrinsic ratesof population increase (see Fig. 3.3) and may be less resilient to �shing than previously thought.

The greatest positive and negative discrepancies in rmax values (extremes in percent di�erence)occurred in species with very low annual fecundity and to a lesser extent low age at maturity (seelower left corner of Fig. 3.2a). The proportional di�erence between updated rmax and previousestimates were greatest in species with low rmax values. Alternatively, greater fecundity, combinedwith late maturity “bu�er” against variation in estimates of rmax (Fig. 3.2a,b right side of plots).When age at maturity is low relative to maximum age (αmat/αmax < 0.3), updated rmax estimateswere much higher than previous estimates. For example, the updated rmax estimate for the LobedStingaree (Urolophus lobatus) is 82% higher than its previous rmax estimate, due to its early relativematuration (αmat/αmax = 0.21, Fig. 3.3). Conversely, when age at maturity is high relative tomaximum age (αmat/αmax > 0.4), updated rmax estimates were lower than previous estimates(Fig. 3.3). For example, the Velvet Belly Lanternshark (Etmopterus spinax) and the Blacktip Shark(Carcharhinus limbatus) have relative maturation ratios of 0.71 and 0.65, respectively, and haveupdated rmax values that are 31% and 28% lower than previously estimated (see Fig. 3.3). Whileour study did not explore the relationship between relative maturation (the αmat/αmax ratio) andrmax values among species, a negative relationship between relative maturation and intrinsic rateof population increase has been previously pointed out in sharks (Liu et al. 2015) and skates (Barnettet al. 2013).

30

-40 -20 0 20 40 60 800

5

10

15

20

25

Percent difference in rm ax estimatesbetween updated and previous equations

Freq

uenc

y

Figure 3.1: Histogram of percent di�erence between updated rmax values (this study) and previousones (from García et al. 2008 and Dulvy et al. 2014b). Dashed and dotted lines indicate median andmean values, respectively.

31

Age

at m

atur

ity (y

ears

)

0

10

20

30

40(a)

Annual fecundity (b female pups per year)

αm

at/α

max

ratio

0.2 0.5 2 5 10 50 200

0.2

0.3

0.4

0.5

0.6

0.7 (b)

70% - Updated r m a x is higher 60% 50% 40% 30% 20% 10%-10%-20%-30% - Updated r m a x is lower

Percent difference in rm a x betweenupdated & previous equations

Age at maturity (years)

1 2 5 10 20 50

(c)

Figure 3.2: Annual fecundity (b, in log-scale) vs (a) age at maturity and (b) the αmat/αmax ratio.(c) Age at maturity vs αmat/αmax ratio. Colour indicates whether the updated model estimates ahigher (red) or lower (blue) rmax than the previous formulation, while point size indicates percentdi�erence in rmax estimates between updated and previous models.

32

Percentage difference in r m a x

α mat

/αm

ax ra

tio

-20 0 20 40 60 80

0.2

0.3

0.4

0.5

0.6

0.7 b

0.25-55-1010-5050+

E. spinax

C. limbatus

Figure 3.3: Comparison of percentage di�erence between updated and traditional rmax and theαmat/αmax ratio across di�erent values of annual reproductive output b. Darker grey and largercircles indicate a higher annual reproductive output (b) value. The grey line is the lowess-smoothedcurve. Species highlighed are: E. spinax = Etmopterus spinax, C. limbatus = Carcharhinus limbatus,R. binoculata = Raja binoculata, R. typus = Rhincodon typus, and U. lobatus = Urolophus lobatus.

33

Previous work comparing chondrichthyan life histories often overestimated the maximum rateof population increase by not accounting for the species-speci�c juvenile mortality rate (Garcíaet al. 2008; Hutchings et al. 2012). Juvenile survival was overestimated for all species, particularlyfor highly fecund and late-maturing species, which in�ated their estimated maximum intrinsicpopulation growth rates.

Our simple method to estimate survival to maturity requires no extra parameters but assumesthat juvenile mortality is equal to adult mortality. This is likely to result in conservative estimatesof M because juveniles tend to have higher mortality rates than adults (Cushing 1975). Futurework could explore using age-speci�c mortality estimators to calculate survival to maturity, butwe caution that these estimators are mostly based on teleost �shes and require additional data suchas von Bertalan�y growth parameters (Chen and Watanabe 1989) or weight-at-age relationships(Peterson and Wroblewski 1984).

We found that species with high fecundity all had lower rmax values than previously estimated,hence our method is more e�ective at representing higher juvenile mortality rates in species withhigh fecundity. Nonetheless, direct estimates of di�erential juvenile mortality are still missingfrom both models, and motivates further research on this topic (Heupel and Simpfendorfer 2002).Our method undoubtedly ignores nuances related to absolute o�spring size and litter size (Smithand Fretwell 1974), but it is still likely to be an improvement over the previous assumption that alljuveniles survive to maturity.

These new insights into the maximum intrinsic rates of increase are relevant for the man-agement of data-poor chondrichthyans. We recommend that scientist and managers using chon-drichthyan rmax estimates reevaluate them using our updated equation and focus on species whosermax values have been consistently overestimated in previous studies: highly fecund species, oftenthought to be more resilient to �shing (Sadovy 2001), and those that only reproduce during a shortspan of their total lifetime. To generalize management and conservation implications beyond thespecies in our study, future work needs to revisit our understanding of life history and ecologicalcorrelates of rmax . Previous work suggest species in deeper (colder) habitat (García et al. 2008)as well as those with late age at maturity (Hutchings et al. 2012) have lower rmax values. Theseand other correlates of rmax can now be re-evaluated with these updated estimates and used inecological risk assessments and other forms of management priority setting.

3.5 Acknowledgements

We are grateful to E.L. Charnov and J. Hutchings for discussions on this topic, and to L.K Davidson,P.M. Kyne, J.M. Lawson, and R.W. Stein for their comments on the manuscript. This research wasfunded by the J. Abbott/M. Fretwell Graduate Fellowship in Fisheries Biology (SAP), NSERC Dis-covery Grants (NKD & JDR), a Canada Research Chair (NKD), and an NSF Postdoctoral Fellowshipin Math and Biology (HKK; DBI-1305929).

34

3.6 Supplementary materials

The Supplementary materials include a more detailed account on deriving rmax which uses manyof the same equations in the main text of the body (here repeated for clarity), details on the con-version of lifetime spawners per spawners to a yearly rate, explanation of why 1/ω means that37% of individuals reach average lifespan, and Supplementary Figures.

The raw data used for our analyses are available on �gshare at https://dx.doi.org/10.6084/m9.figshare.3207697.v1.

3.6.1 Detailed derivation of rmax

The maximum rate of population increase rmax is typically derived from the Euler-Lotka equationin discrete time (Myers et al. 1997):

ω∑t=1

ltmte−r t = 1 (3.8)

Where t is age, ω is maximum age, lt is the yearly survival at age t , mt is fecundity at aget , and r is the rate of population increase. This rate changes with population density, but we areconcerned with the maximum intrinsic rate of population increase rmax , which occurs a very lowdensities in the absense of density dependence. Assuming that after reaching maturity annualfecundity and annual surival are constant (b and p, respectively), we can estimate survival to yeart as survival to maturity lαmat times yearly adult survival p for the years after maturation (Myerset al. 1997):

for(t ≥ αmat )

mt = b

lt = lαmatpt−αmat

(3.9)

where αmat is age at maturity, b is annual fecundity, and p is annual survival of adults and iscalculated as p = e−M where M is the species-speci�c instantaneous natural mortality. This allowsfor survival to maturity lαmat and annual fecundity b to be removed from the sum and the equationto be rewritten as follows (equation 6 in Myers et al. 1997)

lαmatbω∑

t=αmat

pt−αmat e−rmax t = 1 (3.10)

If we assume that ω = ∞ we can then solve the geometric series by �nding the common ratio.Let S be the sum:

S =∞∑

t=αmat

pt−αmat e−rmax t (3.11)

We can break down the summation as:

35

S = p0e−rmaxαmat + p1e−rmax (αmat+1)

+ p2e−rmax (αmat+2) + ...(3.12)

which is equivalent to:

S = e−rmaxαmat + pe−rmaxαmat e−rmax

+ p2e−rmaxαmat (e−rmax )2 + ...(3.13)

The value that would convert the �rst item of the sum into the second one, the second item intothe third one and so on, is the common ratio, which in this case is pe−rmax . Multiplying everythingby pe−rmax gives us:

Spe−rmax = pe−rmaxαmat e−rmax

+ p2e−rmaxαmat (e−rmax )2+ p3e−rmaxαmat (e−rmax )3 + ...

(3.14)

Therefore, the product of S andpe−rmax is equal to S minus the �rst item of the series, e−rmaxαmat .We can then subtract this second series (Spe−rmax ) from S :

S − Spe−rmax = e−rmaxαmat (3.15)

Which allows for estimating S as:

S =e−rmaxαmat

1 − pe−rmax(3.16)

We then replace the summation back in the modi�ed Euler-Lotka equation:

lαmatbe−rmaxαmat

1 − pe−rmax= 1 (3.17)

and �nally isolate lαmatb and rearrange:

lαmatb =1

e−rmaxαmat−

pe−rmax

e−rmaxαmat

= ermaxαmat −permaxαmat

ermax

= ermaxαmat − permaxαmat−rmax

= ermaxαmat − permax (αmat−1)

= ermaxαmat − p(ermax )αmat−1

(3.18)

This results in the same equation used by Hutchings et al. (2012), and is mathematically equiva-lent to the equation used by García et al. (2008) in the case where age of selectivity into the �sheryαsel = 1. Equation 3.18 shows that survival to maturity is only encapsulated in lαmat and that itsomission e�ectively assumes that all recruits survive to maturity.

36

3.6.2 Understanding why spawners per spawners per year α̃ has been equated withannual fecundity b

All calculations of spawners per spawner are derived from the lifetime spawners per spanwner, α̂ .The correct description of α̃ is given in Myers et al. (1997), where it is described as “the numberof spawners produced by each spawner per year (after a lag of αmat years, where αmat is age atmaturity)”. Accounting for that lag is key, as then the lifetime spawners per spawner are divided bythe years of sexual maturity, and therefore it is roughly analogous to annual fecundity in femalestimes survival to maturity. The correct way of calculating α̃ is by solving

α̂ =∞∑

t=αmat

pt α̃ (3.19)

Nonetheless, it has previously been calculated without including the lag of αmat years, here-after de�ned as α̃ ′, and has been estimated by solving α̂ = ∑∞t=0 pt α̃ ′, which is the equation used inMyers et al. (1997; 1999) and Goodwin et al. (2006). When using this equation, we are not removingthe years before maturity e�ectively resulting in a metric more akin average yearly spawners perspawner across all age classes. Solving this geometric series without the lag yields:

α̃ ′ = α̂(1 − p) (3.20)

However, as shown in equation 3.19, we can rewrite the geometric series so that it e�ectivelyremoves immature age classes. Assuming that after reaching maturity annual surival is constant:

α̂ =∞∑

t=αmat

lalphamatpt−αmat α̃

= lalphamat α̃∞∑

t=αmat

pt−αmat

(3.21)

By solving it we obtain the following:

α̃ =α̂(1 − p)lalphamat

(3.22)

which is analogous to average yearly spawners per spawner across adult age classes, and there-fore can be used to estimate rmax instead of lαmatb. It also becomes apparent that α̃ = α̃ ′/lalphamat .Given that this estimate of α̃ is divided by a proportion, it is larger than the previous estimate; thisis expected as lifetime spawners per spawner are partitioned between only by mature age classes(α̃ ) instead of all age classes (α̃ ′).

37

3.6.3 Assumptions ofM = 1/ω

As already mentioned, we assume that natural mortality rate of a cohort is exponentially dis-tributed, thus the mean of that distribution is the reciprocal of that rate. Estimating instantaneousnatural mortality M as the reciprocal of average lifespanω is mathematically equivalent to a givenpercentage of the population reachingω. As previously shown by Hewitt and Hoenig (2005) usingtheir equation as an example, we can calculate that by using M as 1/ω, we are assuming that, onaverage, 36.8% of the population reaches average lifespan:

M = 1/ω (3.23)

We then rearrange and exponentiate:

M ∗ ω = 1

e−Mω = e−1(3.24)

The term e−Mω is equivalent to the survival to age ω, or lω . By then calculating the value ofe−1 we can see that:

e−Mω = lω = 0.3678 (3.25)

Therefore using our method, the average survival to average lifespan is 36.8%, or roughly oneout of three individuals.

38

3.6.4 Supplementary Figures

0.1 10.05 0.5

0.1

1

0.05

0.5

CarchariniformesHexanchiformesLamniformesOrectolobiformesSqualiformesSquatiniformesMyliobatiformesPristiformesRajiformesRhinobatiformesTorpediniformesChimaeriformes

Garcia et al. (2008) rm ax estimates(uses Jensen's M )

Upd

ated

rm

ax e

stim

ates

usin

g M

= 1

ω

Figure 3.4: Comparison of updated rmax estimates with the previous estimates as outlined by Gar-cía et al. (2008) (recalculated using the method outlined in their paper) with our updated estimates.Di�erent symbols denote di�erent chondrichthyan orders. Note that the axes are log-transformed.

39

0.0

0.5

1.0

1.5 (a)

0.0

0.5

1.0

1.5 (b)

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5 (c)

Garcia et al. (2008) rm ax estimates(uses Jensen's M )

Upd

ated

rm

ax e

stim

ates

usin

g M

= 1

ωU

pdat

ed r

max

est

imat

esus

ing

Hew

itt &

Hoe

nig'

s M

Upd

ated

rm

ax e

stim

ates

usin

g Je

nsen

's M

Figure 3.5: Comparison between updated rmax values with natural mortality estimated from (a)reciprocal of average lifespan, (b) Hewitt and Hoenig (2005), and (c) Jensen (1996). The dashed linerepresents the 1:1 relationship. Note that only the updated method using the reciprocal of averagelifespan (a) shows similar values to the previous rmax estimates, while (b) and (c) often producermax estimates equal to zero or negative (both represented here by zeros).

40

0.8

1.0

1.2

1.4

1.6

1.8 (a) b0.25-11-55-1010-5050+

0.8

1.0

1.2

1.4

1.6

1.8 (b) αma t to αma x ratio

< 0.30.3-0.40.4-0.50.5-0.60.6 <

0.0 0.5 1.0 1.5

0.8

1.0

1.2

1.4

1.6

1.8 (c) αma t

1-44-66-99-1313+

Prop

ortio

nal d

iffer

ence

bet

ween

upd

ated

and

prev

ious

rm

ax e

stim

ates

Garcia et al. (2008) rm ax estimates(uses Jensen's M )

Figure 3.6: Proportional di�erence between updated and previous rmax estimates contrasted with(a) annual reproductive output of daughters, (b) αmat/αmax ratio and (c) age at maturity. Thedashed line represents no di�erence between updated and previous estimates.

41

Chapter 4

Quantifying the known unknowns:estimating maximum intrinsic rate ofpopulation increase in the face ofuncertainty3

4.1 Abstract

Scientists are often called upon to provide advice on the consequences of exploiting data-poorand potentially threatened sharks and rays. For species lacking detailed biological information,intrinsic sensitivity is often estimated using simple models that depend upon life history estimates,yet there has been little exploration of the degree to which uncertainty in life history parameterscan in�uence demographic parameter estimates, and therefore �sheries management options. Weestimate the maximum intrinsic rate of population increase (rmax ) for ten coastal carcharhiniformshark populations using an unstructured life history model that explicitly accounts for uncertaintyin life history parameters. We also seek to understand which parameters most in�uenced theestimate of rmax . Although the median rmax estimate proved to be robust to the propagation ofuncertainty of life history parameters when that uncertainty is symmetric around the parameters’point estimates, accounting for uncertainty can lead to a wide range of plausible rmax estimatesfor any given species. While natural mortality M is important when its uncertainty is high, wefocussed on the two directly estimated parameters age at maturity αmat and annual reproductiveoutput b as they are easier to estimate than M . Uncertainty in age at maturity values was low, yetit still resulted in moderate uncertainty in rmax estimates. The model was particularly sensitiveto uncertainty in b when annual reproductive output approached values less than �ve, which isnot unusual for large elasmobranchs and marine mammals. Interestingly, at very low b values

3A.B. Cooper, J.D. Reynolds, and N.K. Dulvy are co-authors on this chapter, which is in preparation for submissionto a journal

42

(<1), there is a threshold that results in implausible rmax estimates when both M and αmat arehigh. Managers and policy makers should be careful to assess and restrict mortality on specieswith annual reproductive outputs <2 females per year. Given that natural mortality is extremelyresource intensive to estimate accurately, we urge elasmobranch biologists to quantify frequencydistributions of litter sizes, how these vary with maternal age and breeding intervals, as well as agesat maturity of data-poor elasmobranchs rather than attempting to improve estimates of naturalmortality.

