A comparison of two predator-prey models with Holling's type I functional response.
In this paper, 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.
Seo
Gunog
G
University of Washington, Applied Mathematics, Box 352420, Seattle, WA 98195-2420, USA. justine@amath.washington.edu
Kot
Mark
M
D000818:Animals; D008954:Models, Biological; D009716:Numerical Analysis, Computer-Assisted; D011157:Population Dynamics; D011235:Predatory Behavior