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Evolutionarily Stable and Convergent Stable Strategies in Reaction–Diffusion Models for Conditional Dispersal

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Abstract

We consider a mathematical model of two competing species for the evolution of conditional dispersal in a spatially varying, but temporally constant environment. Two species are different only in their dispersal strategies, which are a combination of random dispersal and biased movement upward along the resource gradient. In the absence of biased movement or advection, Hastings showed that the mutant can invade when rare if and only if it has smaller random dispersal rate than the resident. When there is a small amount of biased movement or advection, we show that there is a positive random dispersal rate that is both locally evolutionarily stable and convergent stable. Our analysis of the model suggests that a balanced combination of random and biased movement might be a better habitat selection strategy for populations.

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Acknowledgements

This research was partially supported by the NSF grant DMS-1021179 and has been supported in part by the Mathematical Biosciences Institute and the National Science Foundation under grant DMS-0931642. The author would like to acknowledge helpful discussions with Frithjof Lutscher and the hospitality of Center for Partial Differential Equations of East China Normal University, Shanghai, China, where part of this work was done.

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Authors and Affiliations

  1. Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 43210, USA

    King-Yeung Lam

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

    Yuan Lou

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

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Lam, KY., Lou, Y. Evolutionarily Stable and Convergent Stable Strategies in Reaction–Diffusion Models for Conditional Dispersal. Bull Math Biol 76, 261–291 (2014). https://doi.org/10.1007/s11538-013-9901-y

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  • DOI: https://doi.org/10.1007/s11538-013-9901-y

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