# Some regional control problems for population dynamics

## TL;DR

This paper studies regional control strategies for population dynamics, focusing on optimal harvesting and eradication of pest populations, using theoretical analysis and numerical methods to optimize interventions in specific regions.

## Contribution

It introduces a novel iterative numerical method for optimizing regional harvesting efforts and provides new theoretical conditions for population eradication control.

## Key findings

- Optimal harvesting effort structure derived from necessary conditions.
- Iterative method effectively increases total harvest in numerical tests.
- Conditions for zero-stabilizability of pest populations established.

## Abstract

This paper deals with some control problems related to structured population dynamics with diffusion. Firstly, we investigate the regional control for an optimal harvesting problem (the control acts in a subregion $\omega$ of the whole domain $\Omega$). Using the necessary optimality conditions, for a fixed $\omega$, we get the structure of the harvesting effort which gives the maximum harvest; with this optimal effort we investigate the best choice of the subregion $\omega$ in order to maximize the harvest. We introduce an iterative numerical method to increase the total harvest at each iteration by changing the subregion where the effort acts. Numerical tests are used to illustrate the effectiveness of the theoretical results. We also consider the problem of eradication of an age-structured pest population dynamics with diffusion and logistic term, which is a zero-stabilization problem with constraints. We derive a necessary condition and a sufficient condition for zero-stabilizability. We formulate a related optimal control problem which takes into account the cost of intervention in the subregion $\omega$.