The Necessary and Sufficient Condition for Global Stability of a Lotka–Volterra Cooperative or Competition System with Delays

This paper presents the necessary and sufficient condition for the global stability of the following Lotka–Volterra cooperative or competition system with time delays: x t = x t r1 − ax t + αx t − τ11 + βy t − τ12 y t = y t r2 − ay t + βx t − τ21 + αy t − τ22 It is showed that the positive equilibrium of the system is globally asymptotically stable for all delays τij ≥ 0 i j = 1 2 if and only if β < a− α and β ≤ a + α hold.