Hopf Bifurcation Analysis Of Several Species Dynamics Models With Delay

Population dynamics plays an important role in the field of biomathematics.Because of its extensive scope of research and the particularity of its research object,it has been attracting the attention of many scholars.With the passage of time,biological populations are also imperceptibly evolving towards a more complex,detailed,and diverse directions,scholars have studied many realistic factors affecting the population density and population structure,so as to construct rich and diverse population dynamics models.On this basis,in order to make the model more accurately reflect the law of population change,scholars constantly screen out more practical factors,and provide the theoretical cornerstone for the rational utilization of biological resources with the help of the accurate model.The purpose of this paper is to analyze the Hopf bifurcation of population dynamics models with time delays.Based on the stability theory of ordinary differential equations,the Hopf bifurcation of two types of population dynamics models is analyzed by using normal form method and the central manifold theorem.The full text is divided into five chapters.1)Considering that the population is affected by human capture behavior and the pregnancy period of the population itself,after modifying the existing model,a double time delays Lotka-Volterra predator-prey model with a Michaelis-Menten harvest term is studied,and its Hopf bifurcation analysis is performed.Firstly,the existence condition of the positive equilibrium point of the system is discussed.Secondly,taking two delays as parameters,by combining different cases of delays,we analyze the steady-state dynamics changes of the positive equilibrium point under different conditions,as well as the existence conditions of local Hopf bifurcation,and obtain specific expressions that characterize the properties of Hopf bifurcation and the periodic solutions generated by it.Finally,to judge whether the theory is correct,we need to draw an accurate conclusion through numerical simulation.2)In this chapter,considering the impact of the population’s maturity delay and the fear effect of predation,after improving the existing model,we study the double time delays Leslie-Gower predator-prey model with linear harvest term,and analyze and discuss the Hopf bifurcation generated by it.Firstly,we analyze the various equilibrium points that may exist in the system and their stability changes.Secondly,taking time delay as a parameter,dynamic changes in the steady state of the positive equilibrium point of the system and the existence conditions of local Hopf bifurcations under five different combinations of time delays are described.The specific expressions describing the properties of Hopf bifurcations and their periodic solutions are obtained through the central manifold theorem and the normal form method,and the obtained conclusions are extended to the global using the global bifurcation theory.Finally,in order to enrich the results of theoretical analysis,numerical simulation is used to verify.