Abstract
This paper is devoted to study the null controllability properties of a population dynamics model with age structuring and nonlocal boundary conditions. More precisely, we consider a four-stage model with a second derivative with respect to the age variable. The null controllability is related to the extinction of eggs, larvae, and female population. Thus, we estimate a time T to bring eggs, larvae, and female subpopulation density to zero. Our method combines fixed point theorem and Carleman estimate. We end this work with numerical illustrations.
Highlights
Let ui(t, a), 1 ≤ i ≤ 4, be, respectively, the distribution of eggs, larvae, and female and male individuals of age a at time t; Ai, 1 ≤ i ≤ 4, is the life expectancy of an i− stage individual and T is a positive constant
We consider the following population dynamics model based on Fokker–Planck or Kolmogorov-type equations which is written as ztu1(t, ztu2(t, a) a)
We recall that the optimal and exact control problems are widely investigated for age-structured population dynamics by many researchers
Summary
Let ui(t, a), 1 ≤ i ≤ 4, be, respectively, the distribution of eggs, larvae, and female and male individuals of age a at time t; Ai, 1 ≤ i ≤ 4, is the life expectancy of an i− stage individual and T is a positive constant. We recall that the optimal and exact control problems are widely investigated for age-structured population dynamics by many researchers Most of these studies are focused on optimal control problems [3,4,5] and the references therein. Barbu et al considered the exact controllability of the linear Lotka–McKendrick model without spatial structure by establishing an observability inequality for the backward adjoint system [11]. Later on, He and Ainseba investigate the exact null controllability of a stage and age-structured population dynamics system in [12] and the exact null controllability of the Lobesia Botrana model with spatial diffusion in [13].
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