Permanence criteria for general delayed discrete nonautonomous n-species Kolmogorov systems and its applications

In this paper, the permanence for the general nonautonomous n-species discrete Kol- mogorov systems with delays is studied. By using the method of multiple discrete Lyapunov-like functionals, a quite general sufficient condition is established. As application of this result, new sufficient conditions are obtained to ensure the permanence of all species for the discrete nonautonomous pure-delayed n-species Lotka Volterra competitive sys- tems and food chain systems. Particularly, as special cases, a series of new sufficient con- ditions on the permanence for the discrete nonautonomous pure-delayed Lotka Volterra two-species competitive systems, predator prey systems and three-species food chain sys- tems, and the discrete autonomous pure-delayed Lotka Volterra competition systems and food chain systems are obtained.