Bifurcation analysis in a predator–prey system with an increasing functional response and constant-yield prey harvesting

Abstract

In this paper, a Gause type predator–prey system with constant-yield prey harvesting and monotone ascending functional response is proposed and investigated. We focus on the influence of the harvesting rate on the predator–prey system. First, equilibria corresponding to different situations are investigated, as well as the stability analysis. Then bifurcations are explored at nonhyperbolic equilibria, and we give the conditions for the occurrence of two saddle–node bifurcations by analyzing the emergence, coincidence and annihilation of equilibria. We calculate the Lyapunov number and focal values to determine the stability and the quantity of limit cycles generated by supercritical, subcritical and degenerate Hopf bifurcations. Furthermore, the system is unfolded to explore the repelling and attracting Bogdanov–Takens bifurcations by perturbing two bifurcation parameters near the cusp. It is shown that there exists one limit cycle, or one homoclinic loop, or two limit cycles for different parameter values. Therefore, the system is susceptible to both the constant-yield prey harvesting and initial values of the species. Finally, we run numerical simulations to verify the theoretical analysis.