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The subcritical collapse of predator populations in discrete-time predator-prey models. (English) Zbl 0747.92024
Summary: Many discrete-time predator-prey models possess three equilibria, corresponding to (1) extinction of both species, (2) extinction of the predator and survival of the prey at its carrying capacity, or (3) coexistence of both species. For a variety of such models, the equilibrium corresponding to coexistence may lose stability via a Hopf bifurcation, in which case trajectories approach an invariant circle. Alternatively, the equilibrium may undergo a subcritical flip bifurcation with a concomitant crash in the predator’s population. We review a technique for distinguishing between subcritical and supercritical flip bifurcations and provide examples of predator-prey systems with a subcritical flip bifurcation.

MSC:
92D25 Population dynamics (general)
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis
34C23 Bifurcation theory for ordinary differential equations
37G99 Local and nonlocal bifurcation theory for dynamical systems
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