Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

A survey of migration-selection models in population genetics

Pages: 883 - 959, Volume 19, Issue 4, June 2014      doi:10.3934/dcdsb.2014.19.883

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Reinhard B├╝rger - Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria (email)

Abstract: This survey focuses on the most important aspects of the mathematical theory of population genetic models of selection and migration between discrete niches. Such models are most appropriate if the dispersal distance is short compared to the scale at which the environment changes, or if the habitat is fragmented. The general goal of such models is to study the influence of population subdivision and gene flow among subpopulations on the amount and pattern of genetic variation maintained. Only deterministic models are treated. Because space is discrete, they are formulated in terms of systems of nonlinear difference or differential equations. A central topic is the exploration of the equilibrium and stability structure under various assumptions on the patterns of selection and migration. Another important, closely related topic concerns conditions (necessary or sufficient) for fully polymorphic (internal) equilibria. First, the theory of one-locus models with two or multiple alleles is laid out. Then, mostly very recent, developments about multilocus models are presented. Finally, as an application, analysis and results of an explicit two-locus model emerging from speciation theory are highlighted.

Keywords:  Differential equations, recurrence equations, stability, convergence, perturbation theory, evolution, geographic structure, dispersal, recombination.
Mathematics Subject Classification:  Primary: 92D10, 92D15; Secondary: 34D05, 34D30, 37N25, 39A60.

Received: September 2013;      Revised: January 2014;      Available Online: April 2014.