Effects of cross-diffusion on Turing patterns in a reaction-diffusion Schnakenberg model

摘要

In this paper the Turing pattern formation mechanism of a two componentreaction-diffusion system modeling the Schnakenberg chemical reaction coupledto linear cross-diffusion terms is studied. The linear cross-diffusion termsfavors the destabilization of the constant steady state and the mechanism ofpattern formation with respect to the standard linear diffusion case, as shownin Madzvamuse et al. (J. Math. Biol. 2014). Since the subcritical Turingbifurcations of reaction-diffusion systems lead to spontaneous onset of robust,finite-amplitude localized patterns, here a detailed investigation of theTuring pattern forming region is performed to show how the diffusioncoefficients for both species (the activator and the inhibitor) influence theoccurrence of supercritical or subcritical bifurcations. The weakly nonlinear(WNL) multiple scales analysis is employed to derive the equations for theamplitude of the Turing patterns and to distinguish the supercritical and thesubcritical pattern region, both in 1D and 2D domains. Numerical simulationsare employed to confirm the WNL theoretical predictions through which aclassification of the patterns (squares, rhombi, rectangle and hexagons) isobtained. In particular, due to the hysteretic nature of the subcriticalbifurcation, we observe the phenomenon of pattern transition from rolls tohexagons, in agreement with the bifurcation diagram.