Study On The Dynamic Characteristics Of A Mutualistic Model

Population model is the research object of many scholars in the field of mathematical ecology.People abstract the problems in natural ecology into mathematical models and explain the phenomena in ecosystem.With the development of population ecology,the models we study are more practical.One of the most important research directions is the study of population model with time delay.Time delay is a common phenomenon in natural ecology,for example,in predator-prey models,we often assume that young populations do not have predatory capacity,and that there is a growth period from infancy to maturity,which is a time delay phenomenon.Therefore,the first kind of problem studied in this paper is to discuss the stability and Hopf bifurcation in a mutualistic model with time delay on the basis of existing mutualistic model.The other problem that has become a hot issue of our concern is the optimal harvesting policy.For example,in fisheries,we hope to catch as many fish as possible without destroying the natural growth of the fish.Therefore,the second kind of problem in this paper is to study the optimal harvesting policy of the system based on the existing mutualistic model.In the first chapter,combining the background and significance of the research and the current research situation at home and abroad,the content and research methods of this paper are given.In the second chapter,the background knowledge needed by the research object is introduced.In the study of time delay model,the stability theory of differential equation and eigenvalue theory are introduced.In the study of optimal harvesting policy,the upper and lower solution theory and variational calculus are used.In Chapter 3,the stability of time delay system is studied.The boundedness of the system is proved by using the comparison theorem.Sufficient conditions for global stability of positive equilibrium solution of the system are given by constructing Lyapunov functions.In the fourth chapter,the Hopf bifurcation problem of time delay system is studied.Sufficient conditions for the existence of bifurcation values are obtained by using eigenvalue theory,and the correctness of the theory is verified by numerical simulation with Matlab.In the fifth chapter,we study the optimal harvesting policy of constant coefficient harvesting model.According to the theory of upper and lower solutions,we give the sufficient conditions for the existence and uniqueness of periodic solutions,and obtain the optimal harvesting policy of the system by referring to the idea of variational calculus.In Chapter 6,the optimal harvesting policy of the variable coefficient harvesting model is studied.The research method is consistent with Chapter 5,which is a generalization of the constant coefficient harvesting model.Chapter 7 summarizes the contents of the article and looks forward to the possible future research and exploration ideas.