Analysis of non-autonomous systems of reaction-diffusion on continuously deforming domains and surfaces

In this talk, I will briefly present the derivation of reaction-diffusion systems on continuously deforming domains and surfaces. The models are in the form of non- autonomous systems of partial differentialequations whose analysis is not trivial. Using a Lagrangian formulation, assuming uniform isotropic growth, I will then derive a system of reaction-diffusion systems mapped onto stationary domains, with time- dependent coefficients. By employing asymptotic expansions, I will derive a generalisation of conditions for domain-growth diffusion-driven instability. These conditions describe time-dependent Turing spaces. To validate theoretical findings, I will provide numericalexamples for exponential, linear and logistic growth. Furthermore, I will formulate a new activator-activator reaction-diffusion system that can only give rise to pattern formation in the presence of domain growth. This generalisation of Turing’s theory for pattern formation challenges experimentalists to design new experiments whose domains change continuously in time. Under such an experiment, classical long- range activation and short-range inhibition cease to be the sole paradigm for pattern formation. We no longer require systems to be of the form: activator-inhibitor, instead, more generalised systems can be constructed such as inhibitor-inhibitor reaction kinetics or short-range activation and long range inhibition.