4.2 Introduction

Many marine megafauna and predator populations are declining globally and are at increasingrisk of local and regional extinction (Fowler 2005; Christensen et al. 2014; Dulvy et al. 2014a).Sharks, rays, and chimaeras (class Chondrichthyes) play a complex role as marine predators, hencethere is concern for potential indirect impacts of predator declines on marine ecosystems (Kitchellet al. 2002; Heithaus et al. 2008; Heupel et al. 2014). They are often large-bodied, long-lived, late-maturing and produce few o�spring (Compagno 1990; Musick 1999). Consequently, they tend tohave low intrinsic rates of population increase and weak compensatory density dependence injuvenile survival (Compagno 1990; Musick 1999; Forrest and Walters 2009); traits which makethem particularly susceptible to overexploitation (Kindsvater et al. 2016).

There are relatively few stock-assessed elasmobranch species, and they are mostly caught asbycatch in longline �sheries targeting large �n�sh (Kitchell et al. 2002) and trawl �sheries (Elliset al. 2005). Despite the large catch and high value of elasmobranchs, our understanding of species-speci�c catches is poor because many species are not targeted (Stevens 2000) but are consideredvaluable bycatch. Hence, accurate information on population trends is lacking for most species.These challenges to understanding sustainability of elasmobranch �sheries and guiding their ef-fective management to inform policy frameworks that promote precautionary and responsible�sheries are compounded further by indiscriminate �sheries and poor species-speci�c monitoring(FAO 2014; Barker and Schluessel 2005; Lack and Sant 2009).

Many countries have recently adopted policy regulations that require them to assess �sheriesaccording to an Ecosystem-Based Management (EBM) approach (Jennings and Rice 2011; Rogerset al. 2007). The EBM approach requires, among other things, the identi�cation of safe ecologicallimits for bycatch species (Hobday et al. 2011; Salomon and Holm-Müller 2013). These species,which lack detailed stock assessments and are not the focus of targeted commercial extraction,are usually understudied, resulting in a dearth of information on their biology and demography.As such, the usual data-thirsty stock assessment methods are not applicable for a large diversityof bycatch, which has led to a recent increase in the development of tools for the assessment ofdata-poor species.

Identifying which life history traits a�ect resilience to a range of selective pressures (e.g. �sh-ing) is crucial for averting over-exploitation or extinction of data poor species (Reynolds 2003). As

43

life history theory suggests, certain life history parameters are related to one another due to theevolutionary constraints imposed by energy acquisition and processing (Law 1979; Charnov 1993).Some of these relationships, widely known as Beverton-Holt dimensionless ratios, can be usedto predict other life history parameters and tied to population dynamics, albeit with considerableuncertainty (Dulvy and Forrest 2010). The link between life histories and demography allows theuse of life history traits to quantify extinction risk (Musick 1999; Frisk et al. 2001; Dulvy et al. 2004;Reynolds et al. 2005). For example, age- and stage-structured models have been used to estimatethe relative intrinsic sensitivity of numerous shark and ray species (Simpfendorfer 1999; Molletand Cailliet 2002; Frisk et al. 2005; Simpfendorfer 2005; Cortés 2002). Such models depend heav-ily on age- and stage-speci�c estimates of growth, natural mortality and reproductive output, butsuch detailed information is lacking for most elasmobranchs, particularly natural mortality (Milleret al. 2003; Gedamke et al. 2007). Alternatively, unstructured models do not require age- or stage-speci�c life history estimates and instead use species-wide estimates. Unstructured models havethe advantage of relying on simple assumptions on how fertility or survival may vary with age orstage

Recently, there has been increasing awareness of the importance of considering multiple sourcesof uncertainty in demographic parameter estimation (Simpfendorfer et al. 2011). In addition, de-mographic modelling frameworks quantify the degree of caution that should be exercised for theirsustainable management and can have major implications for the conservation of species (Caswellet al. 1998; Cortés 2002; Cortés et al. 2014). The following sources of uncertainty are among thosethat can be easily propagated in a modelling framework: measurement error (or trait error) stemsfrom uncertainty in the empirical estimation of a life history parameter (Harwood and Stokes2003; Quiroz et al. 2010), and coe�cient error is derived from the uncertainty in the values of thecoe�cients of a model (e.g., uncertainty around the intercept of a linear model) (Quiroz et al. 2010).

Studies of elasmobranchs have shown that certain model formulations that account for uncer-tainty result in considerable variability in demographic parameter estimates. Cortés (2002) propa-gated uncertainty in elasmobranch demography using age-structured matrix models by iterativelydrawing life history parameters from both predetermined and observed probability distributions.Quiroz et al. (2010) estimated natural mortality M for two skate species using a range of empiricalmethods while accounting for uncertainty and correlation in trait parameters. A recent reviewby Cortés et al. (2014) highlighted the main sources of uncertainty in elasmobranch stock assess-ments, recommending approaches that explore uncertainty and its e�ects. While multiple sourcesof uncertainty can be readily accounted for in stock assessments, this has not happened to thesame extent in data-poor situations, particularly in commonly used unstructured models.

In this study, we use an unstructured demographic model to address how measurement errorin life history traits a�ects (1) uncertainty in productivity estimates, and (2) sensitivity of theseestimates to uncertainty in each trait. To do this, we propagate measurement error in uncer-tainty based on an unstructured derivation of the Euler-Lotka demographic model, which esti-mates the maximum intrinsic rate of population increase rmax (Myers et al. 1997; Hutchings et al.

44

2012; Pardo et al. 2016; Cortés 2016). In reality rmax is achieved at low population sizes (i.e., in theabsence of density dependence) whereupon it is equivalent to the biological limit reference pointFext—the �shing mortality that will drive the species to extinction (Dulvy et al. 2004; Gedamkeet al. 2007; Cortés et al. 2014). We examine model performance under the estimated uncertaintyof each required life history parameter for ten populations of comparatively well-studied ground(carcharhiniform) sharks. Speci�cally, we calculate uncertainty in rmax estimates through MonteCarlo resampling from probability distributions of the three input parameters required in themodel: annual reproductive output, age at maturity, and instantaneous natural mortality. To as-sess sensitivity, we also compare models that only include uncertainty from individual life historytraits.

4.3 Methods

We used a Monte Carlo simulation model (Fig. 4.1) based on published information on the biology ofa species to iteratively estimate maximum intrisic rate of population increase rmax using a deriva-tion of the Euler-Lotka model (Pardo et al. 2016). The model starts with the data required (Valuesfor age at maturity αmat , maximum age αmax , litter size l , and breeding interval i (“Data” section inFig. 4.1), which are then used to de�ne probability distributions for each parameter (except breed-ing interval whose value is �xed, “Probability distributions” section). Values for age at maturity,maximum age, and litter size are then drawn from these distributions (“Parameters drawn” sec-tion), and used to estimate natural mortality M and annual reproductive output b (“Model inputs”section), which in turn are required to obtain an estimate of rmax (blue box in Fig. 4.1). The drawingof parameters from distributions is repeated 20,000 times to obtain 20,000 rmax estimates (innerloop in Fig. 4.1). Finally, we repeat the whole process after replacing the probability distributionsof each parameter with a �xed value to assess the sensitivity of the model to uncertainty in αmat ,b, and M (outer loop in Fig. 4.1). We apply this model to ten populations of carcharhiniform sharks(order Carcharhiniformes) to examine how the uncertainty in traits underlies uncertainty in rmax .

4.3.1 Data

The population-speci�c life history information required for this simulation model consists of ageat maturity (range of years), maximum age (in years), ranges of litter size (in number of femalepups), and breeding interval (in years) (Table 4.1, Fig. 4.1 “Data” section).

4.3.2 Annual reproductive output

Annual reproductive output (b) estimates were derived from uniform distributions constrained bythe minimum and maximum litter sizes This was necessary because of the empirical distributionsof litter sizes are lacking for most elasmobranchs. published in the literature. We assumed a sexratio of 1:1 to calculate numbers of females only per litter. Therefore the annual reproductive output

45

Dat

aP

roba

bilit

y di

strib

utio

nsP

aram

eter

s dr

awn

Mod

el in

puts

Out

put

Litter size Breeding intervalMaximum age

Litter sizeUniform

Maximum age Uniform

Age at maturity

l i

Annual reproductive

outputb

Instantaneous natural mortality

M

Age at maturity ? mat

maximum intrinsic population growth

rate rmax

Published literature

Maximum intrinsic rate of population increaseSee equation 1

Repeat over 20,000 Monte

Carlo iterations

Repeat by drawing parameters from distributions or

using only means

END

Age at maturity Uniform

? mat ? max

Figure 4.1: Flow chart illustrating the structure of the Monte Carlo simulation model used in thisstudy. The model starts with the data required (Values for age at maturity αmat , maximum ageαmax , litter size l , and breeding interval i), which are then used to de�ne probability distributionsfor each parameter (except breeding interval whose value is �xed). Values for age at maturity,maximum age, and litter size are then drawn from these distributions, and used to estimate natu-ral mortality M and annual reproductive output b, which in turn are required to obtain an estimateof maximum intrisic rate of population increase rmax . The drawing of parameters from distribu-tions is repeated 20,000 times to obtain 20,000 rmax estimates. Finally, we replace the probabilitydistributions of each parameter with a �xed value to assess the sensitivity of the model to uncer-tainty.

46

Table 4.1: Values and sources of life history parameters used to estimate rmax for the ten popula-tions studied. Note that annual reproductive output b is not obtained directly from the literaturebut is estimated from litter size and breeding interval.

Litter size Breeding Mean Age at mat. Max. SourceSpecies min max interval b min max age

Carcharhinusacronotus

1 6 2.0 1.8 4.0 5.0 19.2 Driggers et al. (2004a;b); Barreto et al. (2011);Branstetter (1990)

Carcharhinusbrevipinna

6 10 2.0 4.0 7.0 8.0 16.0 Cortés (2002); Branstetter (1987a)

Carcharhinusisodon

2 6 1.5 2.7 3.3 5.3 8.0 Castro (1993); Carlson et al. (2003); Driggers andHo�mayer (2009)

Carcharhinusleucas

6 12 2.0 4.5 17.0 19.0 31.0 Branstetter (1990); Cli� and Dudley (1991); Cortés(2002); Branstetter and Stiles (1987)

Rhizoprionodonterraenovae

1 12 1.0 6.5 2.8 3.9 9.0 Parsons (1983); Bigelow and Schroeder (1948);Branstetter (1987b); Parsons (1985)

Sphyrna lewini 12 38 1.0 25.0 13.0 15.0 36.0 Branstetter (1987c); Drew et al. (2015); Stevensand Lyle (1989); Cortés (2002)

Sphyrna mokarran 13 42 2.0 13.8 7.4 9.5 31.7 Harry et al. (2011); Compagno (1984b); Stevensand Lyle (1989); Last and Stevens (2009)

Sphyrna tiburo 3 15 1.0 9.0 2.9 4.0 7.5 Lombardi-Carlson et al. (2003); Cortés (2002)Carcharhinuslimbatus ATL

2 10 2.0 3.0 5.7 7.7 21.6 Carlson et al. (2006); Branstetter (1990); Castro(1996)

Carcharhinuslimbatus GULF

2 10 2.0 3.0 4.7 6.7 14.4 Carlson et al. (2006); Branstetter (1990); Castro(1996)

of females b was calculated as b = 0.5 ∗ l/i , where l is litter size (in numbers of males and females)and i is breeding interval (in years).

4.3.3 Age at maturity

Age at maturity (αmat ) estimates were derived from uniform distributions constrained betweenthe minimum and maximum ages at maturity published in the literature (Table 4.1).

4.3.4 Instantaneous natural mortality

Instantaneous natural mortality M was estimated using the reciprocal of average lifespan (M =1/ω), with average lifespan ω de�ned as the midpoint between age at maturity and maximum age(ω = αmat+αmax

2 ). Chapter 3 provides a rationale for this estimate. Given that we obtained a dis-tribution of age at maturity values for each species (see above), we used this uncertainty in age atmaturity as the basis to estimate uncertainty in M , thus uncertainty of M was iteratively estimatedusing the same age at maturity distribution described above. Uncertainty in instantaneous naturalmortality M had very little in�uence on rmax compared to the e�ect of uncertainty in age at matu-rity αmat and annual reproductive output b; however, this is an artifact of the constrained range ofM values produced by our estimation method. When accounting for uncertainty in natural mor-tality using the reciprocal of lifespan equation, we only included uncertainty in age at maturity asit is di�cult to set a plausible range for maximum age. These narrow estimates of M (see Fig. 4.9din Supplementary Materials) resulted in uncertainty in M having a very small e�ect on rmax esti-mates (see Results). We could increase the degree of uncertainty in M estimates, yet this increase

47

would be arbitrary. We explored an alternative scenario where we arbitrarily increase uncertaintyin our M estimate and as expected, its important in rmax estimates increased considerably (seeSupplementary Materials). The e�ect of uncertainty in M on rmax estimates will depend on theM estimator used and the degree of uncertainty associated with it (Then et al. 2014). Because ofthe di�culties of specifying an adequate level of uncertainty in M as well as how resource inten-sive it would be to obtain better estimates of M , we focus our analysis on the e�ects of includinguncertainty in the other parameters required to estimate rmax : αmat and b.

4.3.5 Estimating rmax

We estimated the maximum intrinsic rate of population increase rmax as it is equivalent the �shingmortality required to drive a population to extinction Fext (Dulvy et al. 2004), and thus is also ameasure of vulnerability. Unlike previous estimates of rmax for chondrichthyans (Dulvy et al.2014b; Hutchings et al. 2012; García et al. 2008), this equation accounts for juvenile mortalitywhich has been previously overlooked (Pardo et al. 2016; Cortés 2016):

lαmatb = ermαmat − e−M (erm )αmat−1 (4.1)

where lαmat is survival to maturity and is calculated as lαmat = (e−M )αmat . Because lαmat isderived from M and αmat we did not examine the e�ect of uncertainty in lαmat independently.Equation 4.1 assumes that age at selectivity (i.e., age at which they begin to be captured) is 1. Weused Monte Carlo simulation to propagate uncertainty of input parameters. We drew parametersfrom their respective distributions iteratively 20,000 times, and solved for rmax (inner loop inFig. 4.1)

4.3.6 Separating uncertainty in input parameters

To assess how uncertainty in each parameter a�ected estimated of rmax , we repeated the MonteCarlo simulation for each of the seven possible model combinations with uncertainty in: (i) onlyb, (ii) only αmat , (iii) only M , (iv) b + αmat , (v) b + M , (vi) αmat + M , and (vii) a full model ofb + αmat +M (Fig. 4.1g).

4.3.7 Model sensitivity

In order to visualise the parameter space of rmax values created by ranges of αmat , b, and M , wecreated two-dimensional contour plots, showing rmax estimates along gradients of αmat and b,plotted separately for three levels of M : low M = 0.05 year-1, medium M = 0.1 year-1, and high M =0.2 year-1. We chose these three values of M arbitrarily as they span the natural mortality valueswe estimated for the ten shark species studied.

48

4.3.8 Coding

All models were built in R version 3.2.4 (R Core Team 2016). The rmax equation was solved usingthe nlminb optimisation function by minimising the sum of squared di�erences.

4.4 Results

4.4.1 Estimation of rmax

The median rmax estimates were robust to uncertainty in all populations examined likely due to thesymmetric uncertainty in the underlying parameters. As expected, uncertainty in rmax estimatesvaried considerably among species as a result of uncertainty in the underlying traits (Fig. 4.2).

4.4.2 Comparing uncertainty in input parameters

Estimates of rmax are most sensitive to uncertainty in annual reproductive output b (Fig. 4.2).This is particularly pronounced in the least fecund species (Fig. 4.3b), that is, those with ranges ofannual reproductive output b less than 5 such as the Finetooth Shark C. isodon (b = 2.7; Fig. 4.2b)and Blacknose Shark C. acronotus (b = 1.8; Fig. 4.2c) as these species had larger di�erences incoe�cients of variation between b and αmat (Table 4.2).

