Evolutionarily stable strategy in advective patchy environments

The current study is motivated by a recent work by Jiang, Lam and Lou [Bull. Math. Biol., 2020, Paper No. 131, 42pp], where, to discuss the evolution of dispersal, the authors considered the case of three patches, proposed three models by considering different topology of river networks and found that the slower or faster diffuser may win, or there may appear the evolutionarily singular strategy, depending on given modeling parameters. However, the issue whether there is evolutionarily stable strategy (ESS, a central concept in evolution game theory) is unknown. In this paper, focusing on "Model I" proposed by them, we give a confirmed answer to this unsolved problem, namely, there does exist ESS. Some idea developed in this paper is also useful to treat the other two models proposed by Jiang, Lam and Lou.