# Evolution of dispersal in open advective environments

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## Abstract

We consider a two-species competition model in a one-dimensional advective environment, where individuals are exposed to unidirectional flow. The two species follow the same population dynamics but have different random dispersal rates and are subject to a net loss of individuals from the habitat at the downstream end. In the case of non-advective environments, it is well known that lower diffusion rates are favored by selection in spatially varying but temporally constant environments, with or without net loss at the boundary. We consider several different biological scenarios that give rise to different boundary conditions, in particular hostile and “free-flow” conditions. We establish the existence of a critical advection speed for the persistence of a single species. We derive a formula for the invasion exponent and perform a linear stability analysis of the semi-trivial steady state under free-flow boundary conditions for constant and linear growth rate. For homogeneous advective environments with free-flow boundary conditions, we show that populations with higher dispersal rate will always displace populations with slower dispersal rate. In contrast, our analysis of a spatially implicit model suggest that for hostile boundary conditions, there is a unique dispersal rate that is evolutionarily stable. Nevertheless, both scenarios show that unidirectional flow can put slow dispersers at a disadvantage and higher dispersal rate can evolve.

## Keywords

Evolution of dispersal Advective environments Persistence Invasion analysis Reaction–diffusion–advection## Mathematics Subject Classification (2010)

35K57 92D25## Notes

### Acknowledgments

We thank Odo Diekmann and two anonymous reviewers for their thorough reading of the manuscript and for excellent suggestions. YL was supported by the NSF Grant DMS-1021179 and has been supported in part by the Mathematical Biosciences Institute and the National Science Foundation under grant DMS-0931642. FL was supported by an NSERC Discovery grant and an Early Researcher Award from the MRI, Ontario. A visit of YL at the University of Ottawa was supported by the Mprime Network on Biological Invasions and Dispersal Research (see http://www.unb.ca/bid/bid.php).

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