# American Institute of Mathematical Sciences

2010, 7(1): 17-36. doi: 10.3934/mbe.2010.7.17

## Evolution of dispersal and the ideal free distribution

 1 Department of Mathematics, University of Miami, P. O . Box 249085, Coral Gables, FL 33124-4250, United States 2 Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210

Received  April 2009 Revised  August 2009 Published  January 2010

A general question in the study of the evolution of dispersal is what kind of dispersal strategies can convey competitive advantages and thus will evolve. We consider a two species competition model in which the species are assumed to have the same population dynamics but different dispersal strategies. Both species disperse by random diffusion and advection along certain gradients, with the same random dispersal rates but different advection coefficients. We found a conditional dispersal strategy which results in the ideal free distribution of species, and show that it is a local evolutionarily stable strategy. We further show that this strategy is also a global convergent stable strategy under suitable assumptions, and our results illustrate how the evolution of conditional dispersal can lead to an ideal free distribution. The underlying biological reason is that the species with this particular dispersal strategy can perfectly match the environmental resource, which leads to its fitness being equilibrated across the habitats.
Citation: Robert Stephen Cantrell, Chris Cosner, Yuan Lou. Evolution of dispersal and the ideal free distribution. Mathematical Biosciences & Engineering, 2010, 7 (1) : 17-36. doi: 10.3934/mbe.2010.7.17
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