Equilibrium, observability and controllability in selection-mutation models

In this paper we shortly discuss the problem of the equilibrium in the well-known Fisher type selection-mutation model, also providing a formula for particular three-allele models. The considered continuous-time dynamics is a known extension of the classical model of natural selection given by Fisher. We also extend the existing investigation of the observability of Fisher's model to the case when another evolutionary factor, mutation is also present. Moreover, we prove a result of technical character, which makes it possible to apply the methodology of nonlinear systems with invariant manifold, to models of artificial selection. For an illustration, a class of three-allele systems is presented in which the controllability into equilibrium is guaranteed without any condition on the biological parameters.