Do biological species exist as mathematical solutions?

According to Darwin, biological species can be characterized as groups of individuals with similar morphological characteristics. If we consider humans and take only one such characteristics, for example, their height, then the population can be described by the normal distribution. Such distributions considered for any biological species and their morphological parameters are relatively stable and can be considered as stationary in appropriate time scale. Therefore, we can formulate the question whether population distributions can be described as stable stationary solutions of some relevant models. However, it appears that conventional population models do not have such solutions. In this lecture, we will derive a model for a population distribution with respect to the genotype (and not phenotype, i.e., morphology). We will study the existence and stability of solutions of this equation, in particular, of normal distributions. We will conclude with some biological interpretations including the relation between the genotype and the phenotype.
Joint work with B. Peña and S. Trofimchuk.