Dynamics Of Aphid-aphid Model With Time Delay And Age Structure Of The Analysis

In recent years,with the rapid development of biological science,biologists have gradually realized the great impact of time delay on population biology.In the evolution of species,delay the inevitable,For example,a predator digests prey with a lag in time.The scientists study of time-delay systems caused the wide attention of the world,also become a hot area of research in recent years is delay for the influence of the biological system "will be our future research important topic.In this paper,a predator-prey model with time delay is studied.On the basis of the original system without time delay,time delay parameters τ are added,and the stability of the equilibrium point of the new system and the existence of hopf branches are studied as follows.Firstly,three equilibrium points of the system are obtained,the existence conditions of each equilibrium point are given,the corresponding characteristic equations of the three equilibrium points are solved,the characteristic root distributions of the characteristic equations with delay parameters τ are discussed,and the stability of the equilibrium point of the delay differential system and the theoretical conditions for the existence of hopf branches are obtained.Then,the time-delay differential equation is transformed into an abstract functional differential equation,and by using the central flow pattern theorem and the normative theory of functional differential equation,the calculation formula for judging the direction of hopf branch and the stability of its periodic solution is given.Finally,numerical simulation is conducted to verify the above theoretical results,and the practical significance of each theoretical result is summarized.