title: Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model.
abstract: In this paper we analyze the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population super-diffusion. First, we study how cross super-diffusion influences the formation of spatial patterns: a linear stability analysis is carried out, showing that cross super-diffusion triggers Turing instabilities, whereas classical (self) super-diffusion does not. In addition we perform 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 method that employs shifted Grünwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators. The scheme is 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.
authors.affiliation: Institut de Mathématiques de Bordeaux, Université Victor, Segalen Bordeaux 2, 33076, Bordeaux Cedex, France.
authors.forename: Mostafa
authors.initials: M
authors.lastname: Bendahmane
authors.affiliation: Institute of Earth Sciences, University of Lausanne, 1015, Lausanne, Switzerland.
authors.forename: Ricardo
authors.initials: R
authors.lastname: Ruiz-Baier
authors.affiliation: Department of Basic Sciences, Yancheng Institute of Technology, Yancheng, 224003, China. tiancanrong@163.com.
authors.forename: Canrong
authors.initials: C
authors.lastname: Tian
mesh_terms: D000818:Animals; D058872:Epidemics / Q000706:statistics & numerical data; D020387:Food Chain; D006801:Humans; D016014:Linear Models; D055641:Mathematical Concepts; D008954:Models, Biological; D017711:Nonlinear Dynamics; D016011:Normal Distribution; D011157:Population Dynamics / Q000706:statistics & numerical data*; D011235:Predatory Behavior
keywords: Amplitude equations; Cross-diffusion; Finite volume approximation; Fully adaptive multiresolution; Linear stability; Lévy flights; Pattern formation; Super-diffusion; Turing instability