Evolution of Dispersal in a Stream With Better Resources at Downstream Locations

This paper is concerned with a two-species Lotka–Volterra competition patch model over a stream with better resources at downstream locations. Treating one species as the resident species and the other one as a mutant species, we first show that there exist two quantities $\overline{d}$ and $\underline{d}$ depending on the drift rate: if the dispersal rate of the resident species is smaller (respectively, larger) than $\underline{d}$ (respectively, $\overline{d}$), then a rare mutant species can invade only when its dispersal rate is faster (respectively, slower) than the resident species. Then, we show that there exists some intermediate dispersal rate, which is the unique evolutionarily stable strategy for the resident species under certain conditions. Moreover, the global dynamics of the model is obtained, and both competition exclusion and coexistence can occur. Our method for the patch model can be used for the corresponding reaction–diffusion model, and some existing results are improved.