Eventual Behavior On The Non-Autonomous Lotka-Volterra Competitive Systems With Delay

Recently, mathematical ecology has been studied extensively in pest control, fish catches, neural network, food chain and the prevention and cure of the infectious disease, and so on. For this reason more and more scholars all concentrate on the research of this aspect. Lotka-Volterra systems are very important in population dynamics. There are considerable works on the study of permanence, extinction and global asymptotic behaviors of Lotka-Volterra systems with delays. Scholars study the system not only nonautonomous with nondelay but also nonautonomous with delay, periodic delay and neutral.Extinction of species of the systemis considered in [3]. In this paper, we consider the following nonautonomous N-species Lotka-Volterra type competitive systems with discrete delay and continuous delay where x_i(t) denote the density of ith species at time t. The average conditions on the coefficients are obtained to guarantee that all the species in the systems (1.1) are extinction by constructing proper Lyapunov function. The result of this chapter is a extension to [3]. In [6], extinction of system (1.1) is also considered, but the results we obtained in the chapter are weaker than [6].Now, permanence on nonautonomous Lotka-Volterra system with impure delay have been studied by many scholars. For example [7]-[8],[15]-[18]. In the second part, we discuss permanence of solutions of nonautonomous Lotka-Volterra system with impure delay...