Bifurcation and turing instability analysis for a space- and time-discrete predator–prey system with Smith growth function

In this paper, the dynamic behavior of a space- and time-discrete predator–prey system with Smith growth function is studied. Through the stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Through the bifurcation theory, the expressions of the critical values for the occurrence of Neimark–Sacker bifurcation and flip bifurcation of the system are obtained, and the conditions for the occurrence of Turing bifurcation of the system are given. Finally, through numerical simulation, we can observe some complex dynamic behaviors, such as period-doubling cascade, invariant circles, periodic windows, chaotic dynamics and pattern formation.