Stability And Hopf Bifurcation Analysis For Predator-Prey Model With Time Delay

In the study of ecology,population ecology is the main concern of people,and population dynamics plays an important role in population ecology.In recent years,the predator-prey model has become an important research object in population dynamics,which embodies the perfect combination of mathematics and ecology.It is found that many factors affect the behavior of the population ecosystem,among which the ecosystem with time delay has more complex dynamic behavior,and the ecosystem with stage structure can more accurately reflect the natural growth of the system.Based on the above factors,this paper focuses on the related dynamic behavior of Holling IV predator-prey model with stage structure under time-delay.This paper firstly introduces the research background,significance and research results of predator-prey model at home and abroad,and clarifies the research objectives and main contents,and introduces the theoretical knowledge and methods used in this paper.Secondly,for the staged predator-prey model with single time delay,and combined with the stability theory of delay differential equations,the stability of the system at the positive equilibrium point and the conditions for generating Hopf bifurcation are explored with taking the time-delay as bifurcation parameter.The direction of Hopf bifurcation and the corresponding bifurcating periodic solution are summarized by the center manifold theorem and the normal form method,and finally the conclusions are verified by numerical simulation.Then,on the basis of the above research,the predator-prey model with two delays are further discussed.The stability of positive equilibrium and the Hopf bifurcation are discussed from six different states.Further,the hybrid control is applied to the original system,which delays the original bifurcation and obtains the bifurcation periodic solution and bifurcation direction of the controlled system,and finally the conclusions are verified by numerical simulation.Finally,the main contents of this thesis are reviewed,and proposes the areas for improvement and future research ideas.