# Mathematical analysis of a model of partial differential equations describing the adaptation of mosquitoes facing the usage of insecticides

Abstract : This dissertation is concerned with an age structured problem modelling mosquito plasticity. The main results can be divided into four parts.The first part presents an age structured problem modelling mosquito plasticity in a natural environment. We first investigate the analytical asymptotic solution through studying the spectrum of an operator A which is the infinitesimal generator of a C0-semigroup. Additionally, we get the existence and nonexistence of nonnegative steady solutions under some conditions.In the second part, we study the optimal control of an age structured problem. Firstly, we prove the existence of solutions and the comparison principle for a generalized system. Then, we prove the existence of the optimal control for the best harvesting. Finally, we establish necessary optimality conditions.In the third part, we investigate the local exact controllability of an age structured problem modelling the ability of malaria vectors to shift their biting time to avoid the stressful environmental conditions generated by the use of indoor residual spraying (IRs) and insecticide-treated nets (ITNs). We establish a new Carleman's inequality for our age diffusive model with non local birth processus and periodic biting-time boundary conditions.In the fourth part, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. Firstly, we prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. Finally, we prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation.
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Submitted on : Monday, October 15, 2018 - 2:00:07 PM
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• HAL Id : tel-01895654, version 1

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Linlin Li. Mathematical analysis of a model of partial differential equations describing the adaptation of mosquitoes facing the usage of insecticides. Analysis of PDEs [math.AP]. Université de Bordeaux, 2018. English. ⟨NNT : 2018BORD0097⟩. ⟨tel-01895654⟩

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