Spatial Adaptation of Populations in Ecological Models

Abstract

A discrete dynamic model of populations is described; in this model, spatial migration is specified by a finite-state Markov chain, while growth and nonlinear interactions are defined by convex and concave functions. This makes it possible to efficiently analyze the dynamic process and the behavior of a system based on the theory of monotone operators. The adaptation mechanisms that underlie the evolution of a population migration matrix are studied. It turned out that the final state depends on the choice of the initial state, although the positive eigenvectors (Perron vectors) of all these matrices are almost identical. The components of a Perron vector here correspond to the relative residence time of a population in a certain location. Spatial adaptation indicates the optimization of the residence time of a population in certain regions of its distribution range. The corresponding computations allowed a new interpretation of the phenomenon of spatial coadaptation as an “attraction” (in the case of predation) or a “repulsion” (in the case of competition) of the Perron vectors of populations.