Dynamics of Population Distribution by Patches

Abstract

This paper considers the patch selection by a population without having full information about the utility of the patch, i.e., the amount of its energy resources. This problem is related to the optimal foraging theory. According to U. Dieckmann’s suggestion, the population distribution by patches should be modeled according to the utility function that takes into account the amount of resources per patch, the population–patch distance, and the extent to which the population is aware of the amount of resources in the patch. In this case, the population distribution by patches is described using the Boltzmann distribution. Dieckmann considers a static problem without taking into account the change in the position of the population over time. In this paper, we suggest a dynamic system that describes the population distribution by patches, which depends on the utility of patches that changes over time as a result of variations in the population–patch distance. That said, the Boltzmann distribution is a particular solution of the derived system of ordinary differential equations. The Lyapunov stability condition for the Boltzmann distribution is derived. The patch utility functions dependent on the distance to and the population’s awareness of the patch are introduced. As a result, in the 2D case the R² space in divided in preferential utility domains. This partition generalizes G.F. Voronoy’s diagram.