Turing diffusion-driven instability conditions for reaction-diffusion systems on continuously deforming domains

Anotida Madzvamuse
Mathematics, University of Sussex

(September 8, 2008 10:30 AM - 11:30 AM)

Turing diffusion-driven instability conditions for reaction-diffusion systems on continuously deforming domains

Abstract

In this talk, I will extend the Turing diffusion-driven instability conditions for reaction- diffusion systems from fixed domains to continuously deforming or growing domains for some growth profiles satisfying the condition that the divergence of the domain velocity is a constant. By using the arbitrary Lagrangian-Eulerian formulation (ALE), the model equations on a continuously deforming domain are transformed to a fixed domain at each time, resulting in a conservative system. Linearising the conservative ALE formulation around a time-independent solution u_s we show that u_s is a solution of a system of nonlinear algebraic equations. We further derive and prove the conditions that generalise the classic Turing parameter space inequalities for the case of growing domains and we show that these conditions are a function of not only the model parameters in the reaction terms but also are a function of the divergence of the domain velocity. A fundamental structural difference between our diffusion-driven instability conditions and those obtained on fixed domains, is that our results do not enforce cross/pure kinetics thereby demonstrating that domain growth enlarges the potential range of kinetics which can lead to patterning.