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Study of a Leslie-Gower-type tritrophic population model. (English) Zbl 1031.92027
Summary: A three-dimensional continuous time dynamical system is considered. It is a model for a tritrophic food chain, based on a modified version of the Leslie-Gower scheme [P. H. Leslie and J. C. Gower, Biometrika 47, 219-234 (1960; Zbl 0103.12502)]. We establish and prove theorems on boundedness of the system, existence of an attracting set, existence and local or global stability of equilibria which represent the extinction of the top or intermediate predator. Using intensive numerical qualitative analysis we show that the model could exhibit chaotic dynamics for realistic parameter and state values. Transition to chaotic behavior is established via period doubling bifurcation, and some sequences of distinctive period-halving are found.

MSC:
92D40 Ecology
37N25 Dynamical systems in biology
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
65L99 Numerical methods for ordinary differential equations
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