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

AbstractConsider the permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type system: ẋi(t) = xi(t) {ri − αixi(t) − aixi−1 (t − γi,i−1) − bixi (t − γi,i) − cixi+1 (t − γi,i+1)}, with initial conditions xi(t) = φi(t) ≥ o, t ≤ t0, and φi(t0) > 0. 1 ≤ i ≤n. We define x0(t) = xn+1(t)≡0 and suppose that φi(t), 1 ≤ i ≤ n, are bounded continuous functions on [t0, + ∞) and γi, αi, ci > 0,γi,j ≥ 0, for all relevant i,j.Extending a technique of Saito, Hara and Ma[1] for n = 2 to the above system for n ≥ 2, we offer sufficient conditions for permanence and global asymptotic stability of the solutions which improve the well-known result of Gopalsamy