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Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin, 9001, New Zealand

    Boris Baeumer & Mihály Kovács

  2. Department of Mathematics and Statistics, Michigan State University, East Lansing, MI, 48823, USA

    Mark M. Meerschaert

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  1. Boris Baeumer
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  2. Mihály Kovács
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  3. Mark M. Meerschaert
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Correspondence to Boris Baeumer.

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Partially supported by the Marsden fund administered by the Royal Society of New Zealand, NSF grants DMS-0139927, DMS-0417869 and DMS-0706440 and Postdoctoral grant No.  623-2005-5078 of the Swedish Research Council.

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Baeumer, B., Kovács, M. & Meerschaert, M.M. Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels. Bull. Math. Biol. 69, 2281–2297 (2007). https://doi.org/10.1007/s11538-007-9220-2

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