Cross-diffusion induced spatiotemporal patterns in Schnakenberg reaction–diffusion model

Schnakenberg system is a typical mathematical model to describe activator-depleted kinetics. In this paper, by introducing linear cross-diffusion into Schnakenberg system, we derive cross-diffusion-driven Turing instability conditions. It has been revealed that it is no longer necessary to have long-range inhibition and short-range activation for Turing instability with the help of cross-diffusion. Then, the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which helps us to derive parameter space more specific where certain patterns such as hexagon-like pattern, stripe-like pattern and the coexistence pattern will emerge. Furthermore, the numerical simulations in both Turing instability region and Turing–Hopf region provide an indication of the wealth of patterns that the system can exhibit. Besides, different initial conditions are employed to help better understanding the complex patterns.