The stabilizing effect of growth on pattern formation
Résumé
Based on abundant numerical and experimental evidence, it has been conjectured that growth should have some kind of stabilizing effect on pattern formation. In this paper we answer affirmatively this question: under an isotropic regime, the growth shifts the eigenvalues of the reaction-diffusion system towards the left in the complex plane. Since the real parts of the eigenvalues are smaller, we can can be interpret this fact as a gain of stability. We also prove that growth enhances the possibility of a solution to be global: a local solution (i.e. defined up to a finite time) has more chances to be global (i.e. to exist for all times) on a growing manifold than on a fixed manifold. Moreover, if the growth is fast enough we show that the solutions are always global, regardless to the form of the nonlinearity. We finish with a discussion of the results, showing that the classical linear stability analysis for bifurcations apply to this framework, and pointing out the possible applications of our results to regulatory dynamics in pattern formation, embryogenesis and tumor growth.
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