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Bulletin of Mathematical Biology

, Volume 69, Issue 7, pp 2281–2297 | Cite as

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

  • Boris Baeumer
  • Mihály Kovács
  • Mark M. Meerschaert
Original Article

Abstract

Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. Integro-Difference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1<α ≤ 2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.

Keywords

Reproduction-dispersal equation Integro-difference equation Fractional derivative Anomalous diffusion Operator splitting 

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Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  • Boris Baeumer
    • 1
  • Mihály Kovács
    • 1
  • Mark M. Meerschaert
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of OtagoDunedinNew Zealand
  2. 2.Department of Mathematics and StatisticsMichigan State UniversityEast LansingUSA

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