# American Institute of Mathematical Sciences

April  2019, 24(4): 1875-1887. doi: 10.3934/dcdsb.2018245

## Evolutionarily stable dispersal strategies in a two-patch advective environment

 1 School of Science, Xi'an University of Architecture and Technology, Xi'an 710055, China 2 Institute for Mathematical Science, Renmin University of China, Beijing 100872, China

* Corresponding author: Yihao Fang

Received  December 2017 Revised  March 2018 Published  August 2018

Fund Project: Jing-jing Xiang is partially supported by the Research Foundation of Education Bureau of Shaanxi Province (15JK1433), "The mathematical modeling and analysis of disease spreading in media". Yihao Fang is partially supported by the National Natural Science Foundation of China(11571364).

Two-patch models are used to mimic the unidirectional movement of organisms in continuous, advective environments. We assume that species can move between two patches, with patch 1 as the upper stream patch and patch 2 as the downstream patch. Species disperse between two patches with the same rate, and species in patch 1 is transported to patch 2 by drift, but not vice versa. We also mimic no-flux boundary conditions at the upstream and zero Dirichlet boundary conditions at the downstream. The criteria for the persistence of a single species is established. For two competing species model, we show that there is an intermediate dispersal rate which is evolutionarily stable. These results support the conjecture in [6], initially proposed for reaction-diffusion models with continuous advective environments.

Citation: Jing-Jing Xiang, Yihao Fang. Evolutionarily stable dispersal strategies in a two-patch advective environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1875-1887. doi: 10.3934/dcdsb.2018245
##### References:

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##### References:
Illustration of Lemma 3.6 for $0\leq q < 1$
Illustration of Lemma 3.7 for $1\leq q < 5/4$
PIP for $0<q < 1$ with the sign of $\lambda_1$
PIP for $1\leq q < 5/4$ with the sign of $\lambda_1$
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