Generalization of Controls Bimodality Property in the Optimal Exploitation Problem for Ecological Population with Binary Structure

Abstract

The problem of optimal exploitation of an ecological population with a binary structure is considered (there is an additional criterion for population structuring in addition to age or developmental stage). It is assumed that population state dynamics is described by a nonlinear generalization of the Leslie model. We prove a criterion for the existence of so-called quasi-preserving controls that support the sustainable population dynamics. Moreover, optimal quasi-preserving controls with a minimum number of nonzero coordinates (i.e., controls that preserve unchanged the largest number of structural units of a population) are found explicitly. The proposition about the minimum possible number of nonzero coordinates for optimal vectors is also proved. This proposition is a generalization to the case of a binary population structure of well-known bimodality property of optimal strategies for populations with one-dimensional structure.