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Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules

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Abstract

We study the dynamics of two competing species in three-patch models and illustrate how the topology of directed river network modules may affect the evolution of dispersal. Each model assumes that patch 1 is at the upstream end, patch 3 is at the downstream end, but patch 2 could be upstream, or middle stream, or downstream, depending on the specific topology of the modules. We posit that individuals are subject to both unbiased dispersal between patches and passive drift from one patch to another, depending upon the connectivity of patches. When the drift rate is small, we show that for all models, the mutant species can invade when rare if and only if it is the slower disperser. However, when the drift rate is large, most models predict that the faster disperser wins, while some predict that there exists one evolutionarily singular strategy. The intermediate range of drift is much more complex: most models predict the existence of one singular strategy, but it may or may not be evolutionarily stable, again depending upon the topology of modules, while one model even predicts that for some intermediate drift rate, singular strategy does not exist and the faster disperser wins the competition.

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Acknowledgements

The authors sincerely thank two anonymous referees for their insightful comments and suggestions. HYJ is partially supported by the NSFC grant No. 11571364. KYL and YL are partially supported by the NSF grant DMS-1853561.

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Authors and Affiliations

  1. Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China

    Hongyan Jiang

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

    King-Yeung Lam & Yuan Lou

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  1. Hongyan Jiang
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  2. King-Yeung Lam
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  3. Yuan Lou
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Correspondence to Yuan Lou.

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Jiang, H., Lam, KY. & Lou, Y. Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules. Bull Math Biol 82, 131 (2020). https://doi.org/10.1007/s11538-020-00803-1

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