Application Of Time-delay Dynamic Systems In Biological And Economic Models

Delay differential equation as an important branch of functional differential equations is widely used to describe the fluid mechanics,electronics,population e-cology,biology,economics and cross disciplinary problems.Branch is an important problem in differential equations is usually used to study the unstable structure of the nonlinear system.It has important theoretical and practical significance to study the stability and bifurcation of delay differential equations.The aim of this paper is investigate the application of the delay differential equation in economic and biological models,the results are as follows:In the first chapter,we mainly introduce some basic definition and theorem.In the second chapter,we investigate the stability and Hopf bifurcation for theadvertising-shopping level model with two delays.The locally asymptotic stability for the system and existence for the Hopf bifurcation are discussed by analyzing the distribution of roots associated characteristic equation when the two delays are equal and unequal,respectively.The direction of the Hopf bifurcation and sta-bility of the bifurcating periodic solutions are further investigated by using center manifold theorem and normal form theory.Finally,some numerical simulations are given to illustrate the obtained theoretical results.In the third chapter,papers investigate a Kaldor-Kalecki model of business cycle system with two different delays,which described the interaction of the gross product Y and the capital product K.We derived the conditions for the local sta-bility and the existence of Hopf bifurcation at the equilibrium of the system.By applying the normal form theory and center manifold theory,some explicit formu-lae for determining the stability and the direction of the Hopf bifurcation periodic solutions are obtained.Some numerical simulations by using Mathematica soft-ware supported the theoretical results.Finally,main conclusions are given.In the fourth chapter,the model of hopf bifurcation of Micro-organisms con-tinuous culture and stability with time delay be studied.By applying of the theory of hopf bifurcation and functional differential equation method,we discuss the lin-ear stability of the model and the local hopf bifurcation.And,the numerical simulation shows the effectiveness of the conclusion.