Modeling with Integrodifference Equations
Chapter
First Online:
Abstract
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.
References
- Allee, W. (1949). Principles of animal ecology. Christchurch: Saunders.Google Scholar
- Andersen, M. (1991). Properties of some density-dependent integrodifference equation population models. Mathematical Biosciences, 104, 135–157.CrossRefGoogle Scholar
- Bellows, T. (1981). The descriptive properties of some models for density dependence. Journal of Animal Ecology, 50(1), 139–156.MathSciNetCrossRefGoogle Scholar
- 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
- 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
- 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
- Courchamp, F., Berec, L., & Gascoinge, J. (2008). Allee effects. Oxford: Oxford University Press.CrossRefGoogle Scholar
- Edelstein-Keshet, L. (2005). Mathematical models in biology. Philadelphia: SIAM.CrossRefGoogle Scholar
- 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
- 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
- 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
- Hassell, M. (1975). Density-dependence in single-species populations. Journal of Animal Ecology, 44(1), 283–295.CrossRefGoogle Scholar
- Kot, M. (2001). Elements of mathematical ecology. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
- Kot, M., Lewis, M., & van den Driessche, P. (1996). Dispersal data and the spread of invading organisms. Ecology, 77, 2027–2042.CrossRefGoogle Scholar
- 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
- 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
- May, R. (1975). Biological populations obeying difference equations: Stable points, stable cycles, and chaos. Journal of Theoretical Biology, 51, 511–524.CrossRefGoogle Scholar
- Murray, J. D. (2001). Mathematical biology I: An introduction. Berlin: Springer.Google Scholar
- 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
- 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
- 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
- Ricker, W. (1954). Stock and recruitment. Journal of Fisheries Research Board of Canada, 11, 559–632.CrossRefGoogle Scholar
- Sandefur, J. (2018). A unifying approach to discrete single-species populations models. Discrete & Continuous Dynamical Systems - Series B, 23, 493–508.MathSciNetCrossRefGoogle Scholar
- Schreiber, S. (2003). Allee effects, extinctions, and chaotic transients in simple population models. Theoretical Population Biology, 64, 201–209.CrossRefGoogle Scholar
- 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
- 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
- Wang, M.-H., Kot, M., & Neubert, M. (2002). Integrodifference equations, Allee effects, and invasions. Journal of Mathematical Biology, 44, 150–168.MathSciNetCrossRefGoogle Scholar
Copyright information
© Springer Nature Switzerland AG 2019