The Influence Of Resistant Mosquitoes To Malaria Dynamical System

In this paper,we discuss the chance that resistant mosquitoes will completely replace wild mosquitoes in mathematics.After using different ways to introduce resistant mosquitoes,we set up their corresponding malaria dynamics.Then,conditions of existence and stability about their disease free equilibria will be obtained.There are three chapters in the following.In the first chapter,we simply introduce etiology and epidemiology about malaria.And,we also give some related theorems about epidemic dynamics,Lotka-Volterra competition dynamics, and impulsive differential equations.In the second chapter,we assume that we will do nothing after putting resistant mosquitoes into nature at the first time.On the basis of random pair formation theory and Lotka-Volterra competition dynamics,there is the dynamical system of mosquitoes.Combining human part,we obtain the whole malaria dynamical system.Firstly,by the next generation matrix method,we find its basic reproduction number.Comparing with the basic number of natural situation,it is found that the competition does not influence the relationship between these two reproduction numbers.This method is very helpful to control disease.Furthermore,we find that only when the reproduction probability of the mating between two resistant vectors is bigger than that of the interbreeding between a wild and resistant vector,can two strain vectors coexist stably.It is easy to see that the method is hard to realize.Hence,we will introduce another way in the third chapter.In the last chapter,we consider to put resistant mosquitoes periodically.This method is similar to use predators to control pests.It is used an impulsive differential system to formulate this situation.We discuss the permanence and extinction for two class of two dimensional impulsive competitive model.We obtain the condition guaranteeing that one species will be driven to extinction while the other will stabilize at a certain solution of an impulsive logistic equation. Then,the stability of periodic solution of whole malaria dynamical system is shown by applying Floquet theory.In the end,we get that it is better to introduce resistant mosquitoes periodically than only once.