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.
Highlights
This study takes place in the fight against malaria
Malaria is a serious disease, and our work takes its importance in the strategy to fight against it
Malaria is a vector-borne disease transmitted by an infective female anopheles mosquito
Summary
This study takes place in the fight against malaria. Malaria is a serious disease (in 2017, there were 219 million cases in the world [1]), and our work takes its importance in the strategy to fight against it. A malaria control strategy in Brazil or Burkina Faso consists of releasing genetically modified male mosquitoes (precisely sterile males) in the nature. This can reduce the reproduction of mosquitoes since females mate only once in their life cycle. Very few authors have studied control problems of a two-sex structured population dynamics model. Abstract and Applied Analysis sex structured population dynamics model They first study an approximate null controllability result for an auxiliary cascade system and prove the null controllability of the nonlinear system by means of Schauder’s fixed-point theorem. We are talking about the null controllability as an extinction of the population, but we can see in the sense of exact controllability to the trajectories since there is an equivalence between the null controllability and exact controllability to the trajectories in the linear case
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