Observation and control in models of population genetics

Mathematical systems theory and optimal control have been mostly developed in the context of engineering. In this paper it is shown how these techniques can be applied in population genetics. Based on the classical Fisher's selection model, first a very natural monitoring problem is studied: can the change of the genetic state of a population (described in terms of allele frequencies) be uniquely recovered from the observation of the frequencies of certain phenotypes? We give sufficient conditions for a positive answer to this question in a typical case of heterosis (when mixed genotypes are better than the pure ones, implying stable coexistence of all allele types). The second question is: How to effectively estimate the genetic composition of the population from phenotypic observation? The answer is observer design, which is carried out for two different dominance structures, determining the manifestation of the genetic state. In a model of artificial selection, we show how the population can be steered into equilibrium where maximal mean fitness is attained. Finally, the application of the above methodology is also extended to selection-mutation models, where both fitness parameters and mutation rates are controlled.