Spatial modeling of predator- assisted dispersal Carl Leth Tanner Hill Nichole Zimmerman Colorado State University FEScUE Program, Summer 2008.

Spatial modeling of predator-assisted dispersal

Carl LethTanner Hill

Nichole ZimmermanColorado State University

FEScUE Program, Summer 2008

Lines of Logic

Spatial dispersal of prey species Predator preference We propose to couple these two

ideas through predator-assisted dispersal

Results from Dispersal Studies

Local dispersal has been found to promote the persistence of interacting populations1

Wave-like patterns can occur by dispersing predators and prey2

1. Comins and Hassell 1996

2. Savill and Hogeweg 1999

Results from Preference Studies

Predator preference with switching has been found to promote stability and persistence in some cases1

Preference switching lags behind the optimum for changing prey densities2

Variable interaction strengths can help stabilize a system3

1. Bonsall and Hassell 1999 3. McCann et al. 1998

2. Abrams and Matsuda 2004

Predator-Assisted Dispersal

Combines dispersal and predator preference

Predators may carry their prey to different spatial locations and deposit them there

Empirical studies show that this occurs in nature

Example of Predator-Assisted Dispersal

Dromph looked at collembolans dispersing entomopathogenic fungi

http://en.wikipedia.org/wiki/Image:Isotoma_Habitus.jpg

Dromph 2001

Empirical Studies: Fungi Dispersal Aided by their Predators

Rodents were found likely to be important in the dispersal of vesicular-arbuscular mycorrhizal (VAM) fungus spores1

Australian mammals feeding on hypogeous fungi increased spore dispersal2

1. Janos and Sahley 1995

2. Johnson 1995

Empirical Studies: Fungi Dispersal Aided by their Predators

Mammals were observed to disperse spores of ectomycorrhizal fungi1

Grasshoppers and small mammals transported fungal spores2

1. Cázares and Trappe 1994

2. Warner, Allen, and MacMahon 1987

Our Proposal

We will model predator-assisted dispersal of a two prey system with predator preference

Preliminary results Intended studies

A Brief Overview of the Model

Use spatially explicit mathematical model

Program simulations in Matlab Simplify model to validate

simulation and examine underlying mechanisms

Spatial Model Modeled as a rectangular grid Prey are dispersed locally

Spatial Model

Predators have very high mobility relative to prey, can feed from any patch at any time

Predator-Assisted Dispersal

Prey have a chance to be carried by predators foraging in their patch

Predators deposit prey in a random patch

Questions

1. Given predator-assisted dispersal, how does predator preference affect the final densities of the prey species?

2. How does predator-assisted dispersal affect the resistance of static prey densities in the face of a spatial disturbance?

3. How does predator-assisted dispersal affect the resilience of the system in the face of prey-specific infection?

Question 1 Hypotheses

Given predator-assisted dispersal, how does predator preference affect the final densities of the prey species?

High preference decreases fitness due to increased consumption

High preference increases fitness due to increased dispersal

There is an optimal degree of preference for fitness that balances mortality due to consumption with dispersal

Investigating Question 1: Benefits of Preference

Give predators a constant predation rate between the two species

Vary degree of preference for one species

Measure changes in final densities

Question 2 Hypotheses

How does predator-assisted dispersal affect the resistance of static prey densities in the face of a spatial disturbance?

There is no effect Densities are more resistant to change

than in control cases Densities are less resistant to change

than in control cases

Investigating Question 2: Spatial Disturbance

Vary size and distribution of disturbance Measure recovery time and prey

densities after recovery

Question 3 Hypotheses

How does predator-assisted dispersal affect the resilience of the system in the face of prey-specific infection?

No effect Resilience is decreased because the

predators carry infected individuals Resilience is increased because it

causes patchiness

Patchiness

Investigating Question 3: Infection

Allow prey to fully colonize habitat Introduce a species-specific

infection using an SIR model Measure resilience by how virulent

the infection must be to cause extinction of a species

The Model

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The Model: Mortality

Dispersal Prey undergo local dispersal with

reflective boundary

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,

Comins & Hassell 1996

SIR Model

rRIdt

dR

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Simplifications of the Model Two competing species in absence of a

predator One species in presence of a predator Two competing species in presence of a

predator Predator preference, no assisted

dispersal Predator-assisted dispersal of a single

prey species

The Model: Mortality

Two competing species in absence of a predator

Predator preference, no assisted dispersal

Allows us to measure only the negative effect of preference

Possible outcomes Exclusion due to preference Decreased final density

Predator preference, no assisted dispersal

Predator-assisted dispersal of a single prey species

Allows us to examine the simplest case of predator-assisted dispersal

Possible outcomes Similar outcomes to single predator-

prey simplification Increases the speed of colonization

Predator-assisted dispersal of a single prey species

Complete Model: Predator-assisted dispersal of two prey

Complete Model: Predator-assisted dispersal of two prey

Summary

Predator-assisted dispersal combines independent dispersal models with predator preference

There is a gap in knowledge at the intersection of these two ideas

We propose a mathematical model which investigates these dynamics

Future Work Other Models

Poisson process Alternate equations Discrete time models

Empirical Studies Preference studies Collembolla and fungus

Acknowledgement s

FEScUE and NSF Michael Antolin, Dan Cooley, Don

Estep, Sheldon Lee, Stephanie McMahonn, John Moore, Simon Tavener, Colleen Webb

References Abrams, P.A., Hiroyuki Matsuda. 2004. Consequences of

behavioral dynamics for the population dynamics of predator-prey systems with switching. Popul Ecol 46:13-25.

Bonsall, Michael B. Michael P. Hassell. 1999. Parasitiod-mediated effects: apparent competition and the persistence of host-parasitiod assemblages. Res Popul Ecol 41:59-68.

Cázares, Efrén, James M. Trappe. 1994. Spore dispersal of ectomycorrhizal fungi on a glacier forefront by mammal mycophagy. Mycologia 86:507-510.

Comins, H.N., M.P. Hassell. 1996. Persisence of Multispecies Host-Parasitoid Interactions in Spatially Distributed Models with Local Dispersal. J. theor. Biol. 183:19-28.

Dromph, Karsten M., 2001. Dispersal of entomopathogenic fungi by collembolans. Soil Biology & Biochemistry 33:2047-2051.

References Continued… Janos, David P., Catherine T. Sahley. 1995. Rodent Dispersal of

Vesicular-Arbuscular Mycorrhizal Fungi in Amasonian Peru. Ecology 76:1852-1858.

Johnson, C.N., 1995. Interactions between fire, mycophagous mammals, and dispersal of ectromycorrhizal fungi in Eucalyptus forests. Oecologia 104:467-475.

Krause, A. E., K. A. Frank, D. M. Mason, R. E. Ulanowicz, W. W. Taylor. 2003. Compartments revealed in food-web structure. Nature 426:282-285.

McCann, Kevin, Alan Hastings, Gary R. Huxel. 1998. Weak trophic interactions and the balance of nature. Nature 395: 794-797.

Savill, Nicholas J., Paulien Hogeweg. 1999. Competition and Dispersal in Predator-Prey Waves. Theoretical Population Biology 56: 243-263.

Waren, Nancy J., Michael F. Allen, James A. MacMahon. 1987. Dispersal Agents of Vesicular-Arbuscular Mycorrhizal Fungi in a disturbed Arid Ecosystem. Mycologia 79:721-730.