Null Controllability of a Nonlinear Age Structured Model for a Two-Sex Population

Abstract

This paper is devoted to study the null controllability properties of a nonlinear age and two-sex population dynamics structured model without spatial structure. Here, the nonlinearity and the couplage are at the birth level. In this work, we consider two cases of null controllability problem. The first problem is related to the extinction of male and female subpopulation density. The second case concerns the null controllability of male or female subpopulation individuals. In both cases, if A is the maximal age, a time interval of duration A after the extinction of males or females, one must get the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after Kakutani’s fixed-point theorem.

On a Predator-Prey Model Involving Age and Spatial Structure

“…We believe that it is possible to control the model (2) through the external functions f 1 and f 2 . In other words, by taking the functions f 1 and f 2 as controls in a bounded domain of Ω, it is possible to have the extinction either the population of prey or that of predators or both simultaneously from of a time T as we did in [17].…”

“…A result of existence and uniqueness and positivity of solution is also proved in [16] by Traoré et al where their system models the population dynamics of Callosobruchus Maculatus. Traoré et al have proved an existence result in [17] where the system models a nonlinear age and two-sex population dynamics.…”

Approximate controllability of the semilinear population dynamics system with diffusion

“…One can refer to Traoré et al 46 for detailed explanation on the controllability with suitable modifications.…”

“…For a continuous initial solution $$$ {p}_0 $$$, the analogous discrete system is given by $$$$ \frac{d{U}_l}{d\tau}={G}_l{U}_l+{P}_l+{B}_l{Y}_l,{U}_l(0)={\left({p}_0\left({g}_j\right)\right)}_{1\le j\le n}. $$$$ For the simulation, we take $$$ G=3,\triangle g=\frac{1}{30},\kern3.0235pt b=0.6 $$$ and $$$ n=30 $$$ as discussed by Traoré et al 46 consider the initial condition $$$$ {p}_0(g)=15{e}^{-1/\left(G-g\right)}. $$$$ The fertility $$$ \alpha $$$ (see Traore et al and Traoré et al 47,48 ) is given by …”

“…$$ And the mortality rate is given by $$$$ \sigma (g)=\frac{1}{10\left(G-g\right)}. $$$$ One can refer to Traoré et al 46 for detailed explanation on the controllability with suitable modifications.…”