The Geometric Theory Of The Vector Fields And Application In Biology

Ordinary differential equation is an important tool to use mathematics as a solution on practical problems,and the geometric theory of the vector fields(that is the qualitative theory of ordinary differential equation)is the powerful tool to study ordinary differential equation.Utilizing the qualitative method and theory to study the differential equation models in practical problems,we can obtain significant results for practice.In the geometric theory of the vector fields,there are very abundant methods and results on the planar vector fields.But for the geometric theory of a vector field in space,because of its complexity there is still no systematic and complete methods and results at present.However,most of the models coming from practical problems are higher dimensional systems,especially,the model in biology.Therefor,to better solve practical problems,we must further research the geometric theory of the vector fields in space.In this paper,we at first investigate the problem of space period solution in the geometric theory of the vector fields.We then apply the qualitative theory of differential equation to biology.By analyzing the dynamics of malaria transmission models with sterile mosquitoes,we investigate the impact of releasing sterile mosquitoes on the malaria transmission with the different release strategies.This thesis consists of five chapters.In Chapter 1,we briefly introduce the origins and development of the geometric theory of the vector fields,then reviews the application of ordinary differential equation models in population dynamics and epidemiology in history,and provide a brief introduction for the current research status.Chapter 2 belongs to the geometric theory of the vector fields.We prove that a class of cubic quasi-homogenous vector field in R3 can have two isolated closed orbits on the invariant cone by using the idea of central projection transformation,which sets up a bridge connecting the vector field X(x)in R3 with the planar vector fields.As an application of this result,we show that the 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones,there are two isolated closed orbit on each cone,in suitable conditions.Meanwhile we show that another cubic system has at least 26 isolated closed orbits.The distributions of the 26 isolated closed orbits have two cases:which are located on the 13 invariant cones of the system respectively,there are two isolated closed orbit on each cone;or located on the 26 invariant cones of the system respectively,there is one isolated closed orbit on each cone.Chapter 3 and Chapter 4 belong to the application of the geometric theory of the vector fields in biology.The sterile mosquitoes technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population has been used in preventing the malaria transmission.In Chapter 3,to study the impact of releasing sterile mosquitoes on the malaria transmission,we first formulate a simple SEIR malaria transmission model as our baseline model.We derive a formula for the reproductive number of infection,determine the existence of endemic equilibria and prove the existence of backward bifurcation for the baseline model.We then include sterile mosquitoes into the baseline model and obtain malaria transmission models with sterile mosquitoes,where the releasing rate of sterile mosquitoes is constant.Based on the reproductive number,endemic equilibria and backward bifurcation for the model with the sterile mosquitoes,we determine the releasing thresholds that ascer-tains the disease dies out or spreads.On the other hand,we study the relationship between the population of the infected humans and mosquitoes and the releasing rate of the sterile mosquitoes when the disease spreads.Our results show that we can reduce the components of the infected humans and mosquitoes with the in-crease of.the releases of sterile mosquitoes.It means that the releasing of the sterile mosquitoes is helpful in controlling the spread of disease.In an area where the population size of wild mosquitoes is relatively smal-1,we may let the release rate be proportional to the population size of the wild mosquitoes for cost saving.In Chapter 4,we investigate the impact of releasing sterile mosquitoes on the malaria transmission when the release rate is proportional to the population size of the wild mosquitoes.We still formulate a SEIR malaria transmission model as our baseline model.Because a small mosquito population size may cause possible difficulty in finding mates,we assume an Allee effect for the mating rate of mosquitoes in this model.Then sterile mosquitoes are included into the baseline model,and the release rate is proportional to the population size of the wild mosquitoes.We analyze the dynamics of the baseline model and the model with the sterile mosquitoes.We calculate the reproductive numbers for these models and prove the existence of endemic equilibria and backward bifurcation.Similarly as in Chapter 3,the releasing thresholds that ascertains the disease dies out or spreads are provided.At last,based on theoretical analysis and numerical simulation,we demonstrate the impact of proportional releasing sterile mosquitoes on the malaria transmission in detail.Finally,In Chapter 5,we briefly summarize our work in this thesis and present the research subjects and direction for the future.