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Spreading Speeds in Slowly Oscillating Environments

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Abstract

In this paper, we derive exact asymptotic estimates of the spreading speeds of solutions of some reaction-diffusion models in periodic environments with very large periods. Contrarily to the other limiting case of rapidly oscillating environments, there was previously no explicit formula in the case of slowly oscillating environments. The knowledge of these two extremes permits to quantify the effect of environmental fragmentation on the spreading speeds. On the one hand, our analytical estimates and numerical simulations reveal speeds which are higher than expected for Shigesada–Kawasaki–Teramoto models with Fisher-KPP reaction terms in slowly oscillating environments. On the other hand, spreading speeds in very slowly oscillating environments are proved to be 0 in the case of models with strong Allee effects; such an unfavorable effect of aggregation is merely seen in reaction-diffusion models.

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Authors and Affiliations

  1. LATP, Faculté des Sciences et Techniques, Aix-Marseille Université, Avenue Escadrille Normandie-Niemen, 13397, Marseille Cedex 20, France

    François Hamel

  2. Helmholtz Zentrum München, Institut für Biomathematik und Biometrie, Ingolstädter Landstrasse 1, 85764, Neuherberg, Germany

    François Hamel

  3. UR 546 Biostatistique et Processus Spatiaux, INRA, 84000, Avignon, France

    Julien Fayard & Lionel Roques

Authors
  1. François Hamel
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  2. Julien Fayard
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  3. Lionel Roques
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Correspondence to Lionel Roques.

Additional information

The authors are supported by the French “Agence Nationale de la Recherche” within the projects ColonSGS, PREFERED, and URTICLIM (third author). The first author is also indebted to the Alexander von Humboldt Foundation for its support.

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Hamel, F., Fayard, J. & Roques, L. Spreading Speeds in Slowly Oscillating Environments. Bull. Math. Biol. 72, 1166–1191 (2010). https://doi.org/10.1007/s11538-009-9486-7

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