Cross-diffusion induced spatial patterns in a chemical self-replication model

This paper deals with Turing patterns, spiral wave patterns and patterns in the neighborhood of Turing–Hopf bifurcation point of the reaction-cross-diffusion chemical self-replication model in both one-dimensional and two-dimensional domains. We derive the requirements for homogeneous or inhomogeneous Hopf bifurcations, Turing instability as well as Turing–Hopf bifurcation. Numerical results permit us to investigate the pattern transition in Turing regime, the spiral wave patterns in Hopf regime and complex spatiotemporal patterns arising from interactions between Turing and Hopf bifurcations. We demonstrate Turing patterns or spiral wave patterns can be successfully predicted through the linear theory. Simultaneously, the relevance of initial conditions in pattern formation and the speed of convergence have been emphasized. As for Turing–Hopf bifurcation, the reduction of the model to the normal form adopting the quadratic approximation of parameters allows us to derive parameter spaces more specifically where certain spatiotemporal behaviors arise. Varying two systematic parameters of cross-diffusion coefficient d22 and the rate of enzymatic sink q, we perform a series of simulations to confirm the classification of the spatiotemporal dynamics close to the Turing–Hopf bifurcation point.