Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion

This article presents stability analytical results of a two compo-nent reaction-diffusion system with linear cross-diffusion posed on continuouslyevolving domains. First the model system is mapped from a continuously evolv-ing domain to a reference stationary frame resulting in a system of partialdifferential equations with time-dependent coefficients. Second, by employingappropriately asymptotic theory, we derive and prove cross-diffusion-driven in-stability conditions for the model system for the case of slow, isotropic domaingrowth. Our analytical results reveal that unlike the restrictive diffusion-driveninstability conditions on stationary domains, in the presence of cross-diffusioncoupled with domain evolution, it is no longer necessary to enforce cross norpure kinetic conditions. The restriction toactivator-inhibitorkinetics to inducepattern formation on a growing biological system is no longer a requirement.Reaction-cross-diffusion models with equal diffusion coefficients in the principalcomponents as well as those of theshort-range inhibition, long-range activa-tionandactivator-activatorform can generate patterns only in the presence ofcross-diffusion coupled with domain evolution. To confirm our theoretical find-ings, detailed parameter spaces are exhibited for the special cases of isotropicexponential, linear and logistic growth profiles. In support of our theoreticalpredictions, we present evolving or moving finite element solutions exhibitingpatterns generated by ashort-range inhibition, long-range activationreaction-diffusion model with linear cross-diffusion in the presence of domain evolution.