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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model.

Mostafa Bendahmane1, Ricardo Ruiz-Baier2, Canrong Tian3

  • 1Institut de Mathématiques de Bordeaux, Université Victor, Segalen Bordeaux 2, 33076, Bordeaux Cedex, France.

Journal of Mathematical Biology
|July 30, 2015
PubMed
Summary
This summary is machine-generated.

Introducing fractional-in-space operators into Lotka-Volterra models drives Turing instabilities and pattern formation. A novel adaptive finite volume method efficiently simulates these super-diffusion dynamics.

Keywords:
Amplitude equationsCross-diffusionFinite volume approximationFully adaptive multiresolutionLinear stabilityLévy flightsPattern formationSuper-diffusionTuring instability

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Area of Science:

  • Mathematical Biology
  • Computational Science
  • Nonlinear Dynamics

Background:

  • Population dynamics are often modeled using Lotka-Volterra equations.
  • Super-diffusion describes anomalous spreading patterns in populations.
  • Fractional calculus offers advanced tools for modeling complex diffusion phenomena.

Purpose of the Study:

  • To investigate the impact of fractional-in-space operators on Lotka-Volterra competitive models.
  • To analyze pattern formation and stability in population super-diffusion dynamics.
  • To develop and validate an efficient numerical method for fractional diffusion models.

Main Methods:

  • Linear stability analysis to identify Turing instabilities.
  • Weakly nonlinear analysis to derive amplitude equations.
  • Development of an adaptive multiresolution finite volume method with shifted Grünwald approximations.

Main Results:

  • Cross super-diffusion, unlike classical self super-diffusion, induces Turing instabilities.
  • Amplitude equations confirm the stability of Turing steady states.
  • Numerical simulations near instability boundaries validate analytical predictions.

Conclusions:

  • Fractional-in-space operators are crucial for understanding complex spatial patterns in population dynamics.
  • The proposed finite volume method offers computational efficiency for fractional diffusion models.
  • This work bridges theoretical analysis and numerical simulation in ecological modeling.