Travelling waves in near-degenerate bistable competition models

E. O. Alzahrani, F. A. Davidson, N. Dodds

    Research output: Contribution to journalArticle

    11 Citations (Scopus)

    Abstract

    We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.
    Original languageEnglish
    Pages (from-to)13-35
    Number of pages23
    JournalMathematical Modelling of Natural Phenomena
    Volume5
    Issue number5
    DOIs
    Publication statusPublished - 2010

    Fingerprint

    Competition Model
    Traveling Wave
    Diffusion Coefficient
    Trivial
    Limiting
    Competing Species
    Bistable System
    Traveling Wave Solutions
    Kinetics
    Reaction-diffusion System
    Energy Function

    Keywords

    • Competition
    • Reaction-diffusion
    • Free energy
    • Bistable
    • Travelling waves

    Cite this

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    abstract = "We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.",
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    Travelling waves in near-degenerate bistable competition models. / Alzahrani, E. O.; Davidson, F. A.; Dodds, N.

    In: Mathematical Modelling of Natural Phenomena, Vol. 5, No. 5, 2010, p. 13-35.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - Travelling waves in near-degenerate bistable competition models

    AU - Alzahrani, E. O.

    AU - Davidson, F. A.

    AU - Dodds, N.

    PY - 2010

    Y1 - 2010

    N2 - We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.

    AB - We study a class of bistable reaction-diffusion systems used to model two competing species. Systems in this class possess two uniform stable steady states representing semi-trivial solutions. Principally, we are interested in the case where the ratio of the diffusion coefficients is small, i.e. in the near-degenerate case. First, limiting arguments are presented to relate solutions to such systems to those of the degenerate case where one species is assumed not to diffuse. We then consider travelling wave solutions that connect the two stable semi-trivial states of the non-degenerate system. Next, a general energy function for the full system is introduced. Using this and the limiting arguments, we are able to determine the wave direction for small diffusion coefficient ratios. The results obtained only require knowledge of the system kinetics.

    KW - Competition

    KW - Reaction-diffusion

    KW - Free energy

    KW - Bistable

    KW - Travelling waves

    U2 - 10.1051/mmnp/20105502

    DO - 10.1051/mmnp/20105502

    M3 - Article

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    SP - 13

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    JO - Mathematical Modelling of Natural Phenomena

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    SN - 0973-5348

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    ER -