The Boltzmann distribution in the problem of rational choice by population of a patch under an imperfect information about its resources

The problem of rational choice by the population of a patch containing energy (nutritive) resources is considered. This problem belongs to the theory of optimal foraging, which, in turn of, studies issues related to the behavior of the population when it leaves the patch or chooses the most suitable one. In order to define the optimal patch choice for population, a variational approach, based on the idea of the Boltzmann distribution is proposed. To construct the probability distribution the utility functions are used, that take into account factors that can influence the patch choice of a population: available information about the quality of patches, the energy utility of patches, the cost of moving to the patch, the cost of information about the quality of patches. The main goal of the paper is to investigate the influence of available information about the amount of resources, contained in patches, on a decision-making process generated by the foragers while a suitable patch choosing. The optimal rationality is determined in the cases taking into account the information cost, the average energy utility of all patches, the rationality depending on the patch. The conditions under which the population, with the lack of information, select the “poor” patch, in sense of its resources, are obtained. The latter provides a theoretical justification of experimental observations, according to which a population can choose a patch with worse quality. The obtained results have a general character and may be used not only in behavioral ecology but when constructing any decision making processes.