Spatiotemporal complexity of a discrete space-time predator−prey system with self- and cross-diffusion

In this research, we investigate the spatiotemporal dynamics of a discrete space-time predator−prey system with self- and cross-diffusion. Through stability analysis and bifurcation analysis, Turing pattern formation conditions are derived and two nonlinear mechanisms of pattern formation are found, i.e., pure Turing instability and Hopf-Turing instability. Numerical simulations reveal rich dynamics of the discrete predator−prey system. In spatially homogeneous case, stable homogeneous stationary states, homogeneous periodic, quasiperiodic and chaotic oscillating states are exhibited; in spatially heterogeneous case, a surprising variety of prey and predator patterns are described, including spotted, striped, labyrinth, gapped, spiral, circled patterns and many intermediate patterns. Moreover, sensitivity of spatiotemporal pattern formation to initial conditions is predicted along with Hopf-Turing instability, suggesting the self-organization of diverse patterns under identical parametric conditions. In comparison with former results in literature, the discrete version of reaction-diffusion model developed in this research capture more complicated and richer nonlinear dynamical behaviors, contributing to a new comprehending on the complex pattern formation of spatially extended discrete predator−prey systems.