Global Asymptotic Behavior of a Nonautonomous Competitor-Competitor-Mutualist Model

Abstract

The global asymptotic behavior of a nonautonomous