Discrete kinetic analysis of a general reaction–diffusion model constructed by Euler discretization and coupled map lattices

In this paper, we provide a framework for the Turing instability in a general two-dimensional discrete reaction–diffusion model that utilizes Euler discretization and coupled map lattices. We obtained explicit criterions for the normal forms of Neimark–Sacker bifurcation and flip bifurcation in the absence of diffusion. We derive the general conditions that govern the emergence of pure Turing instability, Neimark–Sacker–Turing instability, Flip–Turing instability, spatially homogeneous stable states, spatially homogeneous periodic oscillation states, and spatially homogeneous period-doubling oscillatory states under spatially inhomogeneous conditions. As an example, we employ a ratio-dependent predation model and verify the derived general conditions through a series of numerical simulations. Additionally, we apply the calculation of Maximum Lyapunov Exponent(MLE) to simulate and demonstrate the path from flip bifurcation to chaos.