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Zbl 1058.92049
Moghadas, S.M.; Alexander, M.E.
Dynamics of a generalized Gause-type predator-prey model with a seasonal functional response. (English)
[J] Chaos Solitons Fractals 23, No. 1, 55-65 (2005). ISSN 0960-0779

Summary: We extend a previous Gause-type predator-prey model [see {\it H. I. Freedman}, Deterministic mathematical models in population ecology. (1980; Zbl 0448.92023)] to include a general monotonic and bounded seasonally varying functional response. The model exhibits rich dynamical behaviour not encountered when the functional response is not seasonally forced. A theoretical analysis is performed on the model to investigate the global stability of the boundary equilibria and the existence of periodic solutions. It is shown that, under certain well-defined conditions, the Poincaré map of the model undergoes a Hopf bifurcation leading to the appearance of a quasi-periodic solution. Numerical results are given for the Poincaré sections and bifurcation diagrams for Holling-types II and III functional responses, using the amplitude of seasonal variation as bifurcation parameter. The model shows a rich variety of behaviour, including period doubling, quasi-periodicity, chaos, transient chaos, and windows of periodicity.

MSC 2000:: *92D40 Ecology
37N25 Dynamical systems in biology
34C60 Applications of qualitative theory of ODE
34D05 Asymptotic stability of ODE
34D23 Global stability
34C23 Bifurcation (periodic solutions)
34C25 Periodic solutions of ODE

Keywords: boundary critical points; period doubling; chaos

Citations: Zbl 0448.92023

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