Study on Bifurcation and Turing Instability in a Temporally and Spatially Discrete Predator–Prey System With Prey Refuge

ABSTRACT This paper focuses on a discrete self‐diffusion predator–prey model with a prey refuge. Firstly, through stability analysis, the parametric conditions for spatially homogeneous steady‐state stability at the equilibrium points of the model are obtained. Subsequently, bifurcation theory is employed to derive the conditions under which flip bifurcation and the Neimark–Sacker bifurcation occur in the model. Furthermore, we delve into chaos control theory and unveil the conditions for the Turing instability in the discrete diffusive model under the influence of overall self‐diffusion. Finally, numerical simulations are conducted to verify the theoretical analysis, and complex dynamical behaviors such as period doubling, limit cycles, periodic windows, chaotic dynamics, and pattern formation are observed.