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Spatial modeling of predatorassisted dispersal
Carl Leth
Tanner Hill
Nichole Zimmerman
Colorado State University
FEScUE Program, Summer 2008
Lines of Logic
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Spatial dispersal of prey species
Predator preference
We propose to couple these two ideas
through predator-assisted dispersal
Results from Dispersal Studies
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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
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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
2. Abrams and Matsuda 2004
3. McCann et al. 1998
Predator-Assisted Dispersal
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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
Dromph 2001
http://en.wikipedia.org/wiki/Image:Isotoma_Habitus.jpg
Empirical Studies: Fungi Dispersal
Aided by their Predators
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Rodents were found likely to be
important in the dispersal of vesiculararbuscular 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
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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
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We will model predator-assisted
dispersal of a two prey system with
predator preference
Preliminary results
Intended studies
A Brief Overview of the Model
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Use spatially explicit mathematical
model
Program simulations in Matlab
Simplify model to validate simulation
and examine underlying mechanisms
Spatial Model
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Modeled as a rectangular grid
Prey are dispersed locally
Spatial Model
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Predators have very high mobility
relative to prey, can feed from any
patch at any time
Predator-Assisted Dispersal
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Prey have a chance to be carried by
predators foraging in their patch
Predators deposit prey in a random
patch
Questions
1.
2.
3.
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?
Question 1 Hypotheses
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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
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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
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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
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Vary size and distribution of disturbance
Measure recovery time and prey
densities after recovery
Question 3 Hypotheses
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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
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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
dX 1
1 X 1   1 X 1 X 2 c1 PX 1
 1 ( X 1 
)
dt
K1
1  X1
2
dX 2
 2 X 2   2 X 1 X 2 c2 PX 2
 2 ( X 2 
)
dt
K2
1 X 2
2
dP a1c1 PX 1 a2 c2 PX 2


 P
dt
1  X1
1 X 2
The Model: Mortality
Dispersal
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Prey undergo local dispersal with reflective
boundary
X i , p  (1  i ) X i , p  i  X i ,q / 8
'
q
Comins & Hassell 1996
SIR Model
dS dX

 iIS  rR
dt
dt
dI
 iIS  I  mI
dt
dR
 I  rR
dt
SIR Model
Simplifications of the Model
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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
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Allows us to measure only the negative
effect of preference
Possible outcomes
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Exclusion due to preference
Decreased final density
Predator preference, no
assisted dispersal
Predator-assisted dispersal of
a single prey species
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Allows us to examine the simplest case
of predator-assisted dispersal
Possible outcomes
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Similar outcomes to single predator-prey
simplification
Increases the speed of colonization
Predator-assisted dispersal of
a single prey species
Complete Model: Predatorassisted dispersal of two prey
Complete Model: Predatorassisted dispersal of two prey
Summary
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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
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Other Models
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Poisson process
Alternate equations
Discrete time models
Empirical Studies
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Preference studies
Collembolla and fungus
Acknowledgement s
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FEScUE and NSF
Michael Antolin, Dan Cooley, Don Estep,
Sheldon Lee, Stephanie McMahonn,
John Moore, Simon Tavener, Colleen
Webb
References
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Abrams, P.A., Hiroyuki Matsuda. 2004. Consequences of
behavioral dynamics for the population dynamics of predatorprey systems with switching. Popul Ecol 46:13-25.
Bonsall, Michael B. Michael P. Hassell. 1999. Parasitiod-mediated
effects: apparent competition and the persistence of hostparasitiod 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…
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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 426:282285.
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.