| Affiliation: | (1) Department of Applied Mathematics, The University of Sheffield, Sheffield, S3 7RH, U.K;(2) Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile;(3) Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, U.S.A;(4) Department of Applied Mathematics, The University of Sheffield, Sheffield, S3 7RH, U.K |
| Abstract: | A non-local model for dispersal with continuous time and space is carefully justified and discussed. The necessary mathematical background is developed and we point out some interesting and challenging problems. While the basic model is not new, a  spread parameter (effectively the width of the dispersal kernel) has been introduced along with a conventional rate paramter, and we compare their competitive advantages and disadvantages in a spatially heterogeneous environment. We show that, as in the case of reaction-diffusion models, for fixed spread slower rates of diffusion are always optimal. However, fixing the dispersal rate and varying the spread while assuming a constant cost of dispersal leads to more complicated results. For example, in a fairly general setting given two phenotypes with different, but small spread, the smaller spread is selected while in the case of large spread the larger spread is selected.S. Martinez was partially supported by Fondecyt 1020126 and Fondecyt Lineas Complementarias 8000010. K. Mischaikow was supported in part by NSF Grant DMS 0107396. Key words or phases: Non-local dispersal – Integral kernel – Evolution of dispersal |
