Phenotypic evolution of hermaphrodites

We consider finite, phenotype-structured population of hermaphrodites, and build an individual based model which describes interactions between the individuals. The model contains such elements as mating of individuals (random or assortative), inheritance of phenotypic traits including mutations, intra-specific competition and mortality. Here offspring’s phenotype depends on parential traits. We consider the limit passage with the number of individuals to infinity, what leads us to continuous distribution of phenotypic traits in the population. The model is described by evolution equation in the space of measures, which contains nonlinear operators. The first of the operators is in charge of mating of individuals and inheritance, the other corresponds to the competition.
The limiting version of the model for random mating is an evolutionary equation, containing bilinear operator. The particular case of the equation is Tjon-Wu equation which appears in the description of the energy distribution of colliding particles. In the case of random mating, under suitable conditions we prove the asymptotic stability result: distribution of the phenotypic traits in the population converges to a stationary distribution. As a by-product we obtain relatively easy proof of Lasota-Traple theorem concerning asymptotic stability of Tjon-Wu equation. Moreover, we show applications of our theorem to some biologically reasonable situations of phenotypic inheritance.