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Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect. (English) Zbl 1126.92062
Summary: Species establishment in a model system in a homogeneous environment can be dependent not only on the parameter setting, but also on the initial conditions of the system. For instance, predator invasion into an established prey population can fail and lead to system collapse, an event referred to as overexploitation. This phenomenon occurs in models with bistability properties, such as strong Allee effects. The Allee effect then prevents easy re-establishment of the prey species.
We deal with the bifurcation analyses of two previously published predator-prey models with strong Allee effects [A. D. Bazykin, Nonlinear dynamics of interacting populations. (1998); A. Kent et al., Ecol. Model 162, 233 ff (2003)]. We expand the analyses to include not only local, but also global bifurcations. We show the existence of a point-to-point heteroclinic cycle in these models, and discuss numerical techniques for continuation in the parameter space. The continuation of such a cycle in a two-parameter space forms the boundary of a region in the parameter space where the system collapses after predator invasion, i.e., where overexploitation occurs. We argue that the detection and continuation of global bifurcations in these models are of vital importance for the understanding of the model dynamics.

MSC:
92D40 Ecology
37N25 Dynamical systems in biology
Software:
AUTO; DSTool; HomCont
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