Dynamics of population patch distribution

The problem of selection by the patch population in the absence of information on the utility of the patch, that is, the volume of its energy resources, is considered. This problem relates to the theory of optimal foraging. U. Dieckman proposed an approach to modeling the population patch distribution. The approach is based on a utility function that takes into account the amount of resources in a patch, the population - patch distance, and the measure of information certainty on patch utility. In this case, the Boltzmann distribution is used to describe the population patch distribution. And U. Dieckman considered a static problem that does not take into account the change in the position of the population with time. In this paper, we propose a dynamic system that describes the population patch distribution, which depends on the utility of the patch. In addition the utility varies with time as a result of distance variations. The Boltzmann distribution is a particular solution of the proposed system of differential equations. The Lyapunov stability condition for the Boltzmann distribution is obtained.The utility functions of the patches, which depend on the population — patch distance and on the measure of the information certainty, are introduced. As a result, in the two-dimensional case, a space $R^2$ is divided into areas of preferred utility. Such a partition is a generalization of the Voronoi diagram.