Degenerate Hopf bifurcation in a Leslie–Gower predator–prey model with predator harvest

Abstract

Abstract In this paper, we investigate the degenerate Hopf bifurcation of a Leslie–Gower predator–prey system with predator harvest. The known work discussed the Hopf bifurcation of this system when the first Lyapunov number does not vanish and gave an example of a stable weak focus with order 2. However, the thorough discussion of center-type equilibrium for all possible parameters has not been completed. In this paper, by computing the first two focal values, we decompose the variety with resultant and prove that the center-type equilibrium is a weak focus with order at most 2 for all the possible parameter values. Moreover, numerical simulations are employed to show the appearance of two limit cycles from degenerate Hopf bifurcation. Our results finish the study of Hopf bifurcation in this system.

Keywords

• Leslie–Gower predator–prey model
• Hopf bifurcation
• Focal value
• Resultant elimination