By focusing on R. terraenovae we see that the 95% quantiles of rmax values for a model onlyincorporating uncertainty in b is approximately twice as large as those in the model only incorpo-rating uncertainty in αmat (Fig. 4.2i). For this species the coe�cient of variation (CV) in rmax esti-mates is 32% when only accounting for uncertainty inb compared to 10% when only accounting foruncertainty in αmat (Table 4.2). While the ranges of plausible age at maturity values were low forall ten populations examined when compared with variation in the other traits (see Fig. 4.9b in Sup-plementary Materials), they still resulted in considerable uncertainty in rmax estimates (Fig. 4.2).Furthermore, CVs of rmax estimates were moderately higher in species with lower estimates ofαmat , which are often thought to be relatively resilient to �shing, than those with higher αmat

estimates (Fig. 4.3a).For all ten populations, the full model incorporating all uncertainties (b + αmat + M) had a

slightly smaller CV than the αmat + b model due to the close correlation between αmat and M

(Table 4.2). This same pattern exists when comparing models with just αmat versus αmat +M .Accounting for uncertainty in life history parameters is important: theoretical biological ref-

erence points based on the 2.5% quantile of rmax (equivalent to Fext ) were on average 60% lowerwhen all sources of uncertainty were accounted for than when the deterministic model was used(Fig. 4.4).

49

Mαmat

ba

Carcharhinus leucas

b

C. isodon

c

C. acronotus

d

C. brevipinna

0 0.2 0.4 0.6

e

C. limbatus ATL

f

C. limbatus GULF

g

Sphyrna lewini

h

S. mokarran

i

Rhizoprionodonterraenovae

0 0.2 0.4 0.6

j

S. tiburo

Maximum intrinsic rate of population increase r m a x

Life

hist

ory

para

met

er

Figure 4.2: Predicted values of maximum intrinsic rate of increase rmax for ten di�erent sharkpopulations when including uncertainty in annual reproductive output b (blue box plots), age atmaturity αmat (yellow box plots), and natural mortality M (red box plots). Boxes indicate median,25% and 75% quantiles, while the lines encompass 95% of the values (2.5% and 97.5% quantiles).

50

αmat

05

1015

2025

3035

5 10 15

(a)

b5 10 15 20 25

(b)

M0.04 0.08 0.12 0.16

(c)C

oeffi

cien

t of v

aria

tion

(CV,

%) i

n r

max

estim

ates

acc

ount

ing

for u

ncer

tain

ty in

:

Figure 4.3: Coe�cient of variation (CV, %) in rmax estimates for ten di�erent shark populationswhen accounting for uncertainty in (a) age at maturity αmat , (b) annual reproductive output b, and(c) natural mortality M , plotted against the median values of the respective life history parameter.Lines are loess-smoothed curves.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

100 150 200 250 300 350 400

C. acrC. breC. iso

C. leu

R. ter

S. lew

S. mok

S. tib

C. lim ATLC. lim GULF

Fishing limit ignoringuncertainty

Fishing limitaccounting for uncertainty

Asymptotic size (L ∞, cm)

Max

imum

intri

nsic

rate

of p

opul

atio

n in

crea

se(r

max

, yea

r− 1)

Figure 4.4: Comparison of potential �shing limits based on rmax when estimated with no uncer-tainty accounted for (grey diamonds) and when accounting for uncertainty from all sources inthe model (using 2.5% quantile, red diamonds). Points were slightly jittered horizontally to avoidoverlap.

51

Tabl

e4.2:

Coe�

cien

tsof

varia

tion

(CV)

inr m

axes

timat

esof

seve

nm

odel

swith

vary

ing

leve

lsof

unce

rtain

tyin

agea

tmat

urity

,nat

ural

mor

talit

yan

dan

nual

repr

oduc

tive

outp

utfo

rthe

ten

shar

kpo

pula

tions

exam

ined

.Mea

ns,m

edia

nsan

dst

anda

rdde

viat

ions

are

also

pres

ente

dfo

rthe

full

mod

elw

hich

acco

unts

foru

ncer

tain

tyin

allt

hree

para

met

ers.

Mod

elCo

e�ci

ento

fVar

iatio

n(C

V,%)

b+αmat+M

(Ful

l)M

odel

Spec

ies

bαmat

Mb+αmat

b+M

αmat+M

b+αmat+M

Mea

nM

edia

nSt

.Dev

Carcharhinu

sisodon

25.0

16.4

4.1

30.0

25.4

12.3

27.9

0.19

70.

188

0.06

1Carcharhinu

slim

batus

GULF

25.2

10.2

1.427

.325

.28.8

26.8

0.19

50.

195

0.01

5Rh

izoprionodon

terraenovae

31.5

9.61.0

33.1

31.6

8.732

.80.

185

0.18

50.

052

Carcharhinu

slim

batus

ATL

21.8

7.90.

723

.221

.87.2

23.0

0.112

0.112

0.00

9Carcharhinu

sacronotus

31.9

5.50.

532

.531

.94.

932

.40.

428

0.40

50.

133

Carcharhinu

sbrevipinna

7.13.9

0.5

8.17.1

3.47.8

0.24

00.

239

0.02

1Carcharhinu

sleucas

7.43.3

0.4

8.17.4

2.87.9

0.31

60.

315

0.03

7Sphy

rnatib

uro

21.0

9.81.1

23.2

21.0

8.722

.80.

462

0.45

30.

103

Sphy

rnamokarran

9.76.

60.

211.

79.7

6.4

11.6

0.20

20.

197

0.04

5Sphy

rnalewini

7.94.

00.

28.8

7.93.8

8.70.

204

0.19

90.

053

52

4.4.3 Model sensitivity

The interactive e�ects of annual reproductive output and age at maturity on rmax are nonlinearand vary based on the values of natural mortality (M = 0.05, 0.2, 0.5 year-1; Fig. 4.5). Overall, rmax

drops steeply at low b values regardless of αmat or M (bottom left corner of all plots in Fig. 4.5). Atmedium to high values of annual reproductive output, the estimate of rmax becomes increasinglysensitive to variation in age at maturity. With increasing M values, there are increasing combina-tions of αmat and b values that result in implausible rmax values (rmax ≤ 0; red areas in Fig. 4.5b& c). This “implausibility” threshold is particularly apparent when natural mortality is moderateto high (M < 0.2 year-1) b is very low (b < 1), and αmat is over 5 years (Fig. 4.5b & c).

4.5 Discussion

The availability of simple methods for estimating key population parameters has opened the doorto comparative risk assessment of a wider range of �shes (Stobutzki et al. 2002; Hobday et al.2011; Dulvy et al. 2014a). One such simple method is the Euler-Lotka approach to estimating themaximum intrinsic rate of population increase using point estimates of three life history traits(García et al. 2008; Hutchings et al. 2012; Dulvy et al. 2014a). However, all life history parametersare estimated with some associated uncertainty. Here, we show that the degree of uncertaintyin life history parameters has a considerable e�ect on the distribution of the resulting range ofmaximum intrinsic rate of population increase, but little e�ect on median values. Fully propagatingthe uncertainty in natural mortality M , age at maturity αmat , and annual reproductive output b,increased the coe�cient of variation of rmax values by between 11 and 46% (Table 4.2).

These �ndings have profound implications for the use of rmax estimates to set �shing limits forsharks and other data poor species (Fig. 4.5). Acknowledging the level of uncertainty associatedwith estimates is crucial when using the precautionary approach, as the degree of risk associatedwith speci�c management practices can be estimated (Harwood and Stokes 2003; Artelle et al.2013).

As we have shown, rmax is particularly sensitive to di�erences in annual reproductive out-put, particularly for species with very low annual reproductive output (b < 5 females per year;Fig. 4.3b). That demography is in�uenced by fecundity of the least fecund species is apparentfrom some demographic models, but it depends on how reproductive output is parameterised. Arecent age-structured model of dogsharks (Order Squaliformes) revealed that biological referencepoints can be strongly in�uenced by their low fecundity (Forrest and Walters 2009). However, wecaution that the rebound potential model (Au and Smith 1997) is agnostic to annual reproductiveoutput, as the values of b cancel out and hence are not considered mathematically in the model(Au et al. 2015). This di�erence in the implementation of annual reproductive output between twosimilar unstructured models may help explain di�erences in species sensitivity when comparisonsare done with di�erent methods. For example, Ward-Paige et al. (2013) used the rebound poten-tial model to compare the sensitivity of manta rays (Manta spp.) to that of other elasmobranchs.

53

5

10

15

20 (a)

M = 0.05 year−1

5

10

15

20 (b)

M = 0.1 year−1

5 10 15 20

5

10

15

20 (c)

M = 0.2 year−1

Implausible

valuesr m a x

Age

at m

atur

ity (α

mat

, yea

rs)

Annual reproductive output (b , n/year)

Figure 4.5: Contour plots of rmax values for varying ranges of age at maturity αmat and annualreproductive output b, with values of instantaneous natural mortalityM set as (a) low (0.05 year-1),(b) medium (0.1 year-1, and (c) high (0.2 year-1).

54

They found that manta rays were intermediate in sensitivity (r ), more similar to Spinner Shark(Carcharhinus brevipinna) or Silky Shark (C. falciformis). However, when the very low annual re-productive output of manta rays is accounted for using rmax , they were found to have one of thelowest population growth rates rmax observed in chondrichthyans (Dulvy et al. 2014a). The annualreproductive output of manta rays is highly uncertain, but with the potential of skipped spawning,reproductive output may be as low as one female pup every second or third year (Couturier et al.2012; Marshall and Bennett 2010), resulting in rmax varying from 0.089 to 0.139 year-1 (Dulvy et al.2014a). Being aware of major di�erences in the implementation of annual reproductive output indi�erent models is important when choosing the model best suited to the research question.

We show that the highest demographic uncertainty occurred in species with very low annualreproductive outputs—less than �ve female pups per year. Many elasmobranchs have this repro-ductive rate (Cortés 2000). Nonetheless, we reached this conclusion based on assuming a uniformdistribution of litter sizes, but they are unlikely to be uniform in the real world. For us to explorethe validity of this assumption requires a better understanding of the empirical distribution of lit-ter sizes. Hence, we urge biologists to report frequency distributions of individual litter sizes andalso to test whether the frequency distribution of annual o�spring number and size varies withmaternal age and size (Hussey et al. 2010).

The updated model for estimating rmax includes juvenile survival which is derived from adultnatural mortality M (Pardo et al. 2016). Yet, because of the known trade-o� between o�spring sizeand litter size (Smith and Fretwell 1974; Hussey et al. 2010), the least fecund species often have thelargest o�spring. As is typical for marine �shes, such larger o�spring will have a greater survivalprobability than the smaller o�spring of species with larger litters. Therefore the di�erence inrmax among elasmobranchs may be lower than suggested by this model. An example of this canbe illustrated by comparing the Spinner Shark (C. brevipinna) with the Scalloped Hammerhead(S. lewini). The Spinner Shark litter size ranges between 3–15 individuals born between 60 and80 cm in length, while the Scalloped Hammerhead has a larger litter size ranging between 13–41individuals but which are born smaller, between 45 and 50 cm in length (Last and Stevens 2009).

Natural mortality is one of the most important parameters in �sheries modelling but one ofthe hardest to estimate (Pope 1975; Vetter 1988; Kenchington 2014). Our estimates of rmax arerelatively insensitive to uncertainty in M for shark-like life histories (Figs. 4.2) as a result of themethod we used for accounting for uncertainty in M . For example, the CV of rmax estimates forR. terraenovae is 1% when accounting for uncertainty in only M yet it increases to 31.5% if onlyuncertainty in annual reproductive output is taken into account (Table 4.2). Our �nding is similarto Au et al. (2015), who showed that M had only a minor role in the estimation of rebound potentialwhen compared with αmat . However, as natural mortality is a di�cult parameter to estimate itis unrealistic for its uncertainty to be narrowly constrained as it was with our method. This begsthe question of how to assess how much uncertainty in M is “enough”. Improving natural mortal-ity estimates would be incredibly di�cult for data-poor chondrichthyans, so research e�orts arelikely better spent improving on life history estimates of the other parameters age at maturity and

55

annual reproductive output. Further increasing uncertainty in natural mortality (by an arbitraryamount) does increase uncertainty in rmax (See Supplementary Materials). While in our studyM is relatively unimportant, more complex age- and stage-structured models consistently showthat juvenile mortality has important contributions to population growth rate (Cortés 2002; Frisket al. 2005). This di�erence in importance of M needs to be borne in mind when comparing acrossdemographic model types.

There has been considerable debate as to which empirical model should be used to estimatenatural mortality M (Hewitt and Hoenig 2005; Quiroz et al. 2010; Then et al. 2014). Here, we usedthe reciprocal of average lifespan; however, other methods, such as Chen and Watanabe (1989)and Peterson and Wroblewski (1984), have been used as they provide varying values of M throughontogeny as required for age-structured demographic modelling (Pardo et al. 2012). While helpfulfor matrix models, only average natural mortality values (and the uncertainty around them) areneeded for unstructured models. Natural mortality estimates in which correlation of parameterscan be accounted for (e.g., Pauly 1980) reduced uncertainty of estimates and their error whenapplied to elasmobranchs (Quiroz et al. 2010). A recent study by Then et al. (2014) suggests thatthe Pauly (1980) mortality model should not be used, and instead a new variant that eliminatestemperature from the equation is preferred. Given that instantaneous natural mortality M hadonly a slight e�ect on estimates of rmax , our results are likely to be robust to the choice of naturalmortality estimator used.

As we have shown, incorporation of uncertainty lead to much more conservative estimates of�shing limits than if uncertainty is ignored. Incorporating uncertainty also considerably increasedthe potential range of maximum population growth rate rmax estimates in these relatively well-studied sharks. Managers and policy makers should be careful to assess and restrict mortalityon species with annual reproductive outputs <2 females per year. This uncertainty in rmax canbe reduced by understanding the correlation in life history parameters. For this to be possible,we urge reproductive biologists to report the distribution of litter sizes and o�spring sizes. Wealso suggest that the variance-covariance matrix of the model should be reported routinely when�tting growth curves. Only then will it be possible to quantify known uncertainty in demographicvulnerability models.

4.6 Acknowledgements

We would like to thank John Carlson, Enric Cortés, Aleksandra Maljković, Holly Kindsvater, andJoel Harding for their thorough reviews of the manuscript, members of the Earth to Ocean Re-search Group for their helpful feedback, JC Quiroz for his helpful insight into uncertainty in de-mographic models, and Nicolás Huerta for his help coding the model.

56

4.7 Supplementary materials

4.7.1 Alternate scenario with a more uncertainM

One shortcoming of our method for accounting for uncertainty in natural mortality M is that it isperfectly correlated with uncertainty in age at maturity, and therefore the distribution of M valuesis narrowly constrained (see Fig. 4.9d). The narrow distribution of M does not represent certaintyabout its value and is not representative of the actual uncertainty around our estimates of M .Unfortunately, this will be the case for any indirect estimate ofM , and therefore uncertainty in thisparameter is highly arbitrary. One simple approach to account for a higher degree of uncertainty inM is to replace the numerator in th equation to estimate M as the reciprocal of life span (M = 1/ω)by a log-normal roughly centered around 1:

M =lnN(0, 0.32)

ω(4.2)

where lnN(0, 0.32) is a log-normal distribution with a mean of 0 and a standard deviation of0.3, and ω is the average life span, de�ned as ω = αmat+αmax

2 . Note that αmat is also drawn from adistribution (see Methods section) while αmax is a single �xed value. The log-normal distributionfor the numerator is centered around 1 and almost entirely constrained between 0.5 and 2. Thesevalues were chosen arbitrarily as they represent a plausible range of values for the numerator inthe M equation.).

The updated distributions ofM for each shark population can be seen in Fig. 4.7d and are muchless constrained than the distributions used previously (Fig. 4.9d). As expected, the resulting rmax

estimates when accounting for uncertainty in M have a much higher uncertainty than before(Fig. 4.6a), which is comparable to the uncertainty in rmax when accounting for uncertainty inage at maturity (Fig. 4.6b). The e�ect uncertainty in M on rmax estimates tended to increase asthe value of M increased. The coe�cients of variation in rmax estimates when accounting for(Table 4.4).

