### On Existence of Optimal Non-destructive Controls for Ecosystem Exploitation Problem Applied to a Generalization of Leslie Model

#### Abstract

We proposed earlier some general formalization for the problem of the nondestructive use of renewable resources and reduced it to some mathematical programming problem. In this article, we study some properties of optimal controls for this mathematical programming problem in the particular case of obtained earlier by authors nonlinear generalization of so-called Leslie model. We establish the existence of optimal controls that preserve the structure of the operated system, assuming the concavity of the functions used. It turns out that there exists a hyperplane containing such controls. We use some generalization of the classical concept of irreducibility for nonlinear maps, the concept of local irreducibility. These results have applications to actual problems of rational exploitation of natural resources.

#### Keywords

Concave programming, Managing of renewable resources, Non-destructive exploitation of ecosystems, Nonlinear Leslie model

DOI

10.12783/dtcse/optim2018/27933

10.12783/dtcse/optim2018/27933

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