Hopf bifurcation analysis in a delayed Leslie–Gower predator–prey model incorporating additional food for predators, refuge and threshold harvesting of preys

Abstract

In this paper, we formulate and analyze a modified Leslie–Gower predator–prey model. Our model incorporates refuge of preys, additional fixed food for predators, harvesting of preys through a continuous threshold policy and a time delay as to account for predators maturity time. We first carry out a qualitative analysis of the model without time delay, showing existence of extinction, prey-free, predator-free and coexistence equilibria. We further study their stability conditions. Relying only on theoretical results of the model, we construct bifurcation diagrams involving refuge and harvest limit parameters. This led to summarize different scenarios for the model including elimination of one species or competition of both species that are proved possible. Furthermore, considering the time delay as bifurcation parameter, we analyze the stability of the coexistence equilibria and prove the system can undergoes a Hopf bifurcation. The direction of that Hopf bifurcation and the stability of the bifurcated periodic solution are determined by applying the normal form theory and the center manifold theorem. Numerical simulations are presented to illustrate our theoretical results.