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.

Automated summary

This paper investigates the Turing instability conditions driven by cross-diffusion in the Schnakenberg system and reveals that long-range inhibition and short-range activation are no longer necessary for Turing instability with the introduction of cross-diffusion. The amplitude equations at the critical value of Turing bifurcation are derived using the multiple scales method, which helps to determine the parameter space where certain patterns emerge. Numerical simulations in the Turing instability region and Turing-Hopf region demonstrate the variety of patterns that the system can exhibit, and different initial conditions are employed to enhance understanding of the complex patterns.