Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model

Mostafa Bendahmane, Ricardo Ruiz-Baier, Canrong Tian

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16 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1441-1465
Number of pages25
JournalJournal of Mathematical Biology
Issue number6
Publication statusPublished - 1 May 2016
Externally publishedYes


  • Amplitude equations
  • Cross-diffusion
  • Finite volume approximation
  • Fully adaptive multiresolution
  • Linear stability
  • Lévy flights
  • Pattern formation
  • Super-diffusion
  • Turing instability

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