Bifurcation and turing instability analysis for a space- and time-discrete predator–prey system with Smith growth function. (January 2023) - Han, Xiaoling; Lei, Ceyu
Abstract: 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. Highlights: This paper explores population diffusion. The diversity of pattern self-organization types of discrete predator–prey system is displayed, which provides a broad idea for the study of pattern dynamics of space- and time-discrete predator–prey system. This paper focuses on the role of flip bifurcation and Neimark–Sacker bifurcation in discrete predator–prey system. Through the corresponding numerical simulation of bifurcation diagram, phase diagram and Lyapunov exponent diagram, the chaotic behavior caused by bifurcation and the dynamic characteristics on the chaotic path are shown. The pattern dynamics of discrete predator–prey system is also studied in this paper. By constructing its space- and time-discrete coupled map lattice model, the conditions of Turing instability in the system are analyzed, and the self-organization formation process of Turing pattern is found through numerical simulation.