Global Dynamics of a Predator-Prey Model with a General Growth Rate Function and Carrying Capacity

Abstract

In this paper, we investigate the global dynamics of a predator-prey model with a general growth rate function and carrying capacity. We prove that the origin is unstable using the blow-up method. Also, by constructing a new Lyapunov function and using LaSalle’s invariance principle, we obtain the global stability of the positive equilibrium state of the system. In addition, the system undergoes the Hopf bifurcation at the positive equilibrium point when the predator birth rate $\delta$ is used as the bifurcation parameter. Finally, two examples are given to verify the feasibility of the theoretical results. One example is given to reconsider the global stability of the positive equilibrium of a Leslie-Gower predator-prey model with prey cannibalism, and the obtained results confirm the conjecture proposed by Lin et al. (Adv Differ Equ 2020, 153, 2020). The other example is given to verify the occurrence of the Hopf bifurcation of a Leslie-Gower predator-prey model with a square root response function, and obtain the Hopf bifurcation diagram by the numerical simulation.