Optimal Birth Control for a Size-Structured Population Model with Diffusion

TL;DR

This paper develops an optimal birth control strategy for a spatial, size-structured invasive species model using semigroup theory, adjoint systems, and variational principles, ensuring existence and uniqueness of solutions.

Contribution

It introduces a novel application of semigroup methods and Ekeland's variational principle to optimize fertility control in a spatial population model.

Findings

Derived optimality conditions for fertility rate control.

Proved existence and uniqueness of the optimal control.

Provided new insights into controlling invasive species spatially.

Abstract

This work addresses the optimal birth control problem for invasive species in a spatial environment. We apply the method of semigroups to qualitatively analyze a size-structured population model in which individuals occupy a position in a spatial environment. With insect population in mind, we study the optimal control problem which takes fertility rate as a control variable. With the help of adjoint system, we derive optimality conditions. We obtain the optimality conditions by fixing the birth rate on three different sets. Using Ekeland's variational principle, the existence, and uniqueness of optimal birth controller to the given population model which minimizes a given cost functional is shown. Outcomes of our article are new and complement the existing ones.