Spatially Nonhomogeneous Neimark–Sacker Bifurcation and New Pattern Formation Mechanism in a Spatiotemporally Discrete Predator–Prey System

This study explores the spatiotemporal dynamics of a discrete reaction–diffusion predator–prey system, with a focus on the spatially nonhomogeneous Neimark–Sacker bifurcation. We rigorously derive the normal form and theorem of the Neimark–Sacker bifurcation, extending the analysis to a two-dimensional space domain. The bifurcation theorem is applied to a spatiotemporally discrete Holling-II predator–prey system. The results uncover a sequence of Neimark–Sacker bifurcation points and the emergence of spatially nonhomogeneous quasi-periodic solutions. Numerical simulations validate the theoretical predictions and illustrate the self-organization of dynamic patterns induced by the bifurcation. Moreover, numerical results suggest that the interplay of multiple spatially nonhomogeneous quasi-periodic modes can produce complex spatiotemporal structures, such as fragmented mosaics and spiral waves. This work provides insights into new nonlinear mechanisms underlying complexity in discrete systems, offering a robust framework for analyzing predator–prey interactions in fragmented habitats.