succession, invasion, & coexistence: pdes in ecology ?· succession, invasion, & coexistence: pdes...

Download Succession, Invasion, & Coexistence: PDEs in Ecology ?· Succession, Invasion, & Coexistence: PDEs in…

Post on 04-Jan-2019

212 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

Succession, Invasion, & Coexistence: PDEs inEcology

Simon Maccracken Stump

Jon Jacobsen, Advisor

Alfonso Castro, Reader

May, 2006

Department of Mathematics

Copyright c 2006 Simon Maccracken Stump.

The author grants Harvey Mudd College the nonexclusive right to make this work availablefor noncommercial, educational purposes, provided that this copyright statement appearson the reproduced materials and notice is given that the copying is by permission of theauthor. To disseminate otherwise or to republish requires written permission from theauthor.

Abstract

We study the behavior of diffusive Lotka-Volterra systems in environments withspatially varying carrying capacities. In particular, we use numeric and analytictechniques to study two similar models for population growth, in order to deter-mine their qualitative differences. Additionally, we investigate competition modelsin the presence of periodic disasters, in order to determine what factors affect com-petitive dominance. We found that under conditions of high spatial heterogeneity,the model for population growth was the main factor determining coexistence. Un-der low spatial heterogeneity, the effect of disturbance on the stronger competitorwas the main factor determining coexistence.

Contents

Abstract iii

Acknowledgments xi

1 Biological Significance 11.1 Succession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Coexistence and Diversity . . . . . . . . . . . . . . . . . . . . . 31.3 Invasive Species . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Ecological Modeling 72.1 Gap Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Markov Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . 82.4 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 112.5 Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Single Species Growth Models 193.1 Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 The Successional Competition Model 294.1 Model Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Conclusions and Future Work 39

Bibliography 41

List of Figures

1.1 The Intermediate Disturbance Hypothesis . . . . . . . . . . . . . 41.2 Invasive Species- Kudzu . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Equilibrium of the Lotka-Volterra Competition Model . . . . . . . 112.2 A Traveling Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Heteroclinic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4 A Model for a Periodic Environment . . . . . . . . . . . . . . . . 16

3.1 Different Environments Used for Testing . . . . . . . . . . . . . . 253.2 Different Heterogeneity Levels Used for Testing . . . . . . . . . . 263.3 Population Response to Heterogeneity . . . . . . . . . . . . . . . 273.4 Population Response to Changes in r/d . . . . . . . . . . . . . . 27

4.1 Monotonicity Problem #1 . . . . . . . . . . . . . . . . . . . . . . 314.2 Monotonicity Problem #2 . . . . . . . . . . . . . . . . . . . . . . 324.3 Monotonicity Problem #3 . . . . . . . . . . . . . . . . . . . . . . 334.4 Differences Between Equations (3.1) and (3.2)- Low Heterogeneity 344.5 Differences Between Equations (3.1) and (3.2)- High Heterogeneity 354.6 Effects of Dispersal Ability on Coexistence . . . . . . . . . . . . 364.7 Effects of Population Growth Rates on Coexistence . . . . . . . . 364.8 Effects of High Disturbance Regimes on Slow Populations . . . . 37

List of Tables

1.1 Species Interaction During Succession . . . . . . . . . . . . . . . 3

Acknowledgments

I would like to thank Professor Adolph for inspiring me to study succession. Iwould like to thank Jeff Hellrung for creating a numerical simulator that was ofinvaluable help early on. I would like to thank Andrea Heald for helping edit thispaper. I would like to thank Professor Yong for help with numerical simulations,and Professor Fowler for help with the biology. Thanks to Professor Castro foragreeing to be my second reader. And most importantly I would like to thankProfessor Jon Jacobsen, for inspiration, guidance, and moral support.

