In this talk, I will present generalised systems of non-autonomous reaction-diffusion equations posed on growing domains. Under suitable assumptions on the evolution law of the domain growth, by using asymptotic analysis, again under suitable biological assumptions, I will demonstrate the generalisation of Turing diffusion-driven instability theory on growing domains. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions; (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Furthermore, by introducing cross-diffusion, reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion and/or domain evolution.
To confirm theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. To support theoretical predictions, I will present evolving finite element solutions exhibiting patterns generated by (i) an activator-activator, and (ii) a short-range inhibition, long-range activation reaction-diffusion model. Such patterns do not exist in the absence of either domain growth or/and cross-diffusion. These theoretical findings are ahead of current experimental hypotheses for pattern formation due to the complex nature of designing experiments on growing domains.
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