Influence of predator–taxis and time delay on the dynamical behavior of a predator–prey model with prey refuge and predator harvesting

Abstract

In this paper, we consider a delay-driven population chemotaxis model with practical significance in biology, incorporating both prey refuge and predator harvesting terms. We first prove the local-in-time existence and uniqueness of classical solutions to this model for $\tau =0$, and the global existence and uniqueness of classical solutions to this model for $\tau >0$. Then, using stability theory, bifurcation theory, and normal form theory, we analyze the local asymptotic stability of the positive constant steady state of the model, as well as the effects of predator chemotaxis and delay on the dynamics of the model. When the delay is zero, through rigorous mathematical analysis, we find that predator–taxis can induce a Turing bifurcation, and the combined effects of predator–taxis and random diffusion of the prey can trigger a Turing–Turing bifurcation. When the delay is nonzero, we show that the delay can induce a Hopf bifurcation, and the joint effects of predator–taxis and delay can result in Hopf–Hopf and Turing–Hopf bifurcations. Furthermore, we rigorously derive the third-order truncated normal form for the Hopf bifurcation of the model, which allows us to determine the direction of the Hopf bifurcation and the stability of the periodic solutions generated by the Hopf bifurcation. At the same time, numerical simulations are conducted to verify the results of the theoretical analysis.