perennial plant models to study species coexistence in a variable environment

Perennial Plant Models to study Species Coexistence in a Variable Environment

Chi Yuan

Species diversity

Coexistence in variable environment

• Species coexistence – Using environment

differently– Puzzling, plants share

similar resources• Variable environment

– Difference in efficiency in using the resource

• Which life history processes vary with environment?

Year 2Year 1

• Which life history processes vary with environment? – Species-specific responses in recruitment

Chesson et al. 2013

• Which life history processes vary with environment? – Species-specific responses in individual growth

Enquist and Leffler, 2001, Long-term tree ring chronologies from sympatric tropical dry-forest trees: individualistic responses to climatic variation

Outline of the research

•Recruitment variation•Contrasting life histories Lottery model

• Variation in individual growth• Demographic schedules

Size-structured

lottery model

• Tradeoff in recruitment and growth• Opposite size-dependency in vital

rates

Life history tradeoff

Non-structure lottery model: recruitment variation only

Reproduction as Environmental Response

Establishment as Competitive Response Survival

A definition of Lottery competition

Picture credit: http://de.sap.info/wp-content/uploads/2013/02/SME_Growing_Plant_iStock.jpghttp://bestclipartblog.com/25-tree-clip-art.html/tree-clip-art-1

http://de.sap.info/wp-content/uploads/2013/02/SME_Growing_Plant_iStock.jpg

• Stability of coexistence – Invasibility analysis

• Resident• Invader

– Stabilizing effect– Equalizing effect

A I N

Residents

invader

• Storage effect– Covariance – Buffer

• Survival

Better environment

Stro

nger

com

petiti

on

Relative nonlinearity • Relative nonlinear growth rates in responses

to competitionLonger-lived species is favored by larger fluctuation in competition

Shorter-lived species is favored by larger fluctuation in competition

Life history characters affecting both mechanisms

• Difference in death rate– Fecundity-mortality

tradeoff • Difference in sensitivity

to environment• Correlation in

environmental responses 0 10 20 30 40 50 60 70 80 90 100

-8

-6

-4

-2

0

2

timeenvi

ronm

enta

l res

pons

es

species 1species 2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.4

0.45

0.5

0.55

0.6

0.65

time

popu

latio

n si

ze

species 1species 2

0 10 20 30 40 50 60 70 80 90 100-8

-6

-4

-2

0

timeenvi

ronm

enta

l res

pons

es

species 1species 2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0.35

0.4

0.45

0.5

0.55

0.6

0.65

time

popu

latio

n si

ze

species 1species 2

Equal sensitivity, zero correlation

Different sensitivity, full correlation

Sensitivity differences between species

Wright 2005

Storage effect and relative nonlinearity

No correlation between speciesHigh correlation between species

Part 1, recruitment variation: summary

• Relative nonlinearity is more restrictive compared with storage effect

• Relative nonlinearity compensates the weakening storage effect

• Differences in death rate have a big affect on coexistence mechanisms only when aligned with sensitivity differences




Size-structured

lottery model


rates


size-structured lottery model

• Introducing the continuous size structure – Explicit post-

recruitment dynamics

– Size dependency in demographic rates

• Difference from other forest models

Age or size

Fecu

ndity

Age or size

Mor

talit

y

Lottery competition in structured model

( )( ) ln( )R tC tA t

( ) ( )

,

( ) { ( ) ( ) [ ( ) ] ( ) }gi gi bjE E t E tj jc j jc js j js js j jc jc

c j

R t e c a s a a c a e c k a e N

( ),

( ) (1 ( ))jc j jc tc j

A t N s a

Resource needed

Resource supply

Cohort Based models:How environment and competition affect the

critical life history process

• Eb environmental response in recruitment

• Eg environmental response in growth.

( )

( )

( )seedling growth

seedling establishment ( )

( )

bj

gi

E t

js j jcc

E tjs j js

C teec k a

a c a e

Year 1

Year 2

Cohort based model:How environment and competition affect the critical

life history process

• Tree growth for cohort c

( )

( )( ) ( ( ))gjE t

jc j jc C t

ea t c a te

death

growth

( )1 ( )j jc ts a

( )( )j jc ts a

( )jca tSize at time t

Size at time t+1

Model overview

Seedling establishmentTree growth

)( ()( 1) bj

jc

E C tjn js a jc

t

c

k eN t c N

( )( 1) gj

js

Ejn js a

tca et a

Seedling Growth

( ) ( )( )( 1) ( ) gj

jc

Ejc jc a

t C ttca a tt e

Two species• Species are identical on averages, except their

responses to the environment

Variation in individual growth can also promote species coexistence

the

stab

ilizin

g ef

fect

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fecundity increases slower with size

Fecundity increases faster with size

• Recruitment variation Vs. variation in individual growth

• Storage effect as the major contributor • Two form of

storage effect

Interaction between reproduction and individual growth

– Variation in reproduction and growth is not additive

• Synergism when growth and reproduction is positively related

• Antagonisms with negative correlation

A

∆ 𝐼

Interaction between reproduction and individual growth

A

∆ 𝐼

Two form of storage effect

• Mean fractional contribution to population growth

𝑝𝑏+𝑝𝑏𝑔+𝑝𝑔+𝑝𝑠=1

p g p g

pbg

pbg

pb pb

increasing increasing

How life history affects the storage effects

• An example increasing pb and Pbg by increasing seedling size

A

∆ 𝐼

A

∆ 𝐼

Variation in reproduction only Variation in growth only

The effect of shapes in demographic schedules

• Changes in overall stabilizing effect

• Storage effect is not sensitive to shapes of the schedules, given p’s are fixed

a. b.

