Abstract:
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 crossdiffusion-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.
