Spatial modeling of predatorassisted dispersal

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Spatial modeling of predator-assisted dispersal

• Carl Leth
• Tanner Hill
• Nichole Zimmerman
• Colorado State University
• FEScUE Program, Summer 2008

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Lines of Logic

• Spatial dispersal of prey species
• Predator preference
• We propose to couple these two ideas through
  predator-assisted dispersal

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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

• Comins and Hassell 1996
• Savill and Hogeweg 1999

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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

• Bonsall and Hassell 1999 3. McCann et al. 1998
• Abrams and Matsuda 2004

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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

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Example of Predator-Assisted Dispersal
Dromph looked at collembolans dispersing
entomopathogenic fungi
http//en.wikipedia.org/wiki/ImageIsotoma_Habitus
.jpg
Dromph 2001
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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

• Janos and Sahley 1995
• Johnson 1995

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Empirical Studies Fungi Dispersal Aided by their
Predators

• Mammals were observed to disperse spores of
  ectomycorrhizal fungi1
• Grasshoppers and small mammals transported fungal
  spores2

• Cázares and Trappe 1994
• Warner, Allen, and MacMahon 1987

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Our Proposal

• We will model predator-assisted dispersal of a
  two prey system with predator preference
• Preliminary results
• Intended studies

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A Brief Overview of the Model

• Use spatially explicit mathematical model
• Program simulations in Matlab
• Simplify model to validate simulation and examine
  underlying mechanisms

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Spatial Model

• Modeled as a rectangular grid
• Prey are dispersed locally

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Spatial Model

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

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Predator-Assisted Dispersal

• Prey have a chance to be carried by predators
  foraging in their patch
• Predators deposit prey in a random patch

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Questions

• Given predator-assisted dispersal, how does
  predator preference affect the final densities of
  the prey species?
• How does predator-assisted dispersal affect the
  resistance of static prey densities in the face
  of a spatial disturbance?
• How does predator-assisted dispersal affect the
  resilience of the system in the face of
  prey-specific infection?

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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

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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

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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

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Investigating Question 2 Spatial Disturbance

• Vary size and distribution of disturbance
• Measure recovery time and prey densities after
  recovery

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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

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Patchiness
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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

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The Model
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The Model Mortality
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Dispersal

• Prey undergo local dispersal with reflective
  boundary

Comins Hassell 1996
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SIR Model
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SIR Model
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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

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The Model Mortality
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Two competing species in absence of a predator
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Predator preference, no assisted dispersal

• Allows us to measure only the negative effect of
  preference
• Possible outcomes
• Exclusion due to preference
• Decreased final density

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Predator preference, no assisted dispersal
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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

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Predator-assisted dispersal of a single prey
species
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Complete Model Predator-assisted dispersal of
two prey
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Complete Model Predator-assisted dispersal of
two prey
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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

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Future Work

• Other Models
• Poisson process
• Alternate equations
• Discrete time models
• Empirical Studies
• Preference studies
• Collembolla and fungus

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Acknowledgement s

• FEScUE and NSF
• Michael Antolin, Dan Cooley, Don Estep, Sheldon
  Lee, Stephanie McMahonn, John Moore, Simon
  Tavener, Colleen Webb

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References

• Abrams, P.A., Hiroyuki Matsuda. 2004.
  Consequences of behavioral dynamics for the
  population dynamics of predator-prey systems with
  switching. Popul Ecol 4613-25.
• Bonsall, Michael B. Michael P. Hassell. 1999.
  Parasitiod-mediated effects apparent competition
  and the persistence of host-parasitiod
  assemblages. Res Popul Ecol 4159-68.
• Cázares, Efrén, James M. Trappe. 1994. Spore
  dispersal of ectomycorrhizal fungi on a glacier
  forefront by mammal mycophagy. Mycologia
  86507-510.
• Comins, H.N., M.P. Hassell. 1996. Persisence of
  Multispecies Host-Parasitoid Interactions in
  Spatially Distributed Models with Local
  Dispersal. J. theor. Biol. 18319-28.
• Dromph, Karsten M., 2001. Dispersal of
  entomopathogenic fungi by collembolans. Soil
  Biology Biochemistry 332047-2051.

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References Continued

• Janos, David P., Catherine T. Sahley. 1995.
  Rodent Dispersal of Vesicular-Arbuscular
  Mycorrhizal Fungi in Amasonian Peru. Ecology
  761852-1858.
• Johnson, C.N., 1995. Interactions between fire,
  mycophagous mammals, and dispersal of
  ectromycorrhizal fungi in Eucalyptus forests.
  Oecologia 104467-475.
• Krause, A. E., K. A. Frank, D. M. Mason, R. E.
  Ulanowicz, W. W. Taylor. 2003. Compartments
  revealed in food-web structure. Nature
  426282-285.
• 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 79721-730.