Research On Dynamics Of Two Kinds Of Predator Prey Models With Delay

Mathematical ecology is a natural science that combines ecology with population dynamics.Scholars use mathematical models to describe the interactions between biological communities and explain the dynamic nature behind natural phenomena.This paper studies the Hopf bifurcation of two types of predator-prey models with different ecological backgrounds,including Michaelis-Menten type harvest term,refuge and time-delay effect.The Hopf bifurcation of two kinds of population dynamics models is analyzed by using the normal form method and the central manifold theorem.The full text consists of the following five chapters.Chapter 1: Introduction.In this chapter,we introduce the research background,significance,and current status of the two population dynamics models studied in this paper.Chapter 2: In this chapter,we describe the basic knowledge of dynamic system involved in this article.Chapter 3: The effect of time delay on the stability of a predator-prey system with harvesting term is studied.Firstly,the conditions for the existence of local Hopf bifurcation in the system are analyzed.Secondly,taking the digestion delay of the predator population as the bifurcation parameter,we analyze the stability of the periodic solutions by reducing the original system on the center manifold.Finally,numerical simulation is conducted to verify the obtained analysis results.The research shows that when the time delay exceeds the threshold,the system will lose stability and generate Hopf bifurcation;When the time delay does not exceed the threshold,the positive equilibrium of the system is locally asymptotically stable.Chapter 4: In this model,the predator’s digestion delay and maturity delay are taken as the research objectives.Firstly,the existence of positive equilibrium point,the stability of the system and the existence of Hopf bifurcation are analyzed;Secondly,the mathematical expression that determines the properties of Hopf bifurcation is calculated;Finally,some numerical examples are given to verify theoretical results.The research shows that the existence of time delay is a sufficient condition to determine the occurrence of Hopf bifurcation of the system.When the feedback time delay is less than the critical value,the system will maintain a stable state,and the number of two species will eventually tend to a fixed value;When the feedback delay is more than the critical value,the stability between the predator and prey populations is destroyed and Hopf bifurcation occurs.In addition,the increase of the refuge coefficient can change the critical value of the delay,thus enhancing the stability of the system.Chapter 5: Summarizes the full text and looks forward to future work.