The evolution of slow dispersal rates: a reaction diffusion model

Abstract. We consider n phenotypes of a species in a continuous but heterogeneous environment. It is assumed that the phenotypes differ only in their diffusion rates. With haploid genetics and a small rate of mutation, it is shown that the only nontrivial equilibrium is a population dominated by the slowest diffusing phenotype. We also prove that if there are only two possible phenotypes, then this equilibrium is a global attractor and conjecture that this is true in general. Numerical simulations supporting this conjecture and suggesting that this is a robust phenomenon are also discussed.

[1]  R. Teague.  A model of migration modification. , 1977, Theoretical population biology.

[2]  Sze-Bi Hsu,et al.  Competitive exclusion and coexistence for competitive systems on ordered Banach spaces , 1996 .

[3]  Alan Hastings,et al.  Can spatial variation alone lead to selection for dispersal , 1983 .

[4]  Pierre-Henri Gouyon,et al.  Metapopulation Genetics and the Evolution of Dispersal , 1995, The American Naturalist.

[5]  M. Hirsch.  Stability and convergence in strongly monotone dynamical systems. , 1988 .

[6]  Xavier Mora,et al.  Semilinear parabolic problems define semiflows on ^{} spaces , 1983 .

[7]  V. Hutson,et al.  LIMIT BEHAVIOUR FOR A COMPETING SPECIES PROBLEM WITH DIFFUSION , 1995 .

[8]  J. Wloka,et al.  Partial differential equations , 1987 .

[9]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[10]  M. Moody.  The evolution of migration in subdivided populations. I. Haploids , 1988 .

[11]  C. Conley.  Isolated Invariant Sets and the Morse Index , 1978 .

[12]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[13]  J. McGregor,et al.  Towards a theory of the evolution of modifier genes. , 1974, Theoretical population biology.

[14]  M. Feldman,et al.  Selection for migration modification. , 1973, Genetics.

[15]  Michael L. Johnson,et al.  Evolution of Dispersal: Theoretical Models and Empirical Tests Using Birds and Mammals , 1990 .

[16]  R. Bürger.  Mutation-selection models in population genetics and evolutionary game theory , 1989 .

[17]  Robert Stephen Cantrell,et al.  Permanence in ecological systems with spatial heterogeneity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[18]  J. M. Ball,et al.  GEOMETRIC THEORY OF SEMILINEAR PARABOLIC EQUATIONS (Lecture Notes in Mathematics, 840) , 1982 .

[19]  Simon A. Levin,et al.  Dispersal in patchy environments: The effects of temporal and spatial structure , 1991 .

[20]  Alan C. Lazer,et al.  On an abstract competition model and applications , 1991 .

[21]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[22]  A. Ōkubo,et al.  Di?usion and ecological problems: mathematical models , 1980 .

[23]  N. Britton.  Reaction-diffusion equations and their applications to biology. , 1989 .

[24]  H. Bruce Stewart,et al.  Generation of analytic semigroups by strongly elliptic operators under general boundary conditions , 1980 .

[25]  T. Nagylaki.  Introduction to Theoretical Population Genetics , 1992 .

[26]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[27]  Anthony W. Leung,et al.  Systems of Nonlinear Partial Differential Equations , 1989 .