Dynamics of Dirac concentrations in the evolution of quantitative alleles with sexual reproduction

Abstract

A proper understanding of the links between varying gene expression levels and complex trait adaptation is still lacking, despite recent advances in sequencing techniques leading to new insights on their importance in some evolutionary processes. This calls for extensions of the continuum-of-alleles framework first introduced by Kimura (1965 Proc. Natl Acad. Sci. USA 54 731–36) that bypass the classical Gaussian approximation. Here, we propose a novel mathematical framework to study the evolutionary dynamics of quantitative alleles for sexually reproducing populations under natural selection and competition through an integro-differential equation. It involves a new reproduction operator which is nonlinear and nonlocal. This reproduction operator is different from the infinitesimal operator used in other studies with sexual reproduction because of different underlying genetic structures. In an asymptotic regime where initially the population has a small phenotypic variance, we analyse the long-term dynamics of the phenotypic distributions according to the methodology of small variance (Diekmann et al 2005 Theor. Popul. Biol. 67 257–71). In particular, we prove that the reproduction operator strains the limit distribution to be a product measure. Under some assumptions on the limit equation, we show that the population remains monomorphic, that is the phenotypic distribution remains concentrated as a moving Dirac mass. Moreover, in the case of a monomorphic distribution, we derive a canonical equation describing the dynamics of the dominant alleles.