Spatial modeling of predator-assisted dispersal

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

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

3. 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

4. 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

5. 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

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

7. 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

8. 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

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

10. A Brief Overview of the Model • Use spatially explicit mathematical model • Program simulations in Matlab • Simplify model to validate simulation and examine underlying mechanisms

11. Spatial Model • Modeled as a rectangular grid • Prey are dispersed locally

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

13. Predator-Assisted Dispersal • Prey have a chance to be carried by predators foraging in their patch • Predators deposit prey in a random patch

14. 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?

15. 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

16. 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

17. 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

18. Investigating Question 2: Spatial Disturbance • Vary size and distribution of disturbance • Measure recovery time and prey densities after recovery

19. 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

20. Patchiness

21. 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

22. The Model

23. The Model: Mortality

24. Dispersal • Prey undergo local dispersal with reflective boundary Comins & Hassell 1996

25. SIR Model

26. SIR Model

27. 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

28. The Model: Mortality

29. Two competing species in absence of a predator

30. Predator preference, no assisted dispersal • Allows us to measure only the negative effect of preference • Possible outcomes • Exclusion due to preference • Decreased final density

31. Predator preference, no assisted dispersal

32. 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

33. Predator-assisted dispersal of a single prey species

34. Complete Model: Predator-assisted dispersal of two prey

35. Complete Model: Predator-assisted dispersal of two prey

36. 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

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

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

39. 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.

40. 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.