Persistence and global stability in Lotka-Volterra delay differential systems Original Research Article

Consider the persistence and the global asymptotic stability of models governed by the following Lotka-Volterra delay differential system: View the MathML sourcedxi(t)dt=xi(t){ci−aixi(t)−∑j=1naijxj(t−τij)},t≥t0,1≤i≤n,xi(t)=φi(t)≥0,t≤t0,andφi(t0)>0,1≤i≤n, Turn MathJax on where each i(t) is a continuous function for t ≤ t0, each ci, ai, and aij are finite and View the MathML sourceai>0,ai+aii>0,1≤i≤n,andτij≥0,1≤i,j≤n. Turn MathJax on In this paper, applying the former results [1], we obtain conditions for the persistence of the system, and extending a technique offered by Saito, Hara and Ma [2] for n = 2 to the above system for n ≥ 2, we establish new conditions for global asymptotic stability of the positive equilibrium which improve the well-known result of Gopalsamy for some special cases.