  We study a family of selection-mutation models of a sexual population
structured by a phenotypical trait. The main feature of these models is the
asymmetric trait heredity or fecundity between the parents : we assume that
each individual inherits mostly its trait from the female or that the trait
acts on the female fecundity but does not affect male. Following previous works
inspired from principles of adaptive dynamics, we rescale time and assume that
mutations have limited effects on the phenotype. Our goal is to study the
asymptotic behavior of the population distribution. We derive non-extinction
conditions and BV estimates on the total population. We also obtain Lipschitz
estimates on the solutions of Hamilton-Jacobi equations that arise from the
study of the population distribution concentration at fittest traits.
Concentration results are obtained in some special cases by using a Lyapunov
functional.
