Dynamics Analysis Of A Two Time Delay Predator-prey Model With Infectious Diseases For Predators

In recent years,with the development of natural science,the research of population dynamics has been widely concerned by many mathematicians and biologists.It has been a long history to study the dynamical system of population by using mathematical models.By establishing a suitable population system model,mathematical analysis of the model is carried out,and the mathematical properties of the system are obtained,so as to get the ecological development rule of the population.It can also predict the future survival and development of population,and help to better regulate and maintain the sustainable development of population.Through investigating the influencing factors of infectious diseases in the population,we further enrich the research contents of the model,and make the model closer to the actual situation.Because d elay differential equations can depict the population ecological model very well.Based on this,we analyze a large number of delayed predator prey models.Through improvement,a class of predator models with infectious diseases is proposed,and the effect of time delay on the stability of the model is also discussed.In the first chapter,we first introduce the main applications of delay differential equations,and briefly describe the current research status of the content at home and abroad.The second chapter introduces some concepts,theorems and lemmas for preparatory knowledge.In the third chapter,we discuss a class of delayed population models with infectious diseases.The existence and regularity of the solution of the model are analyzed.By analyzing the distribution of the corresponding characteristic equations of the linear system,the sufficient conditions for the local stability and the existence of Hopf bifurcation are obtained.By constructing Lyapunov function and applying Lassalle invariance principle,we get the global stability of system equilibrium.By using the central manifold theorem and the normal form theory,we get the formula to determine the Hopf bifurcation direction and the periodic solution of the system.In the fourth chapter,we use Matlab software to do numerical simulation and draw corresponding Hopf bifurcation diagrams,and verify related conclusions.Finally,the research contents are summarized,and the shortcomings and future improvements are given.