57

Tabl

e4.4

:Coe

�ci

ents

ofva

riatio

n(C

V)of

seve

nin

trins

icse

nsiti

vity

mod

elsw

ithva

ryin

glev

elso

func

erta

inty

inag

eatm

atur

ity,n

atur

alm

orta

lity

(incl

udin

gun

certa

inty

inth

enum

erat

orof

Meq

uatio

n)an

dan

nual

repr

oduc

tiveo

utpu

tfor

thet

ensh

ark

popu

latio

nsex

amin

ed.M

eans

,med

ians

and

stan

dard

devi

atio

nsar

eal

sopr

esen

ted

fort

hefu

llm

odel

whi

chac

coun

tsfo

runc

erta

inty

inal

lthr

eepa

ram

eter

s.

Mod

elCo

e�ci

ento

fVar

iatio

n(C

V,%)

b+αmat+M

(Ful

l)M

odel

Spec

ies

bαmat

Mb+αmat

b+M

αmat+M

b+αmat+M

Mea

nM

edia

nSt

.Dev

Carcharhinu

sisodon

25.8

17.0

29.4

31.0

39.7

31.3

41.2

0.19

10.

184

0.06

6Carcharhinu

slim

batus

GULF

25.6

10.4

16.0

27.8

30.4

18.1

31.7

0.19

30.

190

0.03

1Rh

izoprionodon

terraenovae

31.7

9.812

.233

.234

.314

.935

.40.

180

0.17

70.

073

Carcharhinu

slim

batus

ATL

22.2

8.011.

323

.725

.213

.426

.20.

1110.

1100.

016

Carcharhinu

sacronotus

32.4

5.614

.333

.035

.815

.136

.20.

420

0.40

00.

142

Carcharhinu

sbrevipinna

7.24.

014

.38.3

16.1

14.7

16.4

0.23

80.

237

0.02

5Carcharhinu

sleucas

7.43.3

11.8

8.114

.112

.214

.40.

313

0.31

20.

040

Sphy

rnatib

uro

21.3

10.0

12.9

23.6

25.1

15.6

26.7

0.45

20.

445

0.119

Sphy

rnamokarran

9.86.

75.1

11.8

11.1

8.212

.80.

198

0.19

40.

051

Sphy

rnalewini

7.94.

05.4

8.99.6

6.6

10.4

0.19

80.

194

0.06

2

58

Sphyrna tiburo

Rhizoprionodon terraenovae

Sphyrna mokarran

Sphyrna lewini

Carcharhinus limbatus GULF

Carcharhinus limbatus ATL

Carcharhinus acronotus

Carcharhinus brevipinna

Carcharhinus isodon

Carcharhinus leucas (a)M

Sphyrna tiburo

Rhizoprionodon terraenovae

Sphyrna mokarran

Sphyrna lewini

Carcharhinus limbatus GULF

Carcharhinus limbatus ATL

Carcharhinus acronotus

Carcharhinus brevipinna

Carcharhinus isodon

Carcharhinus leucas (b)αmat

Sphyrna tiburo

Rhizoprionodon terraenovae

Sphyrna mokarran

Sphyrna lewini

Carcharhinus limbatus GULF

Carcharhinus limbatus ATL

Carcharhinus acronotus

Carcharhinus brevipinna

Carcharhinus isodon

Carcharhinus leucas

0.0 0.2 0.4 0.6 0.8 1.0

(c)b

Maximum intrinsic rate of population increase r ma x

Figure 4.6: Estimated values of maximum intrinsic rate of increase rmax for ten di�erent sharkpopulations when including uncertainty in (a) natural mortalityM with a log-normally distributednumerator, (b) age at maturity αmat , and (c) annual reproductive output b. Boxes indicate median,25% and 75% quantiles, while lines encompass 95% con�dence intervals.

59

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. isodon

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. limbatus GULF

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +MR. terraenovae

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. limbatus ATL

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. acronotus

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. brevipinna

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. leucas

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M S. tiburo

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M S. mokarran

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8b

αm a tM

b+αm a t

b+Mαm a t +M

b+αm a t +M S. lewini

Mod

eled

unc

erta

inty

0 3 6 9 12 18 0 5 10 15 0.00 0.10 0.20Maximum intrinsic rate of population increase

(rmax, year−1)Annual reproductive

output (b , n/year)Age at maturity

(αmat, years)Natural mortality

(M , year−1)

(a) (b) (c) (d)

Figure 4.7: Estimated values of maximum intrinsic rate of population increase rmax for ten di�erentshark populations with varying levels of uncertainty in age at maturity αmat , natural mortality Mand annual reproductive output b, as shown by y-axis labels. Boxes indicate median, 25% and 75%quantiles, while lines encompass 95% con�dence intervals. Columns on the right show the distri-butions values of (b) annual reproductive output b, (c) age at maturity αmat , and (d) instantaneousnatural mortality M with a log-normally distributed numerator.

60

4.7.2 Supplementary Figures and Tables

Sphyrna tiburo

Rhizoprionodon terraenovae

Sphyrna mokarran

Sphyrna lewini

Carcharhinus limbatus GULF

Carcharhinus limbatus ATL

Carcharhinus acronotus

Carcharhinus brevipinna

Carcharhinus isodon

Carcharhinus leucas (a)M

Sphyrna tiburo

Rhizoprionodon terraenovae

Sphyrna mokarran

Sphyrna lewini

Carcharhinus limbatus GULF

Carcharhinus limbatus ATL

Carcharhinus acronotus

Carcharhinus brevipinna

Carcharhinus isodon

Carcharhinus leucas (b)αmat

Sphyrna tiburo

Rhizoprionodon terraenovae

Sphyrna mokarran

Sphyrna lewini

Carcharhinus limbatus GULF

Carcharhinus limbatus ATL

Carcharhinus acronotus

Carcharhinus brevipinna

Carcharhinus isodon

Carcharhinus leucas

0.0 0.2 0.4 0.6 0.8 1.0

(c)b

Maximum intrinsic rate of population increase r ma x

Figure 4.8: Estimated values of maximum intrinsic rate of increase rmax for ten di�erent shark pop-ulations when including uncertainty in (a) natural mortality M , (b) age at maturity αmat , and (c)annual reproductive output b. Boxes indicate median, 25% and 75% quantiles, while lines encom-pass 95% con�dence intervals. The data shown are identical to Fig. 4.2 but box plots are arrangedby type of uncertainty accounted for instead of by species.

61

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. isodon

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. limbatus GULF

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +MR. terraenovae

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. limbatus ATL

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. acronotus

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. brevipinna

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M C. leucas

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M S. tiburo

bαm a t

Mb+αm a t

b+Mαm a t +M

b+αm a t +M S. mokarran

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8b

αm a tM

b+αm a t

b+Mαm a t +M

b+αm a t +M S. lewini

Mod

eled

unc

erta

inty

0 3 6 9 12 18 0 5 10 15 0.00 0.10 0.20Maximum intrinsic rate of population increase

(rmax, year−1)Annual reproductive

output (b , females/year)Age at maturity

(αmat, years)Natural mortality

(M , year−1)

(a) (b) (c) (d)

Figure 4.9: Estimated values of maximum intrinsic rate of increase rmax for ten di�erent sharkpopulations with varying levels of uncertainty in age at maturity αmat (yellow), natural mortalityM (red), and annual reproductive outputb (blue), as shown by y-axis labels. Boxes indicate median,25% and 75% quantiles, while lines encompass 95% con�dence intervals, while colours denote caseswhere uncertainty stems from a single parameter (as shown in Fig. 4.2). Columns on the rightshow the distributions values of (b) annual reproductive output b, (c) age at maturity αmat , and (d)instantaneous natural mortality M for each of the ten stocks examined.

62

Chapter 5

Growth, productivity, and relativeextinction risk of a data-sparse devil ray4

5.1 Abstract

Devil rays (Mobula spp.) face intensifying �shing pressure to meet the ongoing international de-mand for gill plate. The paucity of information on growth, mortality, and �shing e�ort for devilrays make quantifying population growth rates and extinction risk challenging. Furthermore, un-like manta rays (Manta spp.), devil rays have not been listed on CITES. Here, we use a publishedsize-at-age dataset for the Spinetail Devil Ray (Mobula japanica), to estimate somatic growth rates,age at maturity, maximum age, and natural and �shing mortality. We then estimate a plausibledistribution of the maximum intrinsic population growth rate (rmax ) and compare it to 95 otherchondrichthyans. We �nd evidence that larger devil ray species have low somatic growth rate, lowannual reproductive output, and low maximum population growth rates, suggesting they have lowproductivity. Fishing rates of a small-scale artisanal Mexican �shery were comparable to our esti-mate of rmax , and therefore probably unsustainable. Devil ray rmax is very similar to that of mantarays, indicating devil rays can potentially be driven to local extinction at low levels of �shing mor-tality and that a similar degree of protection for both groups is warranted.

5.2 Introduction

Understanding the sustainability and extinction risk of data-sparse species is a pressing problemfor policy-makers and managers. This challenge can be compounded by economic, social and envi-ronmental actions, as in the case of the mobulid rays (family Mobulidae). This group includes twospecies of charismatic and relatively well-studied manta rays (Manta spp.), which support a cir-cumtropical dive tourism industry with an estimated worth of $73 million USD per year (O’Malley

4A version of this chapter appears as Pardo, S. A., Kindsvater, H. K., Cuevas-Zimbrón, E., Sosa-Nishizaki, O., Pérez-Jiménez, J. C., Dulvy, N. K. 2016. Growth, productivity, and relative extinction risk of a data-sparse devil ray. Scienti�cReports, 6: 33745. http://dx.doi.org/10.1038/srep33745.

63

et al. 2013). The Mobulidae also includes nine described species of devil rays (Mobula spp.). The re-cent international trade regulation of manta ray gill plates under the Convention of InternationalTrade of Endangered Species (CITES) (Mundy-Taylor and Crook 2013) may shift gill plate demandfrom manta rays onto devil rays.

Devil rays are increasingly threatened by target and incidental capture in a wide range of�sheries, from small-scale artisanal to industrial trawl and purse seine �sheries targeting pelagic�shes (Zeeberg et al. 2006; Croll et al. 2016). The meat is sold or consumed domestically, andthe gill plates are exported, mostly to China, to be consumed as a health tonic (Couturier et al.2012). Small-scale subsistence and artisanal �sheries, mainly for meat, have operated throughoutthe world for decades (Couturier et al. 2012). For example, devil rays were caught by artisanal�shermen using harpoons and gill nets around Bahia de La Ventana, Baja California, Mexico, until2007 when the Mexican government prohibited the take of mobulid rays (Poder Ejecutivo Federal2007).

Overall, around 90,000 devil rays are estimated to be caught annually in �sheries worldwide(Heinrichs et al. 2011). Many industrial �eets capture devil rays incidentally. For example, Euro-pean pelagic trawlers in the Atlantic catch a range of megafauna including large devil rays at arate of up to one individual per hour (Zeeberg et al. 2006), while purse seine �eets targeting tunascapture tens of thousands of devil rays each year (Croll et al. 2016). Even if devil rays are handledcarefully and released, their post-release mortality might be signi�cant (Francis and Jones 2016).We do not know whether current �shing pressure and international trade demand for devil rays aresigni�cant enough to cause population declines and increase extinction risk. The degree to whichdevil ray populations can withstand current patterns and levels of �shing mortality depends ontheir intrinsic productivity, which determines their capacity to replace individuals removed by�shing.

Slow somatic growth and large body size are associated with low productivity and elevatedthreat status and extinction risk in marine �shes, including elasmobranchs (Musick 1999; Reynoldset al. 2005; Jennings et al. 1998). Based on these correlations, the American Fisheries Society devel-oped criteria to de�ne productivity and extinction risk: They de�ned four levels of productivity(very low, low, medium, and high) based on four life history traits (age at maturity, longevity,fecundity, and growth rate, which is related to the von Bertalan�y growth coe�cient k) and theintrinsic rate of population increase r (Musick 1999). According to these criteria, manta rays havevery low or low productivity, with some of the lowest maximum rates of population increase (rmax )of any shallow-water chondrichthyan (Musick 1999; Dulvy et al. 2014b).

Here we evaluate the productivity, and hence relative extinction risk of large devil rays, us-ing the only age and growth study available for this group (Cuevas-Zimbrón et al. 2013). Weuse a Bayesian approach to estimate somatic growth rate and a demographic model based on theEuler-Lotka equation to calculate the maximum intrinsic rate of population increase (rmax ) for apopulation of the Spinetail Devil Ray Mobula japanica (Müller & Henle, 1841), which we compareto the productivity of 95 other sharks, rays, and chimaeras.

64

5.3 Methods

We use the only study to measure length-at-age for catch data of M. japanica (Cuevas-Zimbrónet al. 2013). The Spinetail Devil Ray is similar in life history and size to other exploited mobulids,so we assume it is representative of the relative risk of the group. Spinetail Devil Rays examinedin this study were caught seasonally by artisanal �shers using harpoon and gill nets around PuntaArenas de la Ventana, Baja California Sur, Mexico, during the summers of 2002, 2004, and 2005(Cuevas-Zimbrón et al. 2013).

First, we estimate growth parameters using a Bayesian approach that incorporates prior knowl-edge of maximum size and size at birth of this species, using the length-at-age data presented inCuevas-Zimbrón et al. (2013) (Cuevas-Zimbrón et al. 2013) (Part 1). Second, we use the same datasetto plot a catch curve of the relative frequency of individuals in each age-class, from which we caninfer a total mortality rate (Z ) that includes both �shing (F ) and natural mortality (M) (Part 2). Thisplaces an upper bound on our estimate of natural mortality, and allows us to compare the observedrate of mortality for this population with independent estimates of natural mortality rates. Third,we estimate the maximum intrinsic rate of population increase (rmax ) for this devil ray (Part 3)and compare it against the rmax of 95 other chondrichthyans, calculated using the same method(Part 4).

5.3.1 Part 1: Re-estimating VBGPs for the Spinetail Devil Ray

We analyse a unique set of length-at-age data for a single population of M. japanica caught in aMexican artisanal �shery. Individuals in this sample were limited to 1100 and 2400 mm disc width(DW), which falls short (77%) of the maximum disc width reported elsewhere (Notarbartolo-Di-Sciara 1987). Therefore, we use a Bayesian approach to re�t growth curves to this length-at-agedataset (Siegfried and Sansó 2006), using published estimates of maximum size and size at birthto set informative priors. Our aim is to reconstruct the growth rate of the species, rather thanthis local population. Hence, we focus on �nding the growth rate estimate that would be mostconsistent with the species maximum size and the available size-at-age data.

We �t the three-parameter von Bertalan�y equation to the length-at-age data, combiningsexes:

DWt = DW∞ − (DW∞ − DW0)e−kt (5.1)

where DWt is disc width at age t , and the growth coe�cient k , disc width-at-age zero DW0,and asymptotic width DW∞ are the von Bertalan�y growth parameters. These parameters areconventionally presented in terms of length; however disc width is the appropriate measurementfor these rays and we explicitly note all our size estimates as disc width to avoid any confusion.

In order to account for multiplicative error, we log-transformed the von Bertalan�y growthequation and added an error term:

65

loд(DWt ) = loд(DW∞ − (DW∞ − DW0)e−kt ) + ϵt (5.2)

This can be written as:

loд(DWt ) ∼ Normal(loд(DW∞ − (DW∞ − DW0)e−kt ),σ 2) (5.3)

A Bayesian approach allows us to incorporate expert knowledge using prior distributions ofestimated parameters. We based our informative priors on our knowledge of maximum disc widthsand size-at-birth ofM. japanica. Reported size at birth ranges from 880 to 930 mm DW (White et al.2006a;b), while reported maximum size for M. japanica is 3100 mm DW (Paulin et al. 1982). Whilethis reported maximum size is from an individual caught in New Zealand, we use this estimate asgenetic evidence suggests that the Spinetail Devil Rays has little genetic substructure throughoutthe Paci�c Ocean (Poortvliet et al. 2015). Nonetheless, growth and size di�erences could arise fromenvironmental factors (e.g., temperature di�erences), which could lead to biased growth coe�cientestimates in our Bayesian model.

