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