Global attractivity of an integrodifferential model of mutualism (Q1725295)

Summary: Sufficient conditions are obtained for the global attractivity of the following integrodifferential model of mutualism: \(d N_1(t) / d t = r_1 N_1(t) [((K_1 + \alpha_1 \int_0^{\infty} J_2(s) N_2(t - s) d s)/(1 + \int_0^{\infty} J_2(s) N_2(t - s) d s))- N_1(t)]\), \(d N_2(t) / d t = r_2 N_2(t) [((K_2 + \alpha_2 \int_0^{\infty} J_1(s) N_1(t - s) d s)/(1 + \int_0^{\infty} J_1(s) N_1(t - s) d s- N_2(t)]\), where \(r_i, K_i\), and \(\alpha_i\), \(i = 1,2\), are all positive constants. Consider \(\alpha_i > K_i\), \(i = 1,2 \). Consider \(\mathrm{J}_i \in C([0, + \infty)\), \([0, + \infty))\) and \(\int_0^{\infty} \mathrm{J}_i(s) d s = 1\), \(i = 1,2 \). Our result shows that conditions which ensure the permanence of the system are enough to ensure the global stability of the system. The result not only improves but also complements some existing ones.