Journal of Mathematical Biology

, Volume 72, Issue 6, pp 1441–1465 | Cite as

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

  • Mostafa Bendahmane
  • Ricardo Ruiz-Baier
  • Canrong Tian


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.


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

Mathematics Subject Classification

Primary 35B35 Secondary 35B40 47D20 



RRB gratefully acknowledges the support by the Swiss National Science Foundation through the research Grant PP00P2_144922; and CT acknowledges partial support by the PRC Grant NSFC 11201406 and by the Qinglan Project. Finally, we thank the helpful remarks by an anonymous referee, which resulted in substantial improvements to the initial version of this manuscript.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Mostafa Bendahmane
    • 1
  • Ricardo Ruiz-Baier
    • 2
  • Canrong Tian
    • 3
  1. 1.Institut de Mathématiques de BordeauxUniversité VictorBordeaux CedexFrance
  2. 2.Institute of Earth SciencesUniversity of LausanneLausanneSwitzerland
  3. 3.Department of Basic SciencesYancheng Institute of TechnologyYanchengChina

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