The Effect Of Time Delay On The Stability And Branching Of The Predator-prey System

This perper studies a class of Lotka-Volterra predator-prey models with one-time delay and one with two delays,respectively using the Hopf bifurcation theory,we discuss the effects of two kinds of delays on the stability and branching of the predator-prey systems.Finally,we use Matlab software to calculate the values.The simulation is to verify the correctness of the theoretical analyes.The full text is divided into five chapters,the specific arrangement is as follows:The first chapter includes the introduction,which firstly introduces the theoretical significance,practical significance and the historical background of the problem,Then the research status of the problem and the main research work of this thesis are introduced.The second chapter includes the preliminary knowledge introducing the basic concepts of delay differential equations and the Hopf bifurcation theory.In the third chapter,a four-dimensional Lotka-Volterra predator-prey system model with single time delay is discussed.The stability of the positive equilibrium point and the exictance of the Hopf bifurcation of the model are discussed.Applying the central manifold theorem and normative type,we find the specific expression of the Hopf bifurcation,and finally use computer simulation to verify the correctness of theoretical results.In the fourth chapter,a class of four-dimensional Lotka-Volterra predater-prey system model with two time delays is discussed.When the values of two time delays are different,the Hopf bifurcation theory is used to discuss the effects of two time delays on the stability of the system's positive equilibrium point.Also,the sufficient conditions for the Hopf branch of the system are obtained.Finally,the results of theoretical analysis are verified by the numerical simulation.The fifth chapter includes the summary and prospect.By comparing the two models,the influence of time delay on the stability and branching of the predation system are concluded.Finally,the deficiency of this work and the problems that need further to be studied are pointed out.