Permanence for the Discrete Competition Model with Infinite Deviating Arguments

Sufficient conditions are obtained for the permanence of the following discrete model of competition: x(1)(k + 1) = x(1)(k)exp{r(1)(k)[K-1(k) + alpha(1)(k) Sigma(+infinity)(s=0) J(2)(s)x(2)(k - s))/1 + Sigma(+infinity)(s=0) J(2)(s)x(2)(k - s)) - x(1) (k - delta(1)(k))]]; x(2)(k + 1) = x(2)(k)exp [r(2)(k) [K-2(k) + alpha(2)(k) Sigma(+infinity)(s=0) J(2)(s)x(2)(k - s))/(1 + Sigma(+infinity)(s=0) J(1)(s)x(2)(k - s)) - x(2)(k - delta(2)))]} where r(i), K-i, alpha(i), J(i), and delta(i), i = 1,2 are nonnagative sequences bounded above and below by positive constants, and K-i > alpha(i,) i =1,2.