# American Institute of Mathematical Sciences

December  2014, 19(10): 3319-3340. doi: 10.3934/dcdsb.2014.19.3319

## Evolutionarily stable diffusive dispersal

 1 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada, Canada 2 Department of Mathematical and Statistical Sciences, Department of Biological Sciences, University of Alberta, Edmonton AB T6G 2G1, Canada

Received  July 2013 Revised  December 2013 Published  October 2014

We use an evolutionary approach to find most appropriate'' dispersal models for ecological applications. From a random walk with locally or nonlocally defined transition probabilities we derive a family of diffusion equations. We assume a monotonic dependence of its diffusion coefficient on the local population fitness and search for a model within this class that can invade populations with other dispersal type from the same class but is not invadable itself. We propose an optimization technique using numerically obtained principal eigenvalue of the invasion problem and obtain two candidates for evolutionary stable dispersal strategy: Fokker-Planck equation with diffusion coefficient decreasing with fitness and Attractive Diffusion equation (Okubo and Levin, 2001) with diffusion coefficient increasing with fitness. For FP case the transition probabilities are defined by the departure point and for AD case by the destination point. We show that for the case of small spatial variability of the population growth rate both models are close to the model for ideal free distribution by Cantrell et al. (2008).
Citation: Alex Potapov, Ulrike E. Schlägel, Mark A. Lewis. Evolutionarily stable diffusive dispersal. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3319-3340. doi: 10.3934/dcdsb.2014.19.3319
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##### References:
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