How predation can slow, stop or reverse a prey invasion

  • M. R. Owen
  • M. A. Lewis


Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, 238–251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predator-prey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.


Transition Layer Prey Density Logistic Growth Predator Density Prey Model 
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  1. Aronson, D. G. and H. F. Weinberger (1975). Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Lecture Notes in Mathematics, 446, J. A. Goldstein (Ed.), Berlin: Springer-Verlag, pp. 5–49.Google Scholar
  2. Conway, E. D. (1984). Diffusion and (predator-prey) interaction: pattern in closed systems. Res. Notes Math. 101, 85–133.zbMATHMathSciNetGoogle Scholar
  3. Dunbar, S. R. (1986). Traveling waves in diffusive predator-prey equations: periodic orbits and point-to-periodic heteroclinic orbits. SIAM J. Appl. Math. 46, 1057–1078.zbMATHMathSciNetCrossRefGoogle Scholar
  4. Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary succession: Herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, 238–251.CrossRefGoogle Scholar
  5. Fife, P. C. (1976). Pattern formation in Reacting and Diffusing Systems. J. Chem. Phys. 64, 554–564.CrossRefGoogle Scholar
  6. Hadeler, K. P. and F. Rothe (1975). Travelling fronts in nonlinear diffusion equations. J. Math. Biol. 2, 251–263.MathSciNetCrossRefzbMATHGoogle Scholar
  7. Hastings, A., S. Harrison and K. McCann (1997). Unexpected spatial patterns in an insect outbreak match a predator diffusion model. Proc. R. Soc. Lond. B 264, 1837–1840.CrossRefGoogle Scholar
  8. Hosono, Y. (1998). The minimal speed of traveling fronts for a diffusive Lotka Volterra competition model. Bull. Math. Biol. 60, 435–448.zbMATHCrossRefGoogle Scholar
  9. Keener, J. P. (1987). Propagation failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572.zbMATHMathSciNetCrossRefGoogle Scholar
  10. Keener, J. P. (1993). The effects of discrete gap junction coupling on propagation in myocardium. J. Theor. Biol. 148, 49–82.Google Scholar
  11. Keitt, T. H., M. A. Lewis and R. D. Holt (2000). Allee effects, invasion pinning and species borders. Submitted to Amer. Nat. Google Scholar
  12. Lewis, M. A. and P. Kareiva (1993). Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. 43, 141–158.CrossRefzbMATHGoogle Scholar
  13. Lewis, M. A., B. Li and H. F. Weinberger (2000). Spreading speeds and the linear conjecture for two-species competition models. Submitted to J. Math. Biol. Google Scholar
  14. May, R. (1974). Model Ecosystems, Princeton, NJ: Princeton University Press.Google Scholar
  15. Murray, J. D. (1989). Mathematical Biology, Berlin: Springer Verlag.zbMATHGoogle Scholar
  16. Perko, L. (1991). Differential Equations and Dynamical Systems, New York: Springer Verlag.zbMATHGoogle Scholar
  17. Rothe, F. (1981). Convergence to pushed fronts. Rocky Mt. J. Math. 11, 617–633.zbMATHMathSciNetCrossRefGoogle Scholar
  18. Sherratt, J. A., B. T. Eagan and M. A. Lewis (1997). Oscillations and chaos behind predator-prey invasion: mathematical artefact or ecological reality? Phil. Trans. R. Soc. Lond. B 352, 21–38.CrossRefGoogle Scholar
  19. Smoller, J. (1994). Shock Waves and Reaction-Diffusion Equations, New York: Springer Verlag.zbMATHGoogle Scholar
  20. Weinberger, H. F., M. A. Lewis and B. Li (2000). Analysis of the linear conjecture for spread in cooperative models. Submitted to J. Math. Biol. Google Scholar

Copyright information

© Society for Mathematical Biology 2001

Authors and Affiliations

  • M. R. Owen
    • 1
  • M. A. Lewis
    • 2
  1. 1.Nonlinear and Complex Systems Group, Department of Mathematical SciencesLoughborough UniversityLoughboroughUK
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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