Turing pattern dynamics and adaptive discretization for a superdiffusive Lotka-Volterra system
Résumé
We focus our attention on the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population superdiffusion. First, we address the weak solvability of the coupled problem employing the Faedo-Galerkin method and compactness arguments. In addition, we are interested in how cross superdiffusion influences the formation of spatial patterns: a linear stability analysis has been carried out, showing that cross superdiffusion triggers Turing instabilities, whereas classical self superdiffusion suppresses Turing instability. We have also performed a weakly nonlinear analysis yielding a system of amplitude equations, whose study shows the stability of Turing steady states. A second goal of this contribution is to propose a fully adaptive multiresolution finite volume (MRFV) method that employs shifted Grunwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators, and aimed at efficient dynamic mesh adaptation and substantial savings in computational burden. A numerical simulation of the model was performed near the instability boundaries, confirming the behavior predicted by our analysis.
Mots clés
Domaines
Mathématiques [math]
Origine : Fichiers produits par l'(les) auteur(s)
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