Understanding the dual effects of linear cross-diffusion and geometry on reaction–diffusion systems for pattern formation

In this work, we study the dual effects of linear cross-diffusion and geometry on reaction–diffusion systems for pattern formation on rectangular domains. The spatiotemporal dynamics of the reaction–diffusion system with linear cross-diffusion are explored for the case of an activator-depleted model of two chemical species in terms of the domain size and its model parameters. Linear stability analysis is employed to derive the constraints which are necessary in understanding the dual roles of linear cross-diffusion and domain-size in studying the instability of the reaction–diffusion system. The conditions are proven in terms of lower and upper bounds of the domain-size together with the reaction, self- and cross-diffusion coefficients. The full parameter classification of the model system is presented in terms of the relationship between the domain size and cross-diffusion-driven instability. Subsequently, regions showing Turing instability, Hopf and transcritical types of bifurcations are demonstrated using the parameter values of the system. In this work, our theoretical findings are validated according to the proper choice of parameters in order to understand the effects of domain-size and linear cross-diffusion on the long-term spatiotemporal behaviour of solutions of the reaction–diffusion system. For illustrative purposes, numerical simulations showing each of the three types of dynamics are examined for the Schnakenberg kinetics, also known as an activator-depleted reaction kinetics.