Permanence for an integrodifferential model of mutualism

Sufficient conditions are obtained for the permanence of the following integrodifferential model of mutualism dN(1)(t)/dt = r(1)(t)N-1(t) [K-1(t) + alpha(1)(t) integral(infinity)(0) J(2)(s)N-2(t - s)ds/1 + integral(infinity)(0) J(2)(s)N-2(t -s)ds - N-1(t - sigma(1)(t))], dN(2)(t)/dt = r(2)(t)N-2(t) [K-2(t) + alpha(2)(t) integral(infinity)(0) J(1)(.)(s)N-1(t - s)ds/1 + integral(infinity)(0) J(1)(s)N-1(t - s)ds - N-2(t - sigma(2)(t))], where r(i), K-i, alpha(i) and sigma(i), i = 1, 2 are continuous functions bounded above and below by positive constants. alpha(i) > K-i, i = 1, 2. J(i) is an element of C([0, + infinity), [0, + infinity)) and integral(infinity)(0) J(i)(s)ds = 1, i = 1, 2. (c) 2006 Elsevier Inc. All rights reserved.