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.

中文翻译：

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两片式模型就足够了吗？河网模块分布和拓扑的演变

我们研究了三斑块模型中两个竞争物种的动态，并说明了定向河网模块的拓扑结构如何影响扩散的演变。每个模型都假设补丁 1 在上游端，补丁 3 在下游端，但补丁 2 可以是上游、中流或下游，具体取决于模块的特定拓扑。我们假设个体会受到补丁之间的无偏分散和从一个补丁到另一个补丁的被动漂移，这取决于补丁的连通性。当漂移率很小时，我们表明对于所有模型，突变物种可以在稀有时入侵，当且仅当它是较慢的分散者。然而，当漂移率较大时，大多数模型预测分散速度较快者获胜，而有些模型预测存在一种进化上的单一策略。