Asymptotic size can be estimated from maximum size in �shes using the following equation(Froese and Binohlan 2000):

L∞ = 100.044+0.9841∗(loд(Lmax )) (5.4)

where Lmax is maximum size, in centimetres. The data used to estimate this relationship in-cluded some species whose size was estimated in disc width, thus we use for this devil ray. Thisresults in an estimate of DW∞ = 1.01∗DWmax for a value of DWmax = 3100 mm. Instead of settinga �xed value for the conversion parameter, we create a hyperprior for this parameter, de�ned askappa, based on a gamma distribution around a mean of 1.01. We concentrated the probabilitydistribution of kappa between 0.9 and 1.1, and fully constrain it between 0.7 and 1.3 (Froese andBinohlan 2000):

kappa ∼ Gamma(1000, 990) (5.5)

We also constrain our prior forDW0 around size at birth, and use a beta distribution to constrainour prior for growth coe�cient k between zero and one, with a probability distribution that isslightly higher closer to a value of 0.1:

k ∼ Beta(1.05, 1.5)DW∞ ∼ Normal(3100 ∗ kappa, 100)DW0 ∼ Normal(880, 200)

(5.6)

We compared the e�ect of our informative priors on our posteriors with parameter estimateswith weaker priors, in which we maintained the mean of the distributions but increased theirvariance:

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k ∼ Beta(1.05, 1.1)kappa ∼ Gamma(200, 198)DW∞ ∼ Normal(3100 ∗ kappa, 400)DW0 ∼ Normal(880, 300)

(5.7)

We also considered a scenario with uninformative priors, where all prior distributions areuniform:

k ∼ Uni f orm(0, 2)kappa ∼ Uni f orm(0.7, 1.3)DW∞ ∼ Uni f orm(0, 4000)DW0 ∼ Uni f orm(0, 2000)

(5.8)

In all models we set an weakly informative prior for the variance σ 2, such that:

σ 2 ∼ hal f Cauchy(0, 30000) (5.9)

A summary of the priors used can be seen in Table 5.1. Bayesian inference was conducted usingRStan v2.7.0 (Stan Development Team 2015b;a) running in R v3.2.1 (R Core Team 2015).

Table 5.1: Priors used in the three di�erent Bayesian von Bertalan�y growth models.

Parameter Strong priors Weaker priors Uninformative priorsk beta(1.05, 1.5) Beta(1.05, 1.1) Uni f orm(0, 2)DW∞ normal(3100 ∗ kappa, 100) Normal(3100 ∗ kappa, 400) Uni f orm(0, 4000)DW0 normal(880, 200) Normal(880, 300) Uni f orm(0, 2000)kappa дamma(1000, 990) Gamma(200, 198) Uni f orm(0.7, 1.3)σ 2 hal f cauchy(0, 30000) hal f Cauchy(0, 30000) hal f Cauchy(0, 30000)

5.3.2 Part 2. Estimating total mortality using the catch curve

The length-at-age dataset of M. japanica can be used as a representative sample of the number ofindividuals within each age-class if we assume that sampling was opportunistic, and non-selectiveacross each age- or size-class. We also assume that there is limited migration in and out of thispopulation. With these assumptions, counting the number of individuals captured in each age-class represents the population age structure, which can be used to construct a catch curve. Wefurther consider the validity of our assumptions in the Discussion.

Catch curves are especially useful for data-poor species lacking stock assessments (Thorsonand Prager 2011; Hordyk et al. 2015). The frequency of individuals in older or larger classes de-creases due to a combination of natural and �shing mortality. If �shing is non-selective withrespect to size, the total mortality rate Z , which is a combination of both �shing mortality F and

67

natural mortality M , can be estimated using a linear regression as the slope of the natural log ofthe number of individuals in each class (Ricker 1975). This information is very valuable when in-ferring whether �shing mortality F is unsustainable. We calculate Z as the slope of the regressionof the catch curve, including only those ages or sizes that are vulnerable to the �shery. There wereno devil rays aged 12 or 13, and therefore these age-classes were removed from the catch curveanalysis before �tting each regression. Because these age classes are some of the oldest, removingthese points is likely to provide more conservative estimates of total mortality.

We removed age-classes that had zero individuals in our sample to be able to take the naturallogarithm of the count. Because there is uncertainty associated with the dataset (due to its rel-atively small size), we resampled a subset of the dataset 20,000 times, after randomly removing20% of the points. This allowed us to quantify uncertainty in our estimate of Z . For each subset,we computed the age-class with the maximum number of samples, and removed all age-classesyounger than this peak. With the remaining age-classes, we �t a linear regression to estimate theslope which is equivalent to −Z . This method for estimating mortality relies on two assumptionsof the selectivity of the �shery. First, catch is not size-selective once individuals are vulnerableto the �shery. Second, if young age-classes are less abundant than older age-classes, they are as-sumed to have lower catchability. This is why we removed the younger age-classes before the“peak” abundance of each sample, as this will a�ect the steepness of the slope.

Given that the M. japanica length-at-age dataset has very few individuals aged 10 and older, werepeated the bootstrapping approach excluding ages 10 and older in order to avoid overestimatingZ . This means that our range of Z estimates incorporate the possibility that the oldest individualsare missing from our sample because of migration, rather than �shing mortality, with the resultingestimate being a more conservative estimate of total mortality.

5.3.3 Part 3. EstimatingM. japanica maximum population growth rate

Maximum intrinsic population growth rates rmax can be estimated based on a simpli�ed versionof the Euler-Lotka equation (Charnov and Scha�er 1973; Myers and Mertz 1998). We use the an up-dated method for estimating rmax which accounts for juvenile mortality (Pardo et al. 2016; Cortés2016), unlike previous estimates of rmax for chondrichthyan species (Dulvy et al. 2014b; Hutchingset al. 2012; García et al. 2008):

lαmatb = ermαmat − e−M (erm )αmat−1 (5.10)

where lαmat is survival to maturity and is calculated as lαmat = (e−M )αmat , b is the annualreproductive output of daughters, αmat is age at maturity in years, and M is the instantaneousnatural mortality. We then solve equation 5.10 for rmax using the nlm.imb function in R. To ac-count for uncertainty in input parameters, we use a Monte Carlo approach and draw values fromparameter distributions to obtain 10,000 estimates of rmax . Next we describe how we determinedeach parameter distribution.

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Annual reproductive output (b) Adult female mobulid rays only have one active ovary anduterus where a single pup grows. This sets the upper bound of annual fecundity to one pup peryear (Notarbartolo-Di-Sciara 1987), and assuming a 1:1 sex ratio, results in an estimate of b of 0.5female pups per year. It is possible female devil rays have a biennial cycle of reproduction whereone pup is produced every two years, as in manta rays (Dulvy et al. 2014b), so the lower bound forour estimate of b is 0.25 female pups per year. Thus we draw b from a uniform distribution boundbetween 0.25 and 0.5. While it is possible that a very small percentage of litters could consist oftwo pups, there has not been any cases of two pups being observed in any devil ray, and only asingle con�rmed case in the Reef Manta Ray (Manta alfredi)(Marshall and Bennett 2010).

Age at maturity (αmat ) There are no direct estimates of age at maturity for any mobulid ray, butusing age and growth data from Cuevas-Zimbrón et al. (2013) and a size 50% at maturity of 2000mm DW from Serrano-López (2009), we assume female M. japanica individuals reach knife-edgematurity sometime between 5 and 6 years. Thus we draw αmat from a uniform distribution boundbetween 5 and 6.

Natural mortality (M) We estimate natural mortality as the reciprocal of average lifespan: M =1/ω where average lifespan ω is (αmat + αmax )/2). We used this M estimate to calculate survivalto maturity lαmat as (e−M )αmat . This method produces realistic estimates of rmax when accountingfor survival to maturity, unlike many other commonly used natural mortality estimators (Pardoet al. 2016). We also use our estimate of Z from Part 2, which represents both natural and �shingmortality, to contextualize our estimate of M . More speci�cally, our estimate of M needs to belower than our estimate of Z to be credible. We calculated maximum age (αmax ) based on theresults of our analysis in Part 1 and estimated it to be between 15 and 20 years. We thereforecalculate M iteratively by drawing values of αmat (described in the section above) and αmax fromuniform distributions bound between the ranges mentioned.

5.3.4 Part 4. Comparison ofMobula rmax among chondrichthyans

We re-estimate rmax for the 94 chondrichthyans with complete life history data examined in Garcíaet al. (2008), Dulvy et al. (2014b), and Pardo et al. (2016) using equation 5.10. We also updateestimates of rmax for manta rays (Manta spp.; Dulvy et al. 2014b) as a comparison with a closelyrelated species.

5.4 Results

5.4.1 Part 1: Re-�tting the growth curve forMobula japanica

The Bayesian model with strong priors yielded a lower estimate of k (0.12 year-1) and a higherestimate of DW∞(2995 mm DW) than the estimates based on weaker and uninformative priors

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(Table 5.2, Fig. 5.1). The asymptotic size in the model with strong priors was closest to the maximumobserved size for this species (Fig. 5.2). Estimates of k were lowest in the model with strong priorsand highest in the model with uninformative priors (Table 5.2, Fig. 5.1).

Table 5.2: Mean von Bertalan�y growth parameter estimates for the three Bayesian models withdi�ering priors. Values inside square brackets are the 95% credible intervals (CI).

Model Estimate of DW∞ Estimate of k Estimate of σ 2

Strong priors 2999 mm [2711-3295] 0.12 year-1 [0.086-0.169] 0.106 [0.088-0.13]Weaker priors 2515 mm [2232-3018] 0.221 year-1 [0.11-0.353] 0.102 [0.084-0.124]Uninformative priors 2386 mm [2175-2744] 0.268 year-1 [0.144-0.406] 0.102 [0.084-0.124]

D W ∞ = 2993 k = 0.119

D W ∞ = 2479 k = 0.216

D W ∞ = 2361

2000 2400 2800 3200

k = 0.265

0.1 0.4 0.7 1 400 800 1200 0.6 0.8 1.0 1.2 1.4 1.0 1.1 1.2 1.3

DW∞ k DW0 kappa σ2

strongerpriors

weakerpriors

uninformativepriors

prior

posterior

Figure 5.1: Prior and posterior distributions for the Spinetail Devil Ray (Mobula japanica) vonBertalan�y growth parameters (DW∞, k , and DW0) and the error term (σ 2) for the three Bayesianmodels with strong, weaker, and uninformative priors. Median values are shown by the dashedlines, posterior distributions by the black lines, and prior distributions by the red lines. No priordistributions are shown when priors are uninformative (uniform distribution).

5.4.2 Part 2. Estimating total mortality using the catch curve

Our catch curve analysis, using all data points, yielded a median estimate of Z = 0.254 year-1,with 95% of bootstrapped estimates ranging between 0.210 and 0.384 year-1 (Fig. 5.3a,b). Whenremoving individuals aged 10 and older, our catch curve analysis resulted in a median estimate ofZ = 0.196 year-1, albeit with a higher uncertainty than our estimate with all data points (Fig. 5.3c,d).

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0 5 10 15 20

1000

1500

2000

2500

3000Maximum observed size

StrongWeak

UninformativeCuevas-Zimbrón et al. (2013)

Dis

c w

idth

(mm

)

Age (years)

Figure 5.2: Length-at-age data for the Spinetail Devil Ray Mobula japanica showing the Bayesianvon Bertalan�y growth curve �ts for models with strong (grey), weaker (red), and uninformative(blue) priors, as well as the original model �t from Cuevas-Zimbrón et al. (2013). Dashed linesshow the asymptotic size (DW∞) estimates for each model. Dotted line represents the maximumknown size for the species.

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Assuming Z is approximately 0.2 year-1 is therefore a relatively conservative estimate of totalmortality; we infer natural mortality M of M. japanica must be less than 0.2 year-1.

One remaining question is how our assumption of constant selectivity a�ects the catch curvein Fig. 5.3. To explore the e�ects of di�erences in Z given constant selectivity we simulated apopulation with the life history of M. japanica and show that steep declines of older individualsare possible by �shing alone (see Appendix).

By substracting our distribution of natural mortality M estimates from our distribution of totalmortality Z from the catch curve excluding ages 10 and older we obtain a distribution of �shingmortality values, which resulted in a median estimate of F = 0.110 year-1, albeit with a high degreeof uncertainty (95th percentile = -0.034 to 0.610).

5.4.3 Part 3. Maximum population growth rate rmax of the Spinetail Devil Ray

From Part 1, we estimated that maximum lifespan was between 15 and 20 years. Combining thiswith estimated age at maturity, the median estimates of average lifespan for the Spinetail DevilRay was 11.5 years, and therefore the median natural mortality M estimate was 0.087 year-1(95thpercentile = 0.079–0.097). Using this information to create a bounded distribution for naturalmortality in equation 5.10, we found the median maximum intrinsic rate of population increasermax for devil rays is 0.077 year-1 (95th percentile = 0.042–0.108).

5.4.4 Part 4. ComparingMobula rmax to other chondrichthyans

Devil and manta rays have low intrinsic rate of population increase relative to other chondrichthyanspecies (Fig. 5.4). Among species with similar somatic growth rates, the Spinetail Devil Ray hasthe lowest rmax value (black diamond in Fig. 5.4a). This contrast is strongest when excludingdeep-water chondrichthyans (white circles in Fig. 5.4), which tend to have much lower rates ofpopulation increase than shallow-water ones (Simpfendorfer and Kyne 2009). Our estimation ofrmax for manta rays (grey diamond in Fig. 5.4) are comparable with our estimates for the SpinetailDevil Ray, albeit slightly lower (median of 0.068 year-1, 95th percentile = 0.045–0.088). Values ofrmax for other large planktivorous elasmobranchs (Whale and Basking Sharks) are relatively highcompared to manta and devil rays.

5.5 Discussion

In this study, we examined multiple lines of evidence that suggest devil rays have relatively lowproductivity, and hence high risk of extinction compared to other chondrichthyans. The rmax ofthe Spinetail Devil Ray is comparable to that of manta rays, and much lower than that of otherlarge planktivorous shallow-water chondrichthyans such as the Whale Shark and the BaskingShark (Fig. 5.4). We conclude the comparable extinction risks of devil and manta rays, coupledwith the ongoing demand for their gill plates and the potential for increasing exploitation of devil

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0.0

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2.0

2.5

ln(A

bund

ance

)

(a)

2

5 5

7

109

2

8

4

1 1 10 0

All age classes

00.10.20.30.40.50.60.70.80.911.1(b)

Total mortality ( Z

, year − 1)

Median Z = 0.253S.D. = 0.061

1 2 3 4 5 6 7 8 9 10 12 140.0

0.5

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No ages 10+

00.10.20.30.40.50.60.70.80.911.1(d)

Total mortality ( Z

, year − 1)

Median Z = 0.196S.D. = 0.162

Figure 5.3: Estimation of total mortalityZ from bootstrapped catch composition data for the Spine-tail Devil Ray (Mobula japanica) from Cuevas-Zimbrón et al. (2013) using (a,b) all data points and(c,d) exluding ages 10 and older. (a,c) Catch curve of natural log abundance at age. The regressionlines represent the estimated slopes when omitting di�erent age-class subsets (as shown by thehorizontal extent of each line), and resampling 80% of the data. Note that individual estimates ofZ di�er in the number of age-classes included for its computation, resulting in regression linesof di�erent lengths. (b,d) Violin plot of estimated total mortality (Z ) values calculated using thebootstrap resampling method. Estimates from di�erent age-classes suggest an estimate of Z ≈0.25 year-1. The median is shown by the dark grey line.

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0.01 0.10 1.00

0.1

1.0

(a)

Growth coefficient (k )

10 100 1000

0.1

1.0

Max

imum

intri

nsic

rate

of p

opul

atio

n in

crea

se ( r

max

)

(b)

Maximum size (cm)

Figure 5.4: Comparison of maximum intrinsic rate of population increase (rmax ) for 96 elasmo-branch species arranged by (a) growth coe�cient k and (b) maximum size. Small open circlesrepresent deep sea species while small grey circles denote oceanic and shelf species. Four speciesare highlighted using silhouettes and larger symbols: the Spinetail Devil Ray (Mobula japanica) isshown by the black diamond, while the manta ray (Manta spp.), Whale Shark (Rhincodon typus)and Basking Shark (Cetorhinus maximus) are represented by the grey diamond, triangle and circle,respectively. Sources and attributions for the silhouettes used are as follows. Devil ray: PublicDomain. Manta ray by Freepik from Flaticon. Whale Shark by Scarlet23 (vectorized by T. MichaelKeesey), licensed under CC BY-SA 3.0. Basking Shark by Nick Botner, freely available online.

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rays resulting from manta ray regulation, suggest that conferring a similar degree of protection toall mobulids is warranted.

