A Study Of Population Evolution Problems With Time Delays

Population ecology is a subject that describes the relationship between pop-ulation and environment or the interaction between populations.Many biologists and mathematicians will establish the relationship between population and envi-ronment and population into the mathematical model used to describe and predict the development of species,and through human action to regulate and control the survival and development of the population,that made the population lasting and stable.Predator-prey system is a basic structure in population ecology,and it is very important to study predator prey system for understanding the real world.In recent 20 years,many scholars have paid close attention to the stability and Hopf bifurcation of delay differential equations.Especially,the time delay causes the model to produce Hopf bifurcation,thus inducing periodic solutions is one of the subjects of interest to scholars both at home and abroad.In this paper,the dynamical properties of two kinds of differential systems with delay are studied by using the bifurcation theory of differential equations.One of the differential equations is an epidemic model with time delay.By analyzing the equilibrium point of the model,the characteristic roots are obtained to judge the stability of the infectious disease at the equilibrium point and the conditions for the generation of the Hopf bifurcation,so as to effectively cont,rol the infectious diseases.Another differential equation is a predator-prey model with two delays.It is analyzed from two aspects of the dissipation and stability of the system,so as to obtain the conditions for the sustainable development of the system without extinction,and to better predict the development and evolution of the population.The second chapter studies a delayed epidemic model with B-DA function,when the infection rate is of nonlinear and conforms to the Beddington-DeAngelis function,the Q time delay as the bifurcation parameter,using Hopf bifurcation theory,the delay time Q get to meet a series of conditions,the model undergoes a Hopf bifurcation and Hopf-Zero bifurcation.Finally,using Mathematica the software of numerical simulation to verify the theoretical results.In the third chapter,we establish a predator-prey model with two delays and Holling II type functional response.We first.analyze the dissipat.ive system,and then the method of ordinary differential equation stability and qualitative,respectively select two delay as the bifurcation parameter,the stability theory of differential equation,the stability of the unique positive equilibrium analysis;using Hopf bifurcation theory,analysis of the existence of positive equilibrium point of Hopf bifurcation;finally,to verify the correctness of the theory proposed by numerical simulated by Mathematica software.