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Chaos in three species food chains. (English) Zbl 0823.92030

Summary: We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model.

This work extends earlier numerical studies of a particular system by A. Hastings and T. Powell [Ecology 72, No. 3, 896-903 (1991)] by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by J. Guckenheimer and P. Holmes [Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983; Zbl 0515.34001)] on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.

37N99Applications of dynamical systems
37D45Strange attractors, chaotic dynamics
37G99Local and nonlocal bifurcation theory
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