Abstract
Generalising a site-based stochastic model due to Royama, Solé et al. and Sumpter et al., we investigate competition in a single species with discrete, non-overlapping generations. We show that the deterministic limit of the dynamics depends on a few easily interpretable parameters only. Further, we discuss qualitative properties and limit sets of the corresponding difference equations, and we relate these to modes of competition. Moreover, a detailed analysis of stochastic effects in some relevant scenarios indicates that the behaviour of the stochastic model is very sensitive to further details of the model.
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Beverton, R. J. H., & Holt, S. J. (1957). Fisheries investigations, Ser. 2 : Vol. 19. On the dynamics of exploited fish populations. London: H.M. Stationery Office.
Brännström, A., & Sumpter, D. J. T. (2005). The role of competition and clustering in population dynamics. Proc. R. Soc. B, 272, 2065–2072.
Brännström, A., & Sumpter, D. J. T. (2006). Stochastic analogues of deterministic single-species population models. Theor. Popul. Biol., 69, 442–451.
Campbell, N. A., & Reece, J. B. (2004). Biology (7th ed.). San Francisco: Benjamin-Cummings.
Geritz, S. A. H., & Kisdi, E. (2004). On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol., 228, 261–269.
Hassell, M. P. (1974). Density dependence in single species populations. J. Anim. Ecol., 44, 283–296.
Hassell, M. P., Lawton, J. H., & May, R. M. (1976). Patterns of dynamical behavior in single species populations. J. Anim. Ecol., 42, 471–486.
Johansson, A., & Sumpter, D. J. T. (2003). From local interactions to population dynamics in site-based models of ecology. Theor. Popul. Biol., 64, 419–517.
Johst, K., Berryman, A., & Lima, M. (2008). From individual interactions to population dynamics: individual resource partitioning simulation exposes the cause of nonlinear intra-specific competition. Popul. Ecol., 50, 79–90.
Li, T. Y., & Yorke, J. A. (1975). Period three implies chaos. Am. Math. Mon., 82, 985–992.
Lomnicki, A., & Sedziwy, S. (1988). Resource partitioning and population stability under exploitation competition. J. Theor. Biol., 132, 119–120.
May, R. M., & Oster, G. F. (1976). Bifurcations and dynamic complexity in simple ecological models. Am. Nat., 110, 573–599.
Moran, P. A. P. (1950). Some remarks on animal population dynamics. Biometrics, 6, 250–258.
Murray, J. D. (1993). Mathematical biology (2nd ed.). Heidelberg: Springer.
Nasell, I. (2002). Measles outbreaks are not chaotic. In The IMA volumes in mathematics and its applications : Vol. 126. Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory (pp. 85–114). New York: Springer.
Nicholson, A. J. (1954). An outline of the dynamics of animal populations. Aust. J. Zool., 2, 9–65.
Ricker, W. E. (1954). Stock and recruitment. J. Fish. Res. Board Can., 11(5), 559–622.
Royama, T. (1992). Analytical population dynamics. London: Chapman & Hall.
Sharkovski, A. N. (1965). On cycles and structure of a continuous map. Ukr. Math. J., 17, 104–111 (in Russian).
Skellam, J. G. (1951). Random dispersal in theoretical populations. Biometrika, 38, 196–218.
Solé, R. V., Gamarra, J. G. P., Ginovart, M., & Lopez, D. (1999). Controlling chaos in ecology: from deterministic to individual-based models. Bull. Math. Biol., 61, 1187–1207.
Sumpter, D. J. T., & Broomhead, D. S. (2001). Relating individual behavior to population dynamics. Proc. R. Soc. B, 268, 925–932.
Thunberg, H. (2001). Periodicity versus chaos in one-dimensional dynamics. SIAM Rev., 43, 3–30.
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Gotzen, B., Liebscher, V. & Walcher, S. On a Class of Deterministic Population Models with Stochastic Foundation. Bull Math Biol 73, 1559–1582 (2011). https://doi.org/10.1007/s11538-010-9581-9
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DOI: https://doi.org/10.1007/s11538-010-9581-9