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