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A Non-local Model Illustrating Replicator Dynamics

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Non-Local Partial Differential Equations for Engineering and Biology

Part of the book series: Mathematics for Industry ((MFI,volume 31))

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Abstract

The current chapter discusses a utilization from the field of evolutionary game dynamics and in particular from its subarea called replicator dynamics . Considering an infinite continuous strategy space, which for example might be considered as the sampling space of a continuously varying trait of a biological population, as well as payoff functions of Gaussian type we build up a non-local degenerate parabolic problem. As it is appropriate for degenerate problems, a regularized approximation is constructed and then some a priori estimates for its solutions are obtained. Using the derived estimates, we prove that solutions converge to the trivial solution if the initial population is small, whereas they undergo a blow-up in finite time if the initial population is large. In particular, in the latter case, it is shown that the blow-up set coincides with the whole strategy space, i.e. the finite-time blow-up is global.

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Authors and Affiliations

  1. Department of Mathematics, University of Chester, Chester, UK

    Nikos I. Kavallaris

  2. Department of Mathematics, Osaka University, Osaka, Japan

    Takashi Suzuki

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  1. Nikos I. Kavallaris
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  2. Takashi Suzuki
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Correspondence to Nikos I. Kavallaris .

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Kavallaris, N.I., Suzuki, T. (2018). A Non-local Model Illustrating Replicator Dynamics. In: Non-Local Partial Differential Equations for Engineering and Biology. Mathematics for Industry, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-319-67944-0_6

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  • DOI: https://doi.org/10.1007/978-3-319-67944-0_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67942-6

  • Online ISBN: 978-3-319-67944-0

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