A

∆ 𝐼 A

∆ 𝐼

Shift in size structure • Variation in growth:

more larger individuals in invader state than resident state

• Variation in reproduction: more smaller individuals in invader state than resident state

(b) Eb only

(a) Eg only

Invader state

Resident state

Resident state

Invader state

Effect of shift in size structure• ΔS Mean

structure effect under equilibrium environment

• ΔE Changes in mean environment effect due to shift in structure

(a) Eg only

Invader state

Resident state

Being smaller more advantageous

Being larger more advantageous

Effect of shift in size structure • ΔS Mean

structure effect under equilibrium environment

• ΔE Changes in mean environment effect due to shift in structure

(b) Eb only

Resident state

Invader state

Being smaller more advantageous

Being larger more advantegous

Effects of shapes in demographic schedule through shifts in structure

ref flat f flatm flatc

mec

hani

sm p

artit

ion

0.0

0.2

0.4

0.6

E Cs N

ref flat f flatm flatcm

echa

nism

par

titio

n

-0.5

0.0

0.5

E Cs NEb only Eg only

ref

ref

ref

Flat f

Flat m

Flat c

Part two: summary• General theory is compatible with studies of interesting

biological details• Variation in individual growth promote coexistence• Storage effect as the main stabilizing mechanisms

– Only the relative contributions of key processes to population growth in a population matters for storage effect

• Storage effect is strong when processes most sensitive to environment also contribute most strongly on average to population growth

• The effect of size-dependency in life history is determines by shift in structure




Size-structured

lottery model


rates


Part 3 Life history tradeoff

• Difference in life history strategy between species– Formulated as

tradeoff• Tradeoff and species

coexistence– Equalizing effect– Stabilizing? Wright et al. 2010

Jakobsson and Eriksson, 2000

Case 1

• Tradeoff between fecundity and growth– Species 1 with mean advantage in reproduction (solid)– Species 2 with mean advantage in growth (dash)

sp1

sp1sp2

sp2Identicalaverage average

In constant environment

• Equalizing effect of tradeoff in mean demographic properties– No stabilizing effect

alone

Difference in mean environmental responses

sp1sp2

Sp1 winsSp2 wins

Fluctuation dependent mechanisms

• Stable coexistence– ΔS

• Mean structure effect – ΔE

• Mean environment effect– ΔI

• Covariance between environment and competition

• Buffer

• Stabilizing effect • Fitness inequality

i i i iS ES

IA E I

i i

i i

i i

S S S

E E E

I I I

In variable environment• Variation in

reproduction– Species 1 with mean

advantage in reproduction – Species 2 with mean

advantage in growth

• Equalizing effect of the tradeoffs– Compensating

between dE and dI• Small effect of shift in

structure

Strongly asym

Sym

δI1b

δI2b

δE1b

δE2b

δS1b

δS2b

In variable environment

• Variation in reproduction

• Stabilizing effect– Storage effect as

the main stabilizing mechanism

Strongly asym

Sym

∆ 𝐼

∆E ∆S

Alignment between sensitivity and tradeoff

• Species with mean advantage in fecundity (sp1) has fecundity more sensitive to environment, species with mean advantage in individual growth (sp2) has growth more sensitive to environment

Case 1

• Tradeoff in population average properties • No significant effect of shift in structure

sp1

sp1sp2

sp2Identicalaverage average

Case 2

• Ontogenetic tradeoff– An extreme case where shift in structure have bigger effect– two species have contrasting shapes of demographic schedules

sp1 sp1 sp1

sp2 sp2sp2

Asymmetry in sensitivity and shapes

• Sp1: being small has more demographic advantage

• Sp2: being large has more demographic advantage

• Sp1 has only reproduction varies, sp2 has only growth varies sp1 sp2 community

average

mec

hani

sm p

artit

ion

0.0

0.2

0.4

0.6

S E Cs N I

Part 3 Summary

• Tradeoff in demographic traits alone only have equalizing effect

• Tradeoff interact with equalizing effect of the fluctuating dependent mechanisms

• Importance in asymmetry in sensitivity associated with asymmetry in mean life history traits.

• Contrast in population average properties, and shapes of demographic schedules

Implication

• Quantification methods apply in general• Multiple coexistence mechanisms interacting

– Some assumptions holds more easily, others more restrictive

• Life history traits are good predictors of the strength of the mechanisms

• Variation in recruitment and variation in growth• Tradeoffs and sensitivity difference in

environmental responses

Acknowledgment Advisor: Peter Chesson

Committee:Judie BronsteinMike RosenzweigLarry Venable

Jim Cushing Brian McGill

The Lab: Galen Holt, Yue (Max) Li, Pacifica Sommers, Simon Stump, Nick Kortessis, Jessica Kuang, Danielle Ignace, Lina Li, Andrea Mathias, Stephanie Hart, Krista Robinson,Elieza Tang

EEBer:Guan-Zhu Han,Jin Wu, Ginny Fizpatrick, Sara Felker, Jonathan Horst, Lindsey Sloat,Will Driscoll, Xingyue Ge

Liz Oxford, Lili Schwartz, Carole RosenzweigBarry McCabe, Sky Dominguez, Lauren Harrison, Pennie Liebig

FriendsDing Ding,Muhua Wang,Muhan Zhou,Rick and Linda HansonFamily Ying Yu and Jianzhong YuanLi Fan

Funding sourceScience Foundation Arizona NSF research assistantshipEEB departmentGPSCInstitute of EnvironmentHE Carter travel award