Global Attractivity of an Integrodifferential Model of Mutualism

Abstract

Sufficient conditions are obtained for the global attractivity of the following integrodifferential model of mutualism: $d{N}_{\mathrm{1}}\mathrm{}(t)/dt=r_{\mathrm{1}}{N}_{\mathrm{1}}(t)[((K_{\mathrm{1}}+{\alpha }_{\mathrm{1}}\mathrm{}{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{2}}\mathrm{}(s)N_{\mathrm{2}}\mathrm{}(t-s)ds)\mathrm{‍}/(\mathrm{1}+{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{2}}\mathrm{}(s)N_{\mathrm{2}}\mathrm{}(t-s)ds))\mathrm{‍}-N_{\mathrm{1}}(t)]$, $d{N}_{\mathrm{2}}\mathrm{}(t)/dt=r_{\mathrm{2}}{N}_{\mathrm{2}}(t)[((K_{\mathrm{2}}+{\alpha }_{\mathrm{2}}\mathrm{}{\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{}{J}_{\mathrm{1}}(s)N_{\mathrm{1}}\mathrm{}(t-s)ds)\mathrm{‍}/(\mathrm{1}+{\int }_{\mathrm{0}}^{\mathrm{\infty }}{J}_{\mathrm{1}}\mathrm{}(s)N_{\mathrm{1}}\mathrm{}(t-s)ds\mathrm{‍}))-N_{\mathrm{2}}(t)]$, where $r_i,K_i,$ and ${\alpha }_i$, $i=\mathrm{1,2}$, are all positive constants. Consider ${\alpha }_i>K_i$, $i=\mathrm{1,2}.$ Consider $J_i\in C([\mathrm{0},+\mathrm{\infty }),[\mathrm{0},+\mathrm{\infty }))$ and ${\int }_{\mathrm{0}}^{\mathrm{\infty }}\mathrm{‍}{J}_i(s)ds=\mathrm{1}$, $i=\mathrm{1,2}.$ 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.