Modeling with Integrodifference Equations

  • Frithjof Lutscher
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 49)


We derive the basic integrodifference equation and discuss its two main ingredients: the growth function and the dispersal kernel. We introduce several ecological concepts that recur throughout this book and highlight how ecological assumptions are reflected in the mathematical model. This detailed understanding will allow us to formulate ecological insights from the mathematical results and understand the limitations of these insights.


  1. Allee, W. (1949). Principles of animal ecology. Christchurch: Saunders.Google Scholar
  2. Andersen, M. (1991). Properties of some density-dependent integrodifference equation population models. Mathematical Biosciences, 104, 135–157.CrossRefGoogle Scholar
  3. Bellows, T. (1981). The descriptive properties of some models for density dependence. Journal of Animal Ecology, 50(1), 139–156.MathSciNetCrossRefGoogle Scholar
  4. Beverton, R., & Holt, S. (1957). On the Dynamics of Exploited Fish Populations. Fisheries Investigation Series (vol. 2, no. 19). London: Ministry of Agriculture, Fisheries, and Food.Google Scholar
  5. Brännström, A., & Sumpter, D. (2005). The role of competition and clustering in population dynamics. Proceedings of the Royal Society of London B, 272, 2065–2072.Google Scholar
  6. Bullock, J., & Clarke, R. (2000). Long distance seed dispersal by wind: Measuring and modelling the tail of the curve. Oecologia, 124, 506–521.Google Scholar
  7. Courchamp, F., Berec, L., & Gascoinge, J. (2008). Allee effects. Oxford: Oxford University Press.CrossRefGoogle Scholar
  8. Edelstein-Keshet, L. (2005). Mathematical models in biology. Philadelphia: SIAM.CrossRefGoogle Scholar
  9. Etienne, R., Wertheim, B., Hemerik, L., Schneider, P., & Powell, J. (2002). The interaction between dispersal, the Allee effect and scramble competition affects population dynamics. Ecological Modelling, 148, 153–168.CrossRefGoogle Scholar
  10. Fujiwara, M., Anderson, K., Neubert, M., & Caswell, H. (2006). On the estimation of dispersal kernels from individual mark-recapture data. Environmental and Ecological Statistics, 13, 183–197.MathSciNetCrossRefGoogle Scholar
  11. Geritz, S., & Kisdi, É. (2004). On the mechanistic underpinning of discrete-time population models with complex dynamics. Journal of Theoretical Biology, 228, 261–269.MathSciNetCrossRefGoogle Scholar
  12. Hassell, M. (1975). Density-dependence in single-species populations. Journal of Animal Ecology, 44(1), 283–295.CrossRefGoogle Scholar
  13. Kot, M. (2001). Elements of mathematical ecology. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  14. Kot, M., Lewis, M., & van den Driessche, P. (1996). Dispersal data and the spread of invading organisms. Ecology, 77, 2027–2042.CrossRefGoogle Scholar
  15. Lewis, M., Neubert, M., Caswell, H., Clark, J., & Shea, K. (2006). A guide to calculating discrete-time invasion rates from data. In M. Cadotte, S. McMahon, & T. Fukami (Eds.), Conceptual ecology and invasions biology: Reciprocal approaches to nature (pp. 169–192). Berlin: Springer.CrossRefGoogle Scholar
  16. Lutscher, F., & Petrovskii, S. (2008). The importance of census times in discrete-time growth-dispersal models. Journal of Biological Dynamics, 2(1), 55–63.MathSciNetCrossRefGoogle Scholar
  17. May, R. (1975). Biological populations obeying difference equations: Stable points, stable cycles, and chaos. Journal of Theoretical Biology, 51, 511–524.CrossRefGoogle Scholar
  18. Murray, J. D. (2001). Mathematical biology I: An introduction. Berlin: Springer.Google Scholar
  19. Musgrave, J., Girard, A., & Lutscher, F. (2015). Population spread in patchy landscapes under a strong Allee effect. Theoretical Ecology, 8(3), 313–326.CrossRefGoogle Scholar
  20. Nathan, R., Klein, E., Robledo-Arnuncio, J. J., & Revilla, E. (2012). Dispersal kernels: Review. In J. Clobert, M. Baguette, T. G. Benton, & J. M. Bullock (Eds.), Dispersal ecology and evolution (chap. 15.1). Oxford: Oxford University Press.Google Scholar
  21. Neubert, M., Kot, M., & Lewis, M. A. (1995). Dispersal and pattern formation in a discrete-time predator–prey model. Theoretical Population Biology, 48(1), 7–43.CrossRefGoogle Scholar
  22. Ricker, W. (1954). Stock and recruitment. Journal of Fisheries Research Board of Canada, 11, 559–632.CrossRefGoogle Scholar
  23. Sandefur, J. (2018). A unifying approach to discrete single-species populations models. Discrete & Continuous Dynamical Systems - Series B, 23, 493–508.MathSciNetCrossRefGoogle Scholar
  24. Schreiber, S. (2003). Allee effects, extinctions, and chaotic transients in simple population models. Theoretical Population Biology, 64, 201–209.CrossRefGoogle Scholar
  25. Tufto, J., Ringsby, T.-H., Dhondt, A., Adriaensen, F., & Matthysen, E. (2005). A parametric model for estimation of dispersal patterns applied to five passerine spatially structured populations. The American Naturalist, 165, E13–E26.CrossRefGoogle Scholar
  26. Veit, R. R., & Lewis, M. A. (1996). Dispersal, population growth, and the Allee effect: Dynamics of the house finch invasion in eastern North America. The American Naturalist, 148(2), 255–274.CrossRefGoogle Scholar
  27. Wang, M.-H., Kot, M., & Neubert, M. (2002). Integrodifference equations, Allee effects, and invasions. Journal of Mathematical Biology, 44, 150–168.MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Frithjof Lutscher
    • 1
  1. 1.Mathematics and StatisticsUniversity of OttawaOttawaCanada

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