Axioms (Oct 2024)

Bifurcation of a Leslie–Gower Predator–Prey Model with Nonlinear Harvesting and a Generalist Predator

  • Mengxin He,
  • Zhong Li

DOI
https://doi.org/10.3390/axioms13100704
Journal volume & issue
Vol. 13, no. 10
p. 704

Abstract

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A Leslie–Gower predator–prey model with nonlinear harvesting and a generalist predator is considered in this paper. It is shown that the degenerate positive equilibrium of the system is a cusp of codimension up to 4, and the system admits the cusp-type degenerate Bogdanov–Takens bifurcation of codimension 4. Moreover, the system has a weak focus of at least order 3 and can undergo degenerate Hopf bifurcation of codimension 3. We verify, through numerical simulations, that the system admits three different stable states, such as a stable fixed point and three limit cycles (the middle one is unstable), or two stable fixed points and two limit cycles. Our results reveal that nonlinear harvesting and a generalist predator can lead to richer dynamics and bifurcations (such as three limit cycles or tristability); specifically, harvesting can cause the extinction of prey, but a generalist predator provides some protection for the predator in the absence of prey.

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