Seo G, Kot M. A comparison of two predator-prey models with Holling's type I functional response.
Math Biosci 2008;
212:161-79. [PMID:
18346761 DOI:
10.1016/j.mbs.2008.01.007]
[Citation(s) in RCA: 42] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2007] [Revised: 01/29/2008] [Accepted: 01/30/2008] [Indexed: 11/30/2022]
Abstract
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.
Collapse