The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E( x) model

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 E(0) − E(1) s > B(1 − h) , 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.