Optimal contraception control for a nonlinear population model with size structure and a separable mortality

Feng-Qin Zhang,Rong Liu,Yuming Chen

doi:10.3934/dcdsb.2016112

Abstract

This paper is concerned with the problem of optimal contraception controlfor a nonlinear population model with size structure.First, the existence of separable solutions is established, which is crucial in obtainingthe optimal control strategy. Moreover, it is shown that the population densitydepends continuously on control parameters.Then, the existence of an optimal control strategy is proved viacompactness and extremal sequence. Finally, the conditions of theoptimal strategy are derived by means of normal cones and adjoint systems.

Highlights

Wild animals are valuable natural resources and important components of a healthy ecosystem
To the best of our knowledge, so far there is no investigation on the optimal contraception control of size-structured population models
In a similar way as to develop (1), we propose the following model on contraception control for nonlinear size-structured population dynamics

Summary

Introduction

Wild animals are valuable natural resources and important components of a healthy ecosystem. To the best of our knowledge, so far there is no investigation on the optimal contraception control of size-structured population models. Our study is inspired by that of Kato [19], where the author investigated the optimal harvesting problem for the following nonlinear size-structured population model. In a similar way as to develop (1), we propose the following model on contraception control for nonlinear size-structured population dynamics,. Note that (2) is a special case of [19, system (4.1)], which is a general model for size-structured population dynamics with time dependent birth and aging functions. We provide some properties of the solutions, which include boundedness and the continuous dependence of the population density on the control parameter.

For fixed

Choose λ

We now prove that