Our approach provides an alternative to previous data poor methods for estimating rmax thatbetter captures the e�ect di�ering reproductive outputs and juvenile survival have on productivity(Pardo et al. 2016). This is markedly di�erent to other approaches to estimate rmax that do not takeinto account reproductive output such as the demographic invariant method (Niel and Lebreton2005), or the rebound potential method (Au and Smith 1997) which e�ectively ignores di�erencesin fecundity and thus does not account for reproduction-related productivity di�erences (Au et al.2015). Regardless, most of these life history-based approaches perform similarly for slow growingspecies (Cortés 2016). When catch trends are available in addition to life history parameters, otherapproaches can be used (e.g., (Dick and MacCall 2011; Carruthers et al. 2014)).

The comparable low productivities of devil and manta rays, regardless of di�erences in bodysize, are largely due to their very low reproductive rates. Mobulid rays have at most a singlepup annually or even biennially, while the Whale Shark can have litter sizes of up to 300 pups(Joung et al. 1996), thus increasing the potential ability of this species to replenish its populations(notwithstanding di�erences in juvenile mortality). The Basking Shark has a litter size of six pups,which partly explains why its rmax is intermediate between mobulids and the Whale Shark. Whilemobulids mature relatively later with respect to their total lifespan than Basking Sharks and rela-tively earlier than Whale Sharks, they have lower lifetime fecundity than both Whale and BaskingSharks, limiting their productivity.

Our results are consistent with the correlation between low somatic growth rates, later mat-uration, large sizes, and elevated extinction risk (Jennings and Dulvy 2008; Dulvy et al. 2003)(Fig. 5.4a, Fig. 5.5). Similar relationships have been found in other marine �shes. For example,in tunas and their relatives, somatic growth rate is the best predictor of over�shing, such thatspecies with slower growth are more likely to be over�shed as �shing mortality increases thanspecies with faster growth (Juan-Jordá et al. 2013), likely because the species that grow faster ma-ture earlier. Furthermore, species with low fecundities have an elevated extinction risk regardlessof their age at maturity (Fig. 5.5a).

Our method for estimating growth rate for the Spinetail Devil Ray provided lower estimates ofthe growth coe�cientk than was reported in the original study (Cuevas-Zimbrón et al. 2013), espe-cially when we used strongly informative priors. When using strong priors the estimated asymp-totic size was very close to the expectation of it being 90% of maximum size. On the other hand, ourscenario with uninformative priors provided growth coe�cient k estimates that are very similarto the original estimates, which were obtained by nonlinear least squares minimization (Fig. 5.2b).Our growth estimates from the model with strong priors are consistent with our expected valuesof k if length-at-age data were available for larger individuals. Given that the length-at-age dataavailable only includes individuals up to two-thirds of the maximum size recorded for M. japanica,we believe that our approach provides more plausible estimates of growth rates when data aresparse. Our approach provides further evidence that Bayesian estimation is useful for data-sparse

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Age

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ears

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0.2 0.5 2 5 10 50 200

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r m a x

1.10.90.70.50.30.1

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1 2 5 10 20 50

(c)

Figure 5.5: Annual fecundity (b, in log-scale) vs (a) age at maturity and (b) the αmat/αmax ratio. (c)Age at maturity vs αmat/αmax ratio. Circle size represent rmax estimates. Four species are high-lighted by the red circles and silhouettes: the Spinetail Devil Ray (Mobula japanica) is shown bythe black silhouette, while the manta ray (Manta spp.), Whale Shark (Rhincodon typus) and Bask-ing Shark (Cetorhinus maximus) are represented by the grey silhouettes. Sources and attributionsfor the silhouettes used are as follows. Devil ray: Public Domain. Manta ray by Freepik fromFlaticon. Whale Shark by Scarlet23 (vectorized by T. Michael Keesey), licensed under CC BY-SA3.0. Basking Shark by Nick Botner, freely available online.

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species as the available life history information can be easily incorporated in the form of prior dis-tributions, particularly when missing samples of the largest or smallest individuals (Siegfried andSansó 2006). Incorporating prior information when �tting growth curves is an alternative to �xingmodel parameters, which often biases growth estimates (Pardo et al. 2013). In other words, usingBayesian inference allows us to incorporate out-of-sample knowledge of observed maximum sizesand sizes at birth, thus improving our estimates of growth rates and asymptotic size (Siegfried andSansó 2006).

The lower estimate of �shing mortality we calculated from the catch curve (Z −M = F = 0.110year-1) is higher than our estimate of rmax , which also represents the �shing mortality F expectedto drive this species to extinction (Fext = 0.077 year-1) (Myers and Mertz 1998). Even though ourestimates of total and �shing mortality are highly uncertain (Fig. 5.3b,d), our estimate of Z wasderived excluding age classes that might have been underrepresented, resulting in conservativeestimates of both total and �shing mortalities. Hence we infer that before the �shery ceased in2007, the Spinetail Devil Ray population we examined was probably being �shed unsustainably ata rate high enough to lead to eventual depletion. Many teleost �sheries support �shing mortalitiesthat are many times larger than natural mortality. However, mobulids likely have low capacityto compensate for �shing, because their large o�spring and low fecundity suggest weak density-dependent regulation of populations (Forrest and Walters 2009; Kindsvater et al. 2016).

Nonetheless, the steeper decline in abundance of individuals older than age 9 is a pattern thatcould emerge solely from �shing (see Supplementary materials), migration of larger individualsaway from the �shing grounds, or as a result of di�erences in catchability between age groups. Toreduce the likelihood of overestimating Z because of di�erences in catchability or migration, weestimated a slope of the catch curve excluding these age groups. This more conservative estimate ofZ , albeit being more uncertain, better represents the total mortality of the population by discardingthe age classes that could bias our estimate of Z .

The major caveats of using a catch curve analysis to estimate total mortality are that it as-sumes there is no size selectivity in catch, recruitment is constant, the population is closed, andthat the catch is a large enough sample to su�ciently represent population age structure. Theseassumptions are also required in age and growth studies when using length-at-age data. Thus, theuse of catch curves to estimate �shing mortality could be applied to other chondrichthyan growthstudies, assuming that sampling is not systematically size selective and that metapopulation dy-namics are not in�uencing the sample. Whether or not this latter assumption is valid for highlymigratory elasmobranch species has yet to be tested.

Unregulated small-scale artisanal �sheries are targeting mobulids throughout the world (Cou-turier et al. 2012; Croll et al. 2016). Our �ndings hint that there is little scope for unmanagedartisanal �sheries to support sustainable international exports of gill plates or even domestic meatmarkets. Furthermore, the unsustainable �shing mortality stemming from the removal of rela-tively few individuals by an artisanal �shery suggests we urgently need to understand the conse-quences of bycatch of mobulid rays in industrial trawl, long line and purse seine �sheries (Croll

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et al. 2016; Francis and Jones 2016). The combination of high catch rates and low post-release sur-vival suggest �shing mortality rates need to be understood and potentially minimized to ensurethe future persistence of these species.

5.6 Acknowledgements

We thank the �shermen of the �shing camp “Punta Arenas de la Ventana”, Baja California Sur,Mexico, for allowing collection of specimens and biological material. Field and lab work wassupported by F. Galvan, N. Serrano, I. Mendez, A. Medellín, L. Castillo and C. Rodríguez, E. DíÃŋaz,J. M. Alfaro and E. Bravo. We thank S. C. Anderson for statistical advice, and J. M. Lawson for hercomments on the manuscript.

The original �eldwork was supported by the project “Historia Natural, Movimientos, Pesqueríay Criaderos, Administración de Mantas Mobulidas en el Golfo de California” of the Monterey BayAquarium. This research was funded by the J. Abbott /M. Fretwell Graduate Fellowship in FisheriesBiology (SAP), NSERC Discovery and Accelerator Grants (NKD), a Canada Research Chair (NKD),an NSF Postdoctoral Fellowship in Math and Biology (HKK; DBI-1305929), and grants to NKD fromthe John D. and Catherine T. MacArthur Foundation, the Leonardo DiCaprio Foundation, DisneyConservation Fund, and the Wildlife Conservation Society.

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Chapter 6

Temperature and depth shapeproductivity and its mass scaling insharks and rays5

6.1 Abstract

An important challenge in ecology is to develop a mechanistic understanding of what drives aspecies’ productivity as this underpins our understanding of �shing limits, extinction risk, and re-covery potential. Metabolic theory postulates that body mass and temperature drive productivityof species. In sharks and their relatives, hereafter referred to as sharks and rays, productivity isknown be lower in larger species, but also in those living at greater depths. While the deep ocean isassociated with colder temperatures, other factors such as food availability and physiological con-straints may in�uence the low productivities observed in deep sea sharks and rays independentlyof temperature. We use an information theoretic approach while accounting for phylogenetic re-latedness to evaluate the relative importance of temperature and depth on productivity and themass scaling of this relationship. We show that both temperature and depth have separate yetindependent e�ects on productivity of sharks and rays, such that species living in deeper, colderwaters have lower productivities. Furthermore, temperature also correlates with changes in themass scaling coe�cient as cold-water species have mass scaling coe�cients closer to 0. These�ndings underline that there are multiple processes acting in concert which limit the productivityof deep sea sharks and rays and provide a useful template for assessing productivity of speciesbased on easily accessible information such as temperature and depth, particularly for the mostdata poor species.

5N.K. Dulvy is a co-author on this chapter, which is in preparation for submission to a journal

79

6.2 Introduction

The rates at which organisms use and uptake energy vary with body size and temperature (Brownet al. 2004). Metabolic scaling theory postulates that smaller organisms, and those at higher tem-peratures, have higher mass-speci�c metabolic requirements than larger, colder ones as a functionof their vascular arrangements (Brown et al. 2004). This theory predicts that allometric rela-tionships of biological rates should scale in multiples of 1⁄4, a prediction backed by empirical data(Slobodkin 1980; Savage et al. 2004). Over recent years it has become more apparent that scalingcoe�cients are not �xed, but vary as a result of other factors aside from size and temperature(Clarke 2006; Glazier 2005; 2010). For example, in teleosts ecological lifestyles such habitat andtrophic levels also correlate with metabolic rates (Killen et al. 2016). Furthermore, comparisonsacross mammal species also suggest that aside from allometric variation in biological rates, thereis another axis of variation which is driven by di�erences in lifestyle and associated traits (Siblyand Brown 2007, note that temperature is not an axis of variation in endotherms).

Population growth rates, hereafter referred to as productivity, also scale with body size andtemperature with larger organisms having lower productivities than smaller ones as a result ofmetabolic constraints (Slobodkin 1980; Savage et al. 2004; Brown and Sibly 2006). One of theserates is the maximum intrinsic rate of population increase, rmax , which is the theoretical maximumrate at which a population can grow in the complete absence of density dependence, and can occuronly at very low population sizes (Myers et al. 1997). Estimates of rmax are commonly used limitreference points as they are equivalent to the �shing mortality that will drive a species to extinction(Dulvy et al. 2004). While metabolic theory hypothesizes that body size and temperature are theonly factors shaping production rates, energy availability and intake can also limit energy availablefor production: mass-speci�c production rates are higher in mammals with high energy intake andlower in those which maximize survival (Sibly and Brown 2007).

The sharks, rays, and chimaeras (class Chondrichthyes, hereafter referred to as sharks andrays) are an ideal taxon to study allometric relationships as they encompass a broad range ofsizes and inhabit the world’s oceans covering a wide range of temperatures and habitats. Thelack of a pelagic larval stage in sharks and rays allows for the estimation of productivity withlimited life history data (Myers and Mertz 1998; Pardo et al. 2016). Productivity among sharksand rays is known to decrease with increasing size (Hutchings et al. 2012; Dulvy et al. 2014b) anddepths (García et al. 2008; Simpfendorfer and Kyne 2009). However, there is some evidence thatthe relationship between productivity and body size breaks down in the deep sea as even smalldeep-water sharks and rays have very low productivities (Rigby and Simpfendorfer 2015). From ametabolic theory perspective, this suggests there are stronger constraints on the mass scaling ofproduction rates occurring at greater depths than those imposed by body mass and temperature.

In marine ecosystems, exploring variation in productivity across depth presents a unique op-portunity to investigate this lifestyle axis across an energetic gradient. There are multiple ways inwhich energy availability decreases with increasing depth: temperature, light, and consequently

80

primary productivity decrease below the photic zone (Gage and Tyler 1991; Jahnke 1996); metabolicrequirements in species decrease with increasing depth as the distance at which predators and preyinteract is reduced in as light levels drop (Childress et al. 1990; Seibel and Drazen 2007); animalbiomass decreases with increasing depth (Rex et al. 2006); even the unique physiology of sharksand rays can increase the energetic cost of living in the deep, and consequently reduce the energyavailable for production (Treberg and Speers-Roesch 2016). All these potential energetic gradientsare likely to impact the productivity of sharks and rays across a depth gradient, even after tem-perature is accounted for. Nonetheless, while the scaling relationship between population growthrates and temperature (Savage et al. 2004), as well as population growth rate and lifestyle havebeen explored independently (García et al. 2008; Simpfendorfer and Kyne 2009), little work hasbeen done and teasing these di�erences apart from each other.

Comparative studies of metabolic rate in marine organisms suggest that the lower mass-speci�cmetabolic rate seen in deep water (>1000 m) species is a result of the “visual-interactions hy-pothesis” (where the distance at which predators and prey interact is reduced as light levels dropChildress et al. 1990; Seibel and Drazen 2007), instead of food availability or other factors. Whilethere are a number of studies that explore the relationship between metabolic rate and depth whileaccounting for body size and temperature (see Drazen and Seibel 2007), there are no studies ex-ploring how productivity across species scales with both depth and temperature. The processesshaping metabolic rate evolution are likely to di�er from those shaping productivity: for exam-ple, evidence suggests that food availability does not drive metabolic rate in deep sea organisms(Drazen and Seibel 2007; Seibel and Drazen 2007), yet it would be expected from simple energeticsmodels that food availability should drive productivity regardless of its e�ect on metabolic rate.

In this study, we aim to examine the separate role of temperature and depth on the productivityand mass scaling relationship among sharks and rays by using an information theoretic approachwhile accounting for phylogenetic non-independence. We also examine the role of temperatureaand depth on the somatic growth rate (as described by the von Bertalan�y growth coe�cient k),as this parameter is often used as a proxy of productivity (e.g. Musick 1999).

6.3 Methods

6.3.1 Data

The data used in this study were obtained from multiple sources (see Supplementary Materials forcomplete dataset). Maximum reported body mass (in grams) were obtained from FishBase (Froeseand Pauly 2016) using the rfishbase package (Boettiger et al. 2012). When body mass data wereunavailable, maximum length data were converted to body mass using species-speci�c length-to-weight conversions also sourced from FishBase. Data for rmax and growth coe�cient k wereobtained from a modi�ed Euler-Lotka model following Pardo et al. (2016, see supplement), mediandepth estimates for each species as reported from Dulvy et al. (2014a), and species distributions

81

maps were obtained from Aquamaps (Kaschner et al. 2015), which is an online resource of globalspecies distribution models for over 25,000 aquatic species.

The core distributions for each species (where probability of occurrence >= 0.9) were over-laid with the International Paci�c Research Center’s interpolated dataset of gridded mean annualocean temperatures across 27 depth levels (0-2000 m below sea level), which is based on mea-surements from the Argo Project (see http://apdrc.soest.hawaii.edu/projects/Argo/data/statistics/On_standard_levels/Ensemble_mean/1x1/m00/index.html for data and more in-formation). This temperature interpolation covers most of the world’s oceans, but has incompletecoverage of some shallow coastal areas, such as the Indo-Paci�c Triangle and the southeast coastof South America.

To calculate a temperature for each species, we selected the depth level from the grid that wasclosest to the species’ median depth, and from that grid extracted all the temperature grid pointsthat overlaid the species’ core distribution. From that distributions of temperatures, the medianwas calculated and set as the temperature value for each species. For example, for a species witha median depth of 130 m, we used the 150 m layer to estimate mean annual temperature as 130 mis closer to 150 than 100 m. In species that are known to be endothermic (e.g. family Lamnidae)we added a correction factor of 3.5 Kelvin.

The phylogenetic trees were obtained from Stein et al. (2017) and we followed their scienti�cnomenclature. Their analyzes did not provide a single tree but a distribution of possible trees withthe same topology but di�ering branch lengths. We ran our analyses by using 20 di�erent treessampled from their distribution. The results were almost identical regardless of which tree wasused, and therefore we only report our �ndings when using only a single tree (see Table 6.6 in theSupplementary materials).

