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Evolution of dispersal in spatial population models with multiple timescales

Journal of Mathematical Biology volume 80, pages 3–37 (2020)Cite this article

Abstract

We study the evolutionary stability of dispersal strategies, including but not limited to those that can produce ideal free population distributions (that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium). The environment is assumed to be variable in space but constant in time. We assume that there is a separation of times scales, so that dispersal occurs on a fast timescale, evolution occurs on a slow timescale, and population dynamics and interactions occur on an intermediate timescale. Starting with advection–diffusion models for dispersal without population dynamics, we use the large time limits of profiles for population distributions together with the distribution of resources in the environment to calculate growth and interaction coefficients in logistic and Lotka–Volterra ordinary differential equations describing population dynamics. We then use a pairwise invasibility analysis approach motivated by adaptive dynamics to study the evolutionary and/or convergence stability of strategies determined by various assumptions about the advection and diffusion terms in the original advection–diffusion dispersal models. Among other results we find that those strategies which can produce an ideal free distribution are evolutionarily stable.

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Acknowledgements

Some of the ideas in this paper arose from discussions in workshops at the Banff International Research Station: Multiscale Analysis of Self-Organization in Biology (09w5070) July 12–July 17, 2009, organizers B. Perthame and T. Hillen, and Emerging Challenges at the Interface of Mathematics, Environmental Science and Spatial Ecology (11w5106) July 3–July 08, 2011, organizers R. S. Cantrell, R. Holt, and M. A. Lewis. CC thanks Odo Diekmann for discussions that helped motivate this paper.

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Authors and Affiliations

  1. Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, People’s Republic of China

    Robert Stephen Cantrell & Yuan Lou

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Robert Stephen Cantrell & Chris Cosner

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Mark A. Lewis

  4. Department of Biological Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Mark A. Lewis

  5. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Yuan Lou

Authors
  1. Robert Stephen Cantrell
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  2. Chris Cosner
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  3. Mark A. Lewis
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  4. Yuan Lou
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Corresponding author

Correspondence to Chris Cosner.

Additional information

The paper should be listed for the special issue S.I. : In honor of Alan Hastings’ 65th birthday.

Research partially supported by NSF Grants DMS-1118623 and DMS-1514752 (RSC, CC), NSF Grant DMS-1411476 (YL), NSFC Grant No. 11571364 (YL), and an NSERC Discovery Grant (ML)

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Cantrell, R.S., Cosner, C., Lewis, M.A. et al. Evolution of dispersal in spatial population models with multiple timescales. J. Math. Biol. 80, 3–37 (2020). https://doi.org/10.1007/s00285-018-1302-2

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  • DOI: https://doi.org/10.1007/s00285-018-1302-2

Keywords

  • Multiple timescales
  • Ideal free distribution
  • Evolutionarily stable strategy
  • Evolution of dispersal

Mathematics Subject Classification

  • 92D15
  • 92D20