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Persistence, Competition, and Evolution

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The Dynamics of Biological Systems

Part of the book series: Mathematics of Planet Earth ((MPE,volume 4))

Abstract

In this chapter we discuss some reaction–diffusion models for single and multiple populations in spatially heterogeneous environments and advective environments. Our goal is to illustrate some interesting, and perhaps surprising, effects of spatial heterogeneity and diffusion on the population dynamics. Specific topics include the logistic model, linear eigenvalue problem with indefinite weight, Lotka–Volterra competition models, reaction–diffusion models in advective environments, and the evolution of dispersal. We will introduce some basic tools for reaction–diffusion equations such as the super-sub solution method, the variational principle for principal eigenvalues, Lyapunov functionals, comparison principles for parabolic equations and systems, etc. Some recent developments will be discussed. In addition, problems with various difficulties ranging from elementary exercises to open research questions will be presented.

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Acknowledgements

We sincerely thank the referee for his comments and suggestions which help improve the presentation. KYL and YL were partially supported by the NSF grant DMS-1411476 and DMS-1853561. Part of the work was done during the visit of YL to the University of Alberta to participate in the 2016 Séminaire de Mathématiques Supérieures: Dynamics of Biological Systems Summer School, and he thanks the organizers for the warm hospitality.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, USA

    King-Yeung Lam & Yuan Lou

  2. Institute for Mathematical Sciences, Renmin University of China, Beijing, People’s Republic of China

    Yuan Lou

Authors
  1. King-Yeung Lam
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  2. Yuan Lou
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Correspondence to Yuan Lou .

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Editors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Arianna Bianchi

  2. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Thomas Hillen

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Mark A. Lewis

  4. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Yingfei Yi

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Lam, KY., Lou, Y. (2019). Persistence, Competition, and Evolution. In: Bianchi, A., Hillen, T., Lewis, M., Yi, Y. (eds) The Dynamics of Biological Systems. Mathematics of Planet Earth, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-030-22583-4_8

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