6.3.2 Metabolic scaling expectations

Metabolic scaling theory predicts that biological rates related to productivity, in this case rmax , arerelated with body mass and temperature as follows (Brown et al. 2004; Savage et al. 2004):

rmax ∝ i0Mβe−E/kBT (6.1)

where rmax is the maximum intrinsic rate of population increase (in year-1), i0 is a taxon-speci�c normalization constant, M is is maximum body mass (in grams), β is the scaling exponent,E is the activation energy, kB is the Boltzmann constant (8.617 × 10−5 eV), and T is temperature(in Kelvin).

We do not aim to estimate the scaling exponent of this relationship as we do not have a largeenough sample size to do so; instead we use this framework to compare hypotheses about thecorrelates of productivity among sharks and rays. First, the equation above is simpli�ed whentransformed to log-space:

82

loд(rmax ) ∝ loд(i0) + β ∗ loд(M) + −E ∗ 1/kBT (6.2)

The equation above is equivalent to a simple linear model:

loд(rmax ) ∝ β0 + β1 ∗ loд(M) + β2 ∗ 1/kBT (6.3)

where the intercept β0 is the log-transformed normalization constant, and the coe�cients β1and β2 are the scaling coe�cient β and the negative activation energy −E, respectively. We donot use a taxon-speci�c normalization constant (i0 = β0) as we are using a phylogenetic covari-ance matrix instead of a taxonomic nested structure. For ease of interpretation we will write thefollowing in the simpler pseudo-code formula notation:

loд(rmax ) ∼ loд(M) + 1/kBT (6.4)

This simple linear model in eq. 6.4 is our “null hypothesis” against which we compete ourother models. Thus, if an alternative hypothesis has higher support than our null this indicatesthat there are other variables that correlate with productivity (rmax ) aside from what would beexpected from metabolic theory alone.

6.3.3 Does depth play a role in the metabolic scaling relationship of rmax?

We compared di�erent hypotheses using an information theoretic approach (Burnham and Ander-son 2002). By assessing the relative quality of each model we can identify the model with the mostsupport. Nine models were chosen as they all represent potential hypotheses of the relationshipsamong body mass, depth, and temperature with rmax (Table 6.1).

We only included a single interaction term as this is easier to interpret than multiple interactionterms in a model and to avoid model over�tting. We also considered a hypothesis just based onbody mass to assess whether temperature data were informative. We compared all models usingthe corrected Akaike Information Criterion (AICc). If including a parameter improved a model’sAICc value by less than 2, we considered that parameter to be uninformative (Arnold 2010).

To account for non-independence among closely related species, we �tted phylogenetic linearmodels using the pgls function in the caper package (Orme et al. 2013), running on R version 3.3.2(R Core Team 2016). The phylogenetic covariance matrix, which is contrasted with the residualsfrom each model, was adjusted using Pagel’s λ. This allows for encompassing a wide range ofcovariance structures corresponding to di�erent trait phylogenetic signal values, ranging from nosignal at all (i.e. classic OLS; λ = 0) to Brownian motion (λ = 1 Revell 2010). We tested for collinear-ity between parameters, and as expected depth and inverse temperature are positively correlated(Pearson’s r = 0.64); this is lower than the threshold of |r | > 0.7, which is when collinearityseverely distorts model estimation (Dormann et al. 2013).

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Table 6.1: Productivity (rmax ) models tested in our analysis and the hypotheses associated witheach. The model labelled with MTE denotes the model proposed by the Metabolic Theory ofEcology (Brown et al. 2004)

Model Hypothesis

loд(rmax ) ∼ loд(M) Productivity only varies with body massloд(rmax ) ∼ loд(M) + depth Productivity varies with body mass and depthloд(rmax ) ∼ loд(M) + 1/kBT Productivity varies with body mass and temperature (MTE)loд(rmax ) ∼ loд(M) + 1/kBT + depth Productivity varies with body mass, temperature, and

depthloд(rmax ) ∼ loд(M) + 1/kBT ∗ depth Productivity varies with body mass, temperature, and

depth, and the e�ect of temperature varies with depthloд(rmax ) ∼ loд(M) ∗ depth Productivity varies with body mass and depth, and the ef-

fect of mass (i.e. scaling coe�cient) varies with depthloд(rmax ) ∼ loд(M) ∗ 1/kBT Productivity varies with body mass and temperature, and

the e�ect of mass (i.e. scaling coe�cient) varies with tem-perature

loд(rmax ) ∼ loд(M) ∗ depth + 1/kBT Productivity varies with body mass, temperature, anddepth, and the e�ect of mass (i.e. scaling coe�cient) varieswith depth

loд(rmax ) ∼ loд(M) ∗ 1/kBT + depth Productivity varies with body mass, temperature, anddepth, and the e�ect of mass (i.e. scaling coe�cient) varieswith temperature

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A model using the residuals from a regression of median depth on temperature (i.e., sequentialregression of the form invtemp ∼ depth) instead of median depth values produced identical resultsas our model including both variables (invtemp + depth).

6.3.4 Metabolic scaling of growth coe�cient k

We also examined the e�ect of mass, temperature, and depth on the von Bertalan�y growth coef-�cient k (von Bertalan�y 1957) using the same nine models described on Table 6.1 but using log(k)as a response variable instead of loд(rmax ). Note that this k value is unrelated to the Boltzmannconstant kB mentioned in the models above.

6.4 Results

We collected a complete dataset on temperature, body mass, depth, rmax , and k for 63 chon-drichthyan species including 40 sharks, 20 rays, and three chimaeras (Fig. 6.1).

6.4.1 Metabolic scaling of rmax

The model with the highest support is the one including both temperature and depth as well as aninteraction term between body mass and temperature (Table 6.2). This model also had the highestadjusted R2 value (0.33). The two other models that had marginal support (5 ≤ ∆AICc ≤ 7) in-cluded both temperature and depth, as well as an interaction term between body mass and depth.The e�ect size of body mass in all three best models is around −0.3 (see shaded areas in Fig. 6.2),which overlaps to the expectation of − 1/4 predicted by metabolic theory (Brown et al. 2004). Thebest model included a positive interaction term between body size and inverse temperature (Ta-ble 6.3), indicating that the mass-scaling coe�cient becomes closer to zero with decreasing tem-peratures. Only the two best models, both of which included an interaction term between bodysize and temperature, resulted in plausible temperature e�ect sizes. While the e�ect of depth wasnegative in all three best models, the e�ect of the inverse of temperature was negative in the twobest model but positive in the third best model (see shaded areas in Fig. 6.2, Table 6.3). The ex-pectation from metabolic theory is that this coe�cient should be negative as it is equivalent to−E and E has to be positive (see eq. 6.2), thus only the two best models provide realistic ranges ofvalues for this coe�cient, likely due to the poor �t resulting from ignoring the interaction term.There was a strong phylogenetic signal the residuals of rmax in all nine models tested. Estimatedλ ranged between 0.78 and 0.87 with the best model having the lowest λ value (Table 6.3).

6.4.2 Metabolic scaling of growth coe�cient k

There were four models that were relatively similarly supported by the data (∆AICc ≤ 2, Ta-ble 6.4). The model with the lowest AICc included body mass, temperature, and an interactionterm between these two variables. However, the model with body mass and temperature with no

85

Callorhinchus miliiCallorhinchus capensisChimaera monstrosaLeucoraja naevusLeucoraja erinaceaLeucoraja ocellataAmblyraja radiataRaja miraletusRaja montaguiRaja microocellataRaja brachyuraRaja asteriasRaja clavataBeringraja binoculataTorpedo californicaTorpedo marmorataTorpedo torpedoRhinobatos productusAetobatus flagellumMyliobatis californicusManta birostrisDasyatis americanaPteroplatytrygon violaceaNotorynchus cepedianusDeania calceaCentrophorus squamosusSqualus acanthiasEtmopterus spinaxRhincodon typusAlopias superciliosusCarcharias taurusCetorhinus maximusLamna ditropisLamna nasusCarcharodon carchariasIsurus oxyrinchusScyliorhinus caniculaGaleorhinus galeusTriakis semifasciataFurgaleus mackiMustelus manazoMustelus antarcticusMustelus canisGaleocerdo cuvierSphyrna lewiniSphyrna tiburoRhizoprionodon terraenovaeNegaprion brevirostrisCarcharhinus limbatusCarcharhinus tilstoniCarcharhinus signatusCarcharhinus amblyrhynchosCarcharhinus falciformisPrionace glaucaCarcharhinus plumbeusCarcharhinus sorrahCarcharhinus longimanusCarcharhinus obscurusCarcharhinus galapagensisCarcharhinus leucasCarcharhinus brevipinnaCarcharhinus brachyurusCarcharhinus acronotus

50 My

1 102 104 0.0 0.4 0.8 1.2 0 5 20

* ** *

0 500 1500

Max. weight (kg) rmax (year−1) Median depth (m)Temperature (C)

Figure 6.1: Phylogeny, body mass, productivity, temperature, and depth in sharks, rays, and chi-maeras Phylogenetic tree is based on Stein et al. (2017), body mass estimates were sourced fromFishBase (Froese and Pauly 2016), rmax estimates from Pardo et al. (2016), median depth valuesfrom Dulvy et al. (2014a), and mean annual temperature values (at median depth) were estimatedbased on species distribution maps from Aquamaps (Kaschner et al. 2015) and global temperaturegrids from the Argo database. Asterisks (*) denote temperature values corrected for endothermy.Horizontal dotted lines indicate separate taxonomic orders.

86

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●

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●

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●

Intercept

log(M)

invtemp

depth

log(M):depth

log(M):invtemp

−3 −2 −1 0 1Effect size

Coe

ffici

ent

Model●

●

●

●

log(M) * invtemplog(M) * invtemp + depthlog(M) * depth + invtemplog(M) + invtemp + depth

Figure 6.2: Coe�cient plots for the four models of loд(rmax ) with lowest AICc values. Lightercolours indicate models with decreasing support based on ∆AICc. Error bars show the 95% con-�dence intervals, and e�ect sizes were considered signi�cant when con�dence intervals do notoverlap zero. Shaded areas show the expected e�ect sizes for body mass (-0.33 to -0.25) and in-verse temperature (-1.0 to -0.6) based on metabolic theory.

87

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10 m depth

5 C, 1000 m depth

5 C, 10 m depth

0.1

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1 10 100 1000 10000Maximum weight (kg)

r max

(ye

ar−1

)

Mediandepth (m)

●

●

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100

500

1000

1500

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20

Mediantemperature (°C)

Figure 6.3: Relationship between maximum weight and maximum intrinsic rate of populationincrease rmax , in log space. Di�erences in depth and temperature are shown by the point size andcolour, respectively. Light blue line denotes mean predicted log(rmax ) for shark and ray speciesat a temperature of 20 C and a median depth of 10 metres, while dark blue lines represent meanpredicted log(rmax ) for species at a temperature of 5 C at a median depth of 10 metres (solid line)or 1000 metres (dashed line). Median temperatures are corrected for species which have bodytemperatures that are higher than their surroundings.

88

Table 6.2: Comparison of loд(rmax ) models using standard and corrected Akaike Information Cri-teria (AICc), number of parameters (n), negative log-likelihood (-LL), adjusted R2, and Akaikeweights. The model with lowest AIC is shown in bold, while the models with ∆AICc ≤ 2 arehighlighted in grey.

loд(rmax ) ∼ n -LL AICc Adj. R2 ∆AICc Weightsloд(M) 2 -45.5 95.1 0.17 10 0.005loд(M) + depth 3 -44.8 96.1 0.17 11 0.003loд(M) + invtemp 3 -44.2 94.9 0.19 9.8 0.006loд(M) + invtemp + depth 4 -41.3 91.2 0.24 6.1 0.038loд(M) + invtemp ∗ depth 5 -41.2 93.5 0.23 8.4 0.012loд(M) ∗ depth 4 -42.8 94.3 0.21 9.2 0.008loд(M) ∗ invtemp 4 -40.7 90.1 0.26 5 0.066loд(M) ∗ depth + invtemp 5 -39.7 90.6 0.27 5.5 0.052loд(M) ∗ invtemp + depth 5 -37 85.1 0.33 0 0.809

interaction had almost the same support (∆AICc = 0.1, Table 6.4). The model including both tem-perature and depth was also within 2 ∆AICc. However adding an extra parameter (either depth oran interaction term) did not improve model parsimony so these parameters were not consideredinformative (Arnold 2010). Thus the model most supported by the data is the one only includingbody mass and temperature; accounting for depth did not improve model �t. The adjusted R2 ofthe best model of growth coe�cient k was considerably lower (R2 = 0.25) than that of the bestmodel of rmax (R2 = 0.33). The phylogenetic signal of the residuals of k was low in all the modelstested (λ between 0 and 0.23; Table 6.5).

6.5 Discussion

Our �ndings strongly support the hypothesis that, across species, productivity increases at greatertemperatures and decreases at greater depths. These temperature-related metabolic limitationsand depth-related di�erences that are independent of temperature, happen concurrently and shapeproductivity of chondrichthyans. Both these processes result in deep-sea species having some ofthe lowest productivities among chondrichthyans (Simpfendorfer and Kyne 2009). In a similarmanner as metabolic rate, inter-speci�c productivity varies with depth, independently of bodysize and temperature (Drazen and Seibel 2007; Seibel and Drazen 2007). Depth can be thoughtof as a third axis of the slow-fast life history continuum (after body mass and temperature in ec-totherms) as described by Sibly and Brown (2007), and could be partially driven by di�erences inenergy availability in the deep sea when compared with shallow, photosynthetically active wa-ters. However, these patterns could also arise from physiological constrains imposed by living inthe deep, that are particular to chondrichthyans, that limit their ability to use available energyfor production. Treberg and Speers-Roesch (2016) hypothesized there are two separate physio-logical constraints limiting energy availability in deep water chondrichthyans: the energetic costof lipid accumulation for buoyancy, and nitrogen limitation due to their osmoregulatory strategy.

89

Tabl

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es(9

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ases

timat

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anda

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rors

show

nin

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log(

M):i

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mp

log(

M):d

epth

invt

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1.00)

-0.31

(-0.4

8--

0.14

)-

--

--

0.85

(0.59

-0.96

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dep

th-0

.03

(-1.0

0-0

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-0.30

(-0.4

6--

0.13

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-0.19

(-0.50

-0.13

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0.81

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0.02

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94)

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)∗in

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0.07

(-0.79

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0.78

(0.47-0

.93)

90

These physiological limitations could be acting in concert with energetic limitations of low foodavailability to reduce even further the capability of deep-sea chondrichthyans to grow in size andproduce o�spring. Another potential mechanism that could result in the observed pattern is thevisual-interaction hypothesis: deep-sea species interact with predators and prey on much smallerscales than shallow water ones, which potentially result in lower basal metabolic rates (Drazenand Seibel 2007), which could also be driving the observed patterns in productivity. While thisstudy does not unpack the mechanism causing this relationship, it encourages further explorationinto the mechanism behind the relationship between productivity and depth. Taking into accountadditional spatial datasets that provide information on nutrient availability in the deep sea, such assedimentation rates or organic carbon burial rates (Jahnke 1996) could provide a better understand-ing of the degree to which food availability limits productivity among deep-sea sharks and rays.Alternatively, exploring whether the e�ect of depth on rmax levels o� after a certain threshold (asit does for basal metabolic rate in pelagic �shes, supporting the visual-interactions hypothesis) orit is continuous would help elucidate the mechanism driving the observed relationship with depth.

The two best models of productivity included an interaction term between maximum sizeand temperature (Table 6.2), which indicates that the mass-scaling coe�cient is not �xed, butrather changes across a temperature gradient. This suggests that the observation by Rigby andSimpfendorfer (2015) on the absence of a relationship between size and productivity in deep-seasharks and rays is probably due to the low temperatures found in the deep sea rather than depth it-self. In other words, it is not depth that “�attens” the productivity mass-scaling in chondrichthyansas has been suggested (Rigby and Simpfendorfer 2015) but temperature, so that species that livein colder environments have a mass-scaling relationship with productivity that is closer to 0 thanthose that live in warmer waters. Further research might also see this shallower mass-scaling ofproductivity in shallow, cold-water ecosystems in higher latitudes. Further exploration of the howthe mass-scaling relationship in chondrichthyans from solely shallow environments varies withtemperature would help elucidate this further.

