Uniform persistence for Lotka-Volteffa-type delay differential systems

Consider the uniform persistence (permanence) of models governed by the following Lotka-Volterra-type delay differential system: dx(i)(t)/dt = x(i)(t)r(i)(t){c(i)-a(i)x(i)(t)-Sigma(j=1)(n)Sigma(k=0)(m)a(ij)(k)x(j)(tau(ij)(k)(t))}, t greater than or equal to t(0), 1less than or equal to i less than or equal to n, m x(i)(t) = phi(i)(t) greater than or equal to 0, t less than or equal to t(0) and phi(i)(t(0)) > 0, 1 less than or equal to i less than or equal to n, where each r(i)(t) is a nonnegative continuous function on [0, + infinity), r(i)(t) not equivalent to 0, each a(i) greater than or equal to 0 and tau(ij)(k)(t) less than or equal to t, 1 less than or equal to i, j less than or equal to n, 0 less than or equal to k less than or equal to m. In this paper, we establish sufficient conditions of the uniform persistence and contractivity for solutions (and global asymptotic stability). In particular, we extend the results in Wang and Ma (J. Math. Anal. Appl. 158 (1991) 256) for a predator-prey system and Lu and Takeuchi (Nonlinear Anal. TMA 22 (1994) 847) for a competitive system in the case n = 2, to the above system with n greater than or equal to 2. (C) 2003 Elsevier Science Ltd. All rights reserved.