Qualitative Analysis Of Some Variable Coefficient Two Population Models

Based on Mathematics and biology,biomathematics establishes and perfects its own theoretical system in application.Population model is an important research object in biomathematics.Scholars abstract the problems existing in natural environment into mathematical models,and further explain the phenomena in ecosystem.With the development of research process,The population model has developed many research directions,one of which is about the optimal harvest of the population,which has a very important value in the rational utilization and protection of biological resources and other aspects.The other is the research on the delayed population,which has been considered in many literatures that the species has a process of growth from infancy to maturity in real life,Based on the above two research directions,the first problem of this paper is the optimal harvesting strategy of the variable coefficient predator-prey model.The second problem of this paper is the optimal harvesting strategy of the variable coefficient reciprocal model,The third problem discussed in this paper is the existence of Hopf bifurcation in the time-delay mutualism model with variable coefficients.The first chapter introduces the background and significance of the research,and obtains the development history of the three main models in this paper.By considering the research content in combination with the current research situation,the preparatory knowledge is given.In the second chapter,we mainly discuss the variable coefficient predator-prey model.Firstly,we use the upper and lower solution theory of differential equation to obtain the existence and uniqueness of periodic solution;secondly,we obtain the sufficient conditions for the persistence and persistence of the model;finally,we give the optimal harvesting strategy of the predator-prey model with two harvesting terms through the variational principle,The persistence of predator and prey in the model is verified by numerical simulation.In Chapter 3,we study the mutualism model with variable coefficients,and obtain the sufficient conditions for the existence and uniqueness of the periodic solution of the model;then we give the sufficient conditions for the persistence of two mutualism populations;finally,we obtain the optimal harvesting strategy of mutualism model with variable coefficients,and verify the persistence theory of mutualism populations with numerical simulation.In Chapter 4,we study the mutualism model with variable coefficient and time delay.Firstly,we obtain the boundedness of the model solution by comparison theorem.Secondly,we construct Lyapunov function to obtain the sufficient condition for the global stability of the positive singularity.Secondly,we give the sufficient condition for the existence of the bifurcation by combining the eigenvalue theory.Finally,we verify the validity of the conclusion by numerical simulation.