Permanence and global stability in a Lotka-Volterra predator-prey system with delays

Consider the permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type.system: x(i)(t)=x(i)(t) {r(i)-a(i)x(i)(t)-a(i)x(i-1) (t-tau(i,i-1))-b(i)x(i) (t-tau(i,i))-c(i)x(i+1)(t-tau(i),(i+1))}, tgreater than or equal to t(0), 1less than or equal toiless than or equal ton, with initial conditions x(i) (t)=phi(i) (t)greater than or equal to0, tless than or equal tot(0), and phi(i)(t(0))>0, 1less than or equal toiless than or equal ton. We define x(0)(t)=x(n+1)(t)equivalent to0 and suppose that phi(i)(t), 1less than or equal toiless than or equal ton, are bounded continuous functions on (t(0), +infinity) and r(i), alpha(i), c(i)>0, tau(i,j)greater than or equal to0, for all relevant i,j. Extending a technique of Saito, Hara and Ma [1] for n=2 to the above system for ngreater than or equal to2, we offer sufficient conditions for permanence and global asymptotic stability of the solutions which improve the well-known result of Gopalsamy. (C) 2003 Elsevier Ltd. All rights reserved.