Stationary pattern and bifurcation of a Leslie–Gower predator–prey model with prey-taxis

Abstract

In this paper, we consider a Leslie–Gower predator–prey model with prey-taxis subject to the homogeneous Neumann boundary condition. First, the stability of positive constant solution for corresponding ODE and PDE models is studied, respectively. It turns out that small prey-taxis can cause Turing instability and produce spatio-temporal patterns; large prey-taxis can stabilize the instability caused by diffusion. Then, in spatial one dimension, we use the center manifold theory to analyze the direction of Hopf bifurcation and stability of the bifurcation periodic solutions. By regarding prey-tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of non-constant solutions bifurcating from the positive equilibrium and find the stability criterion of such bifurcation solutions. Finally, some numerical simulations are shown to verify and expand our theoretical findings.