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Chaos in periodically forced discrete-time ecosystem models. (English) Zbl 0964.92044
Summary: Natural populations whose generations are non-overlapping can be modelled by difference equations that describe how the populations evolve in discrete time-steps. These ecosystem models are, in general, nonlinear and contain system parameters that relate to such properties as the intrinsic growth-rate of a species. Typically, the parameters are kept constant. In this study, in order to simulate cyclic effects due to changes in environmental conditions, periodic forcing is applied to system parameters in four specific models, comprising three well-known, single-species models due to May, Moran-Ricker, and Hassell, and also a Maynard Smith predator-prey model. It is found that, in each case, a system that has simple (e.g., periodic) behavior in its unforced state can take on extremely complicated behavior, including chaos, when periodic forcing is applied, dependent on the values of the forcing amplitudes and frequencies. For each model, the application of forcing is found to produce an effective increase in the parameter space over which the system can behave chaotically. Bifurcation diagrams are constructed with the forcing amplitude as the bifurcation parameter, and these are observed to display rich structurc, including chaotic bands with periodic windows, pitch-fork and tangent bifurcations, and attractor crises.

37N25Dynamical systems in biology
37D45Strange attractors, chaotic dynamics
Full Text: DOI
[1] May, R. M.: Simple mathematical models with very complicated dynamics. Nature 261, 459-467 (1976)
[2] Hassell, M. P.; Comins, H. N.: Discrete time models for two-species competition. Theoret population biol 9, 202-221 (1976) · Zbl 0338.92020
[3] Rogers, T. D.: Chaos in systems in population biology. Progr theoret biol 6, 91-146 (1981)
[4] May, R. M.: Chaos and the dynamics of biological populations. Proc roy soc London A 413, 27-44 (1987) · Zbl 0656.92012
[5] Murray, J. D.: Mathematical biology. (1989) · Zbl 0682.92001
[6] Holton D, May RM. Chaos and one-dimensional maps. In: Mullin T, editor. The nature of chaos. Oxford: Clarendon Press, 1993:95--119
[7] Holton D, May RM. Models of chaos from natural selection. In: Mullin T, editor. The nature of chaos. Oxford: Clarendon Press, 1993:120--48
[8] Cushing, J. M.: Stable limit cyles of time dependent multispecies interactions. Math biosci 31, 259-273 (1976) · Zbl 0341.92011
[9] Cushing, J. M.: Periodic time-dependent predator--prey systems. SIAM J appl math 32, 82-95 (1977) · Zbl 0348.34031
[10] Cushing, J. M.: Two species competition in a periodic environment. J math biol 10, 385-400 (1980) · Zbl 0455.92012
[11] Sabin, G. C. W.; Summers, D.: Chaos in a periodically forced predator--prey ecosystem model. Math biosci 113, 91-113 (1993) · Zbl 0767.92028
[12] Rinaldi, S.; Muratori, S.; Kuznetsov, Y.: Multiple attractors, catastrophes and chaos in seasonally perturbed predator--prey communities. Bull math biol 55, 15-35 (1993) · Zbl 0756.92026
[13] Foster, E. A. D.: Boundary crises arising from saddle-node bifurcations in periodically forced models. Int J bifurc chaos 6, 647-659 (1996) · Zbl 0879.58052
[14] Kot, M.; Schaffer, W. M.: The effects of seasonality on discrete models of population growth. Theoret population biol 26, 340-360 (1984) · Zbl 0551.92014
[15] Grafton, R. Q.; Silva-Echenique, J.: How to manage nature? strategies, predator--prey models, and chaos. Marine res economics 12, 127-143 (1997)
[16] Smith, J. Maynard: Mathematical ideas in biology. (1968)
[17] May, R. M.: On relationships among various types of population models. Am nat 107, 46-57 (1972)
[18] Krebs, C. J.: Ecology: the experimental analysis of distribution and abundance. (1972)
[19] Moran, P. A. P.: Some remarks on animal population dynamics. Biometrics 6, 250-258 (1950)
[20] Ricker, W. E.: Stock and recruitment. J fish res bd can 11, 559-623 (1954)
[21] Cook, L. M.: Oscillation in the simple logistic growth model. Nature 207, 316 (1965)
[22] May RM. Ecosystem patterns in randomly fluctuating environments. In: Rosen R, Snell F, editors. Progress in theoretical biology. New York: Academic Press, 1974:1--50
[23] Hassell, M. P.: Density-dependence in single-species populations. J anim ecol 44, 283-295 (1975)
[24] Hassell, M. P.; Lawton, J. H.; May, R. M.: Patterns of dynamical behaviour in single-species populations. J anim eco 45, 471-486 (1976)
[25] Neubert, M. G.; Kot, M.: The subcritical collapse of predator populations in discrete-time predator--prey models. Math biosci 110, 45-66 (1992) · Zbl 0747.92024
[26] Kocak, H.: Differential and difference equations through computer experiments. (1989)
[27] Gumowski I, Mira C. Recurrences and discrete dynamic systems. Lecture Notes in Mathematics, vol. 809. New York: Springer, 1980 · Zbl 0449.58003
[28] Lauwerier HA. Two-dimensional iterative maps. In: Holden AV, editor. Chaos. Princeton: Princeton University Press, 1986:58--95
[29] Parker, T. S.; Chua, L. O.: Practical numerical algorithms for chaotic systems. (1989) · Zbl 0692.58001
[30] Grebogi, C.; Ott, E.; Yorke, J. A.: Chaotic attractors in crisis. Phys rev lett 48, 1507-1510 (1982)
[31] Grebogi, C.; Ott, E.; Yorke, J. A.: Crises, sudden changes in chaotic attractors and chaotic transients. Physica D 7, 181-200 (1983) · Zbl 0561.58029
[32] Grinfeld, M.; Knight, P. A.; Lamba, H.: On the periodically perturbed logistic equation. J phys A 29, 8035-8040 (1996) · Zbl 0898.58013
[33] Farmer, J. D.; Ott, E.; Yorke, J. A.: The dimension of chaotic attractors. Physica D 7, 153-180 (1983) · Zbl 0561.58032
[34] Hastings, A.; Hom, C. L.; Ellner, S.; Turchin, P.; Godfray, H. C. J.: Chaos in ecology: is mother nature a strange attractor?. Ann rev ecol syst 24, 1-33 (1993)
[35] Costantino, R. F.; Desharnais, R. A.; Cushing, J. M.; Dennis, B.: Chaotic dynamics in an insect population. Science 275, 389-391 (1997) · Zbl 1225.37103
[36] Godfray, C.; Hassell, M.: Chaotic beetles. Science 275, 323-324 (1997)