The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model |
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Authors: | Paul R Levitt |
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Institution: | Department of Sociology, Harvard University, Cambridge, Massachusetts 02138 U.S.A. |
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Abstract: | The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if
, where s 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit. |
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