Turing conditions for pattern forming systems on evolving manifolds

The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises a variety settings, including applications developmental biology, spatial ecology, and experimental chemistry. Analyzing such is complicated, as there strong dependence any spatially homogeneous base states time, the resulting structure linearized perturbations used to determine onset instability inherently non-autonomous. We obtain general conditions for diffusion driven which evolve terms time-evolution Laplace-Beltrami spectrum domain functions specify evolution. Our results give sufficient diffusive phrased differential inequalities are both versatile straightforward implement, despite generality studied problem. These generalize large number known literature, algebraic commonly criterion Turing static domains, approximate asymptotic valid specific types growth, domains. demonstrate our with different evolution laws, particular show how insight can be gained even when changes rapidly state oscillatory, case Turing-Hopf instabilities. Extensions higher-order also included way demonstrating approach.