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A comparison of two predator-prey models with Holling’s type I functional response. (English) Zbl 1138.92033
Summary: We analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. We show, numerically, that our two models can both possess two limit cycles that surround a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit super-critical and discontinuous Hopf bifurcations. We then present and analyze a new functional response, built around the arctangent, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclic-fold, and Bautin bifurcations. We use our analyses to characterize predator-prey systems that may exhibit bistability.

MSC:
92D40 Ecology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
Software:
AUTO; AUTO-86; XPPAUT
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