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Biased random walks and propagation failure.

Vicenç Méndez1, Sergei Fedotov, Daniel Campos

  • 1Grup de Física Estadística, Departament de Física, Facultat de Ciències. Edicifi Cc, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Spain.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 16, 2007
PubMed
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This study determines the critical reaction rate for invasive front propagation in biased random walks. The critical rate, dependent on waiting time and jump distribution, is underestimated by diffusion approximations, with invasion succeeding if the reaction rate exceeds jump frequency.

Area of Science:

  • Statistical Physics
  • Mathematical Biology
  • Complex Systems

Background:

  • Invasive front propagation is crucial in fields like ecology and epidemiology.
  • Understanding the conditions for sustained invasion is key to predicting spread.
  • Non-Markovian processes and biased movement complicate invasion dynamics.

Purpose of the Study:

  • To derive the critical reaction rate sustaining invasive front propagation.
  • To analyze the influence of non-Markovian dynamics and bias on invasion.
  • To compare analytical predictions with diffusion approximations.

Main Methods:

  • Analytical derivation using the Hamilton-Jacobi equation for the mean-field equation.
  • Analysis of non-Markovian biased random walks with reactions.

Related Experiment Videos

  • Comparison with diffusion approximations and validation through numerical simulations.

    • Main Results:

    • The critical reaction rate depends on mean waiting time and jump length distribution.
    • Diffusion approximations consistently underestimate the critical reaction rate.
    • Invasion is guaranteed if the reaction rate exceeds the jump frequency, irrespective of bias.

    Conclusions:

    • The study provides a precise condition for sustained invasion in complex random walks.
    • Non-Markovian effects and bias significantly alter invasion dynamics compared to diffusion models.
    • Analytical predictions are robustly supported by numerical simulations.