BIFURCATION AND TURING PATTERN ANALYSIS FOR A SPATIOTEMPORAL DISCRETE DEPLETION TYPE GIERER-MEINHARDT MODEL WITH SELF-DIFFUSION AND CROSS-DIFFUSION

This paper presents a study on spatiotemporal dynamics and Turing patterns in a space-time discrete depletion type Gierer-Meinhardt model with self-diffusion and cross-diffusion based on coupled map lattices (CMLs) model. Initially, the existence and stability conditions for fixed points are determined through linear stability analysis. Secondly, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation, and Turing bifurcation are derived by means of the center manifold reduction theorem and bifurcation theory. The results indicate that there exist two nonlinear mechanisms, namely flip-Turing instability and Neimark–Sacker-Turing instability. Additionally, some numerical simulations are performed to illustrate the theoretical findings. Interestingly, a rich variety of dynamical behaviors, including period-doubling cascades, invariant circles, periodic windows, chaotic regions, and striking pattern formations (plaques, mosaics, curls, spirals, and other intermediate patterns), are observed. Finally, the evolution of pattern size and type is also simulated as the cross-diffusion coefficient varies. It reveals that cross-diffusion has a certain influence on the growth of patterns.