Chapter 1

Biological Significance

This thesis concerns the mathematical modeling of species interaction and succes-sion. The first chapter is a short review of the biological processes that will bemodeled throughout the remainder of the thesis, and is written so as to be accessi-ble to any reader. The second chapter examines a variety of models that are used tomodel species interaction, succession, and the spread of invasive species. The thirdchapter considers two commonly used equations that model species interactions,and analyze the differences between them. The fourth chapter looks at a simulationfor competing species in a temporally varying environment. The final chapter is alist of potential future research projects, growing out of this work.

1.1 Succession

Succession is the process of ecological community change. During succession,species are displaced by better competitors. These species are then either displacedby even stronger competitors, or grow to dominate a particular ecosystem. Succes-sion is mainly studied in communities of sessile (non-mobile) organisms, namelyplant communities and inter-tidal communities.

There are two basic types of succession: primary and secondary [Ricklefs andMiller, 2000]. During primary succession, all traces of previous life are lost, orwere not there to begin with. The classic example of primary succession is lifeforming on volcanic pumice, as occurred on Mt. Saint Helens [del Moral and Jones,2002]. Primary succession is often characterized as being more stochastic, as aresult of species needing to migrate in from other areas. During secondary succes-sion, traces of former life remain, such as in the seed bank. Classic examples ofsecondary succession include forest fires, clear-cutting, and other natural disasters[Ricklefs and Miller, 2000]. Any event which causes a local extinction is known

2 Biological Significance

as a disturbance event.There is a general pattern to the evolution of ecological communities during

succession. Shortly after a disaster has occurred, the landscape is colonized bypioneer species [Ricklefs and Miller, 2000]. The classic example of a pioneerspecies in terrestrial habitats is grass. Pioneer species tend to thrive under highlight conditions, and in heterogeneous landscapes. They spend most of their energygrowing and reproducing rapidly (usually at least one reproductive bout per year),and as such spend very little energy on permanent structures. Because of this, theyare eventually invaded and out-competed by species which do not reproduce ordiffuse quickly, but put more energy into being better competitors. This processrepeats itself until the landscape is in its climax state. A climax state is one whichis approximately stable, as long as the environment is stable. Climax species tendto be those that are shade tolerant, live a long time, and do not reproduce until manyyears into their life. Because the best competitors are the slowest plants to enter anew area, it suggests that there is a trade-off between competitive ability and fastdispersal. This is likely due to the fact that forming permanent woody structuresand growing to a great enough size to be a good light competitor takes time andenergy, and could instead be used producing a huge number of seeds.

Currently there are three models for competitive interaction during succes-sion [Ricklefs and Miller, 2000] [Sanchez-Velasquez, 2003]. The first is calledthe Facilitation Model. Under the Facilitation Model, organisms in each succes-sional stage allow for the introduction of species in later stages. One example ofthis would be nitrogen-fixing plants, without whom trees would be unable to gettheir necessary amount of nutrients. The second model is known as the InhibitionModel. Under the inhibition model, when a species begins inhabiting an area, itprevents later species from invading, through methods such as allelopathy (the useof chemicals to alter soil, making it more difficult for other plants to grow there).A famous example of this was discovered by Sousa [Sousa, 1979], who studiedsuccession on coastal boulders, and found that a particular algae, when it becameattached to a boulder, was able to prevent any other algae from becoming attached.Often, it was not until the algae was damaged by predators that other species couldgrow on a boulder (for more details, see [Ricklefs and Miller, 2000]). The finalmodel is known as the Tolerance Model. Under the Tolerance Model, timing ofcolonization does not affect an organisms ability to colonize (i.e., if species A col-onizes an area before species B, it has the same end result as if species B colonizesan area before species A). Unless otherwise noted, species in my thesis will interactunder the Tolerance Model. See Table 1.1 for a diagram of each model.

Coexistence and Diversity 3

Table 1.1: Three classic methods for succession. Under the Facilitation Model,early successional plants make it possible for later plants to colonize an area. Underthe Inhibition Model, early successional plants prevent later plants from colonizing.Under the Tolerance Model, the timing of colonization does not affect the overallr