Global Attractivity And Oscillation Of Differential Systems With Multi-delays

In recent years, Persistence and global attractivity of ecological system have been concerned by many authors. However, most of the systems that were studied are Predator-Prey systems,Competitive systems or two dimensional,three dimensional systems. The study about multi-dimensional Predator-Competition system with multi-delays can be rarely found. As for the oscillation of differential equations, many results have been obtained about the oscillation of functional differential equations with delay. But the study about neutral equations is not much. Especially, the study about oscillation of neutral equations with multi-delays only can be rarely found.In this paper, we study the global attractivity and oscillation of differential systems with multi-delays. In Chapter one, we discuss n+2 dimensional Lotka-Volterra Predator-Competition system with multi-delays.By Comparison theorem and V function, we obtain sufficient conditions that guarantee the persistence of system and global attractivity of positive periodic solution. In Chapter two, we discuss the oscillation of first order neutral nonlinear functional differential equations with multi-delays and obtain some sufficient oscillation criteria.Chapter IPersistence and Global Attractivity of n + 2 Dimensional Lotka-Volterra Predator-Competition System with Multi-delaysIn this chapter, we discuss n+2 dimensional Lotka-Volterra Predator-Competition System with Multi-delays :where x1(t),x2(t),y1(t) denote the density of species x1,x2 and yi at time t1respectively. Species x1, x2 and yi make up a predator-prey chain. x2 is the predator of x1, yi is the predator of x2. yi compete with each other. a1(t), a2(t), b1(t),b2(t),c1(t),C2(t),p1(t),p2(t),dk(t) ri(t),ei(t),ht(t) qik(t)(i,k = 1,2, .....n) are positive,continuous,periodic functions.are positive con stants. Denoting T = max{ max Tij max Tk, max ai, max denote the Banach space of all non-negative continuous functions.ForInitial function For non-negative continuous bounded functionMain results:Assume (Ho) (/) Consider equationequation (3.2) has positive -periodic solution (t),which is globally asymptotically stable. (//) Consider equationequation (3.3) has positive -periodic solution (t), which is globally asymptotically stable.(III) Consider equationequation (3.4) has positive periodic solution (t),which is globally asymptotically stable.(IV) Consider equationthen equation (3.5) has positive periodic solution (t),which is globally asymptotically stable. (V) Consider equationthen equation (3.6) has positive periodic solution (t),which is globally asymptotically stable.(VI)Consider equationthen equation (3.7) has positive periodic solution ,which is globally asymptotically stable. Theorem 1. If (Ho), (H!)>(H2), (H3) hold, then system (1) is uniformly persistent.Theorem 2. If (Ho), (H!)>(H2), (H3) hold, and there exist positive constants , such thatThen system (1) has a unique positive periodic solution, which is globally attractive.Chapter IIOscillation of First Order Neutral Nonlinear Functional Differential Equations with Multi-delaysIn this chapter, we discuss first order neutral nonlinear functional differential equations with multi-delays:We study the oscillation of the proper solutions which are defined on , and obtain oscillation criteria of equations (1) and (2).For equation (2), Assume (H0) , and is nondecreasing, and For equation (1), Assume (H2) Qi(t) is positive,continuous, periodic function with periodand for any x, the following inequalities hold:Main results:Theorem 1. If (H2),(H3) hold, and satisfies (H4) equation (1) is oscillatory.Theorem 2. If (H0),(H1) hold, and assume thatthere exist , such thatthere exist , such that (when t , and Then equation (2) is oscillatory.