Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules
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Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules Hongyan Jiang1 · King-Yeung Lam2 · Yuan Lou2 Received: 26 June 2020 / Accepted: 7 September 2020 © Society for Mathematical Biology 2020
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. Keywords River network module · Patch model · Network topology · Evolution of dispersal Mathematics Subject Classification 34D20 · 92D15 · 92D40
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Yuan Lou [email protected] Hongyan Jiang [email protected] King-Yeung Lam [email protected]
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Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China
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Department of Mathematics, Ohio State University, Columbus, OH 43210, USA 0123456789().: V,-vol
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H. Jiang et al.
1 Introduction The dynamics of population models in advective habitats such as rivers have received increasing attention in recent years. These studies covered a wide range of topics, including flow reactors (Ballyk et al. 1998), persistence (Lam et al. 2016; Lutscher et al. 2005, 2006, 2007; Pachepsky et al. 2015; Vasilyeva and Lutscher 2011), benthicdrift modeling (Huang et al. 2016; Jin et al. 2019), seasonal environment (Jin and Lewis 2011, 2012; Jin et al. 2014), competition models (Lou et al. 2016, 2019; Vasilyeva and Lutscher 2012a, b; Vasilyeva 2017; Zhao and Zhou 2016; Zhou 2016), Allee effect (Wang and Shi 2019; Wang et al. 2019; Wang and Shi 2020), among others. Organisms in advective environment are often subject to both unbiased dispersal and passive drift (Speirs and Gurney 2001). These two modes of dispersal focus on different niches. On the one hand, passive drift pushes individual to a relatively downstream habitat, which can sometimes be less desirable. e.g. when a river meets the ocean, the downstream end of the river could be an ecological sink for fresh water organisms. On the other hand, unbiased
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