The high phylogenetic signal in the residuals of rmax indicates that productivity is overall anevolutionarily conserved trait, thus by knowing the phylogenetic placement of a species one canhave a rough idea of its productivity. This is not the case with growth coe�cient k , likely becausek varies considerably among closely related species as a function of body size. Estimates of rmax

are more strongly in�uenced by reproductive output (see Chapter 4), which unlike size is stronglyconserved among closely related species (Dulvy and Reynolds 1997). While many phylogeneticmethods like pgls are not meant to be used as predictive tools, our models could potentially beadapted for predictive purposes. A phylogenetic eigenvector framework (Guénard et al. 2013)could be used to develop predictive models for most data poor chondrichthyan species as an almostcomplete phylogeny has recently become available (Stein et al. 2017). Some degree of validationof predictive models would be required for these to be useful (e.g., through cross-validation, jack-kni�ng, etc.). However, the predictive ability of these models is also limited by the moderatecollinearity observed between depth and temperature, which is in great part unavoidable, yet an

91

important caveat of this analysis. One important limitation of this analysis is there is considerablycollinearity between depth and temperature. Some degree of collinearity is unavoidable as warmerwaters are almost chie�y found in shallow depths, while the deep is consistently cold. As the model�t improves when both these variables are included rather, the correlation between parametersdoes not seem to “mask” the e�ect of depth on rmax .

6.6 Acknowledgements

We are grateful to J. Bigman, F. Mazel, P.S.D. Kyne, and D. Greenberg for their insightful commentson the manuscript and help with the �gures.

92

Table 6.4: Comparison of loд(k) models using standard and corrected Akaike Information Criteria(AICc), adjusted R2, and Akaike weights. The model with lowest AIC is highlighted in bold, whilethe models with ∆AICc ≤ 2 are highlighted in grey.

loд(k) ∼ n -LL AICc Adj. R2 ∆AICc Weightsloд(M) 2 -62.3 128.8 0.18 5.7 0.016loд(M) + depth 3 -59.5 125.4 0.21 2.3 0.086loд(M) + invtemp 3 -58.4 123.2 0.24 0.1 0.259loд(M) + invtemp + depth 4 -58 124.7 0.23 1.6 0.122loд(M) + invtemp ∗ depth 5 -57.5 126 0.23 2.9 0.064loд(M) ∗ depth 4 -59.5 127.7 0.2 4.6 0.027loд(M) ∗ invtemp 4 -57.2 123.1 0.25 0 0.272loд(M) ∗ depth + invtemp 5 -57.7 126.5 0.23 3.4 0.05loд(M) ∗ invtemp + depth 5 -57 125 0.25 1.9 0.105

93

Tabl

e6.

5:Co

e�ci

ente

stim

ates

(95%

CIas

estim

ated

from

stan

dard

erro

rssh

own

inbr

acke

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oral

lmod

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floд(k)

.The

mod

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ithth

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t∆A

ICcv

alue

ism

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bold

and

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mod

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are

high

light

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grey

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M)

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dept

hlo

g(M

):inv

tem

plo

g(M

):dep

thin

vtem

p:de

pth

Page

l’sλ

loд(M)

-0.55

(-1.4

9-0

.40)

-0.38

(-0.57

--0.

19)

--

--

-0.

23(0

.00

-0.78

)loд(M)+

dep

th-0

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.07)

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(-0.4

7--0

.13)

--0

.41

(-0.73

--0.

09)

--

-0.

00(0

.00

-0.54

)loд(M)+

invtemp

-0.52

(-1.32

-0.28

)-0

.35(-0

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0.18

)-0

.48

(-0.80

--0.

16)

--

--

0.00

(0.0

0-0

.36)

loд(M)+

invtemp+dep

th-0

.57(-1

.38-0

.24)

-0.34

(-0.51

--0.

16)

-0.36

(-0.78

-0.0

6)-0

.18(-0

.59-0

.23)

--

-0.

00(0

.00

-0.35

)loд(M)+

invtemp∗dep

th-0

.47

(-1.31

-0.36

)-0

.34(-0

.51--

0.17

)-0

.60

(-1.22

-0.0

3)0.

35(-0

.77-1

.48)

--

-0.56

(-1.6

7-0.

55)

0.00

(0.0

0-0

.30)

loд(M)∗

dep

th-0

.71(-1

.56-0

.13)

-0.31

(-0.4

9--

0.13

)-

-0.52

(-2.11

-1.0

7)-

0.02

(-0.32

-0.37

)-

0.00

(0.0

0-0

.55)

loд(M

)∗in

vtemp

-0.25

(-1.12

-0.62)

-0.40

(-0.59

--0.22)

-1.90

(-3.78

--0.02)

-0.29

(-0.09

-0.66)

--

0.00

(0.00-0

.29)

loд(M)∗

dep

th+invtemp

-0.4

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.33-0

.41)

-0.36

(-0.55

--0.

18)

-0.4

1(-0

.85-0

.03)

-0.71

(-2.29

-0.86

)-

0.12

(-0.23

-0.4

8)-

0.00

(0.0

0-0

.33)

loд(M)∗

invtemp+dep

th-0

.31(-1

.19-0

.58)

-0.39

(-0.58

--0.

20)

-1.72

(-3.6

8-0

.24)

-0.14

(-0.55

-0.27

)0.

27(-0

.11-0

.65)

--

0.00

(0.0

0-0

.29)

94

6.7 Supplementary materials

6.7.1 Supplementary Tables

95

Tabl

e6.

6:D

i�er

ence

sin

corr

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dA

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nCr

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(∆A

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.

loд(r

max)∼

12

34

56

78

910

1112

1314

1516

1718

1920

loд(M)

10.70

10.4

09.3

010

.5010

.60

10.30

10.4

09.3

010

.5010

.40

9.30

9.80

10.6

010

.60

10.50

10.50

10.50

10.70

10.70

10.4

0loд(M)+

dep

th11.

4011.

2010

.60

11.30

11.40

11.20

11.20

10.6

011.

3011.

2010

.60

11.00

11.20

11.30

11.30

11.30

11.30

11.40

11.50

11.20

loд(M)+

invtemp

10.6

010

.309.0

010

.3010

.5010

.1010

.309.0

010

.3010

.309.0

09.4

010

.60

10.4

010

.3010

.3010

.40

10.6

010

.60

10.30

loд(M)+

invtemp+dep

th6.

506.

406.

306.

406.

406.

206.

306.

306.

306.

206.

306.

406.

406.

206.

306.

306.

306.

306.

406.

30loд(M)+

invtemp∗dep

th8.8

08.7

08.6

08.7

08.8

08.5

08.6

08.6

08.7

08.6

08.6

08.8

08.7

08.5

08.6

08.6

08.6

08.6

08.7

08.7

0loд(M)∗

dep

th9.4

09.1

09.0

09.3

09.3

09.2

09.3

09.0

09.3

09.2

09.0

09.3

09.0

09.3

09.3

09.3

09.2

09.2

09.4

09.2

0loд(M)∗

invtemp

5.50

5.30

4.20

5.30

5.40

5.30

5.30

4.20

5.30

5.40

4.20

4.30

5.50

5.60

5.30

5.30

5.40

5.60

5.50

5.20

loд(M)∗

dep

th+invtemp

5.60

5.40

5.80

5.50

5.50

5.40

5.50

5.80

5.50

5.40

5.80

5.80

5.40

5.50

5.40

5.40

5.40

5.30

5.60

5.50

loд(M

)∗in

vtemp+depth

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

96

Chapter 7

General discussion

Throughout the process of completing this thesis I became increasingly aware of the myriad of po-tential ways in which I could explore the speed of life among sharks, rays, and chimaeras through alife history theory lens. It was quite di�cult to narrow down all the potential options into �ve the-sis chapters; ultimately I attempted to cover diverse aspects of life history theory but maintaininga central focus around methods that can be used for by decision-makers for assessing data-poorspecies. This is why I limited my thesis work to two key life history parameters: growth coe�cientk and maximum intrinsic rate of population increase rmax . After all, one of the main reasons thatdrove me to pursue this PhD was to do research that informs decision-making in a scienti�cally-sound manner.

7.1 Main �ndings

Very broadly speaking, in this thesis I clari�ed methodological problems regarding the estimationof growth and productivity, explored the e�ect of uncertainty when using one of these methods,applied these insights to a species of special conservation concern, and tested hypotheses about thedrivers behind growth and productivity using a metabolic theory framework. More speci�cally,I �rst show that �xing the L0 parameter in the von Bertalan�y growth function (VBGF, which isoften done in chondrichthyan age and growth studies) often causes considerable bias in growthcoe�cient k estimates, and recommend the use of the three-parameter VBGF instead. I then pointout an important omission in the method commonly used for estimating rmax in chondrichthyansand clarify the correct way to estimate it, followed by an exploration of the e�ect of uncertainty ininput parameters; I show that species with low annual reproductive outputs are bound to have verylow productivities, and that researchers should focus on accurately estimating litter sizes, breedingintervals, and the variability of these, in order to improve productivity estimates for sharks andtheir relatives. As an example of how to apply the novel insights gained in the previous threechapters, I then use these methods to better estimate growth and productivity for a data sparsespecies of conservation concern, the Spinetail Devil Ray (Mobula japanica), and show it has a much

97

lower somatic growth rate than previously thought and one of the lowest productivities whencompared to other chondrichthyans. Finally, I take a di�erent approach to explore life historyrelationships among chondrichthyans by testing hypotheses based on the metabolic theory ofecology (Brown et al. 2004) to show that productivity in chondrichthyans varies with temperatureas well as depth, and that the scaling of this relationship changes with temperature according tothe expectation from Bergman’s rule. I have already discussed in detail the relevance of these�ndings in the Discussion section of each chapter, so here I will instead discuss some the generalrelevance of my thesis and some broad areas of future work that I think will be the most promising.

7.2 Signi�cance

If there is a single message that I hope my thesis can portray is that applied conservation tools neednot be complicated to inform and contribute toward policy. Of course, they should always take intoaccount implicit assumptions and associated uncertainty to be scienti�cally sound, thus addressinga valid criticism of the use of life history theory tools: that they are very rough representationsextremely noisy processes, and thus are far from perfect. Perhaps this is why, in my opinion, thereis a tendency in �sheries science to develop more complex and quantitatively-intensive tools asthese are thought to be more useful or to be better at informing policy decisions. I think this isnot necessarily true, especially when working with data poor species and in data poor parts of theworld. On this subject, I borrow Jeremy Prince’s concept of a “barefoot ecologist”; in essence apragmatic researcher with a broad knowledge in multiple disciplines and the ability to adapt towork e�ectively in di�erent situations (Prince 2003). I believe that a large part of an ecologist’stoolkit should consist of life history theory-based methods.

One aspect of chondrichthyan biology that allows for these tools to be useful is that species’ranges tend to be very large and populations are often interconnected; even in cases where dis-tinct subpopulations do exist the variations in life history strategies between populations varypredictably; for example, latitudinal variation in Bonnethead (Sphyrna tiburo) life histories (Par-sons 1993a;b; Carlson and Parsons 1997; Lombardi-Carlson et al. 2003). In this thesis, I illustratehow the temperature component relates to important life history metrics such as productivity,which most likely accounts for the variation seen in life history between populations of a singlespecies. Thus, I lay the groundwork for understanding, and accounting for, regional variations inlife history traits.

The work pursued in Chapter 5 on devil rays is a clear example of how simple life historytheory-based methods can be used to e�ectively inform policy. In 2016, all devil rays (Mobula

spp.) were proposed for listing under CITES, which after accepted, would result in countries be-ing legally bound to regulate their international trade. The research presented in Chapter 5 waspublished, in an open access peer-reviewed journal, just prior to the meeting. I was watching theonline video feed of the talks, and as the discussion on devil rays began I was taken aback whenthe opening statement delivered by the proposing country, Fiji, included ad verbatim statements

98

which I wrote to summarise the publication! Furthermore, the Food and Agriculture Organiza-tion (FAO) Expert Advisory Panel, which at CITES provides advice based on the best-availableevidence, echoed the �ndings of my work showing the low productivity in devil rays and thussupported the listing of devil rays under CITES. As a side note, this was the only elasmobranchproposal that was supported by the FAO panel during this meeting. In my slightly biased opinion,this anecdote clearly demonstrates that life history theory-based tools can be extremely importantfor policy decisions that have real on the ground repercussions for the conservation and sustain-able management of imperilled species.

One interesting observation based on the di�erent approaches to �tting growth curves inChapters 2 and 5 is that Bayesian growth models may provide a better and more �exible alternativefor estimating growth in data-poor species. Bayesian priors allow the curve-�tting algorithm to�uctuate in a way that �xing model parameters does not, so incorporating empirical data throughthe use of Bayesian priors can help constrain growth parameters while reducing bias that resultswhen those same estimates are included though the mathematical �xing of growth parameters (asis the case when �xing the size-at-age zero L0 in the two-parameter von Bertalan�y growth func-tion). These methods could be easily compared using the same framework I developed in Chapter2, and would further help elucidate which growth models we should be using for ageing sharksand rays.

One of the important implications of Chapter 6 is that it illustrates the feasibility of usingcoarse global spatial datasets for extracting important life history information (in this case, meanannual temperature at depth) that can be used to test novel hypotheses related to variation inlife history traits. Even when this information extracted is fairly “rough around the edges”, itstill is informative as it considerably improved model �t and thus supported one of the speci�chypotheses being tested.

7.3 Future directions

While my thesis work opens many potential avenues of further research, there are two areas whichI believe would be the most fruitful: estimating the productivity and extinction risk of data-poorspecies using the best available information that are of special conservation concern, and gainingfurther insights into the role of temperature in shaping chondrichthyan life histories.

In Chapter 5 I have exempli�ed an approach for assessing the relative extinction risk of datapoor species using basic life history information. This type of analyses can be adapted for usein most chondrichthyan species, particularly now that chondrichthyan phylogenies are readilyavailable (Stein et al. 2017). I foresee these analyses being used to inform debate about the fateof particular species when these discussions arise in the future. A relevant chondrichthyan ex-ample would be the freshwater stingrays of South America (family Potamotrygonidae), which areincreasingly being �shed to supply the live aquarium trade and been proposed for listing underCITES Appendix II twice in the last half decade; these proposals have been rejected or withdrawn

99

because of lack of adequate data to inform this listing. Providing the best-available scienti�c in-formation on their growth and productivity prior to the next meeting (if proposed again) wouldcontribute substantially to the discussion.

I have shown in Chapter 6 that species-speci�c temperature can be extracted from coarseoceanographic datasets using simple spatial analyses, and that it is useful in describing variationof key life history traits, such as productivity. This chapter lays the groundwork for creating mapsof chondrichthyan productivity across the world’s oceans: if the location, depth, and maximumsize of a species is known we could have a rough idea of its productivity. To my knowledge, this isthe �rst time temperature-at-depth data has been included in comparative chondrichthyan stud-ies, opening the door for further exploring how environmental variables relate with life historiesusing broad-scale oceanographic dataset. Aside of temperature, future studies can use other globaloceanographic variables that likely relate to energy availability, for example primary productivityor biomass estimates, to explore further axes of variation in the allometry of productivity.

7.4 Concluding thoughts

“Science can help ensure that decisions are made with thebest available information, but ultimately the future ofbiodiversity will be determined by society.”

— Millennium Ecosystem Assessment, 2005

We know that the degradation of our planet’s ecosystems is ultimately driven by the way wehave organized globally as a society. As researchers working in academia, we are often part ofthe privileged few who bene�t from this societal arrangement. Even when trying to reduce ourecological footprint we still consume a disproportionate share of the world’s resources: we thusare unwilling contributors and accomplices of environmental degradation. While research shouldalways strive to be as objective and unbiased as possible, I believe that as academics we can, andshould, aim at least part of our intellectual pursuits to answering the speci�c questions being askedby decision-makers about the environmental issues we face. These questions will be constantlychanging, and we will have to be strategic to aim our scienti�c arrows at them. These questions,particularly regarding species outside developed nations, will rarely require complex, data hungryquantitative methods to be answered. Rather, the intersection of clear scienti�c analyses (oftensimple ones) based on the best available data, political will driven by public pressure, global-scalereduction in consumption, and the provision of well though-of and truly sustainable alternativesfor those whose livelihoods are a�ected by the implementation of responsible management mea-sures, will be the only way to e�ect meaningful, lasting societal change. It is my sincere hope thatthis thesis contributes a small piece toward solving this crucial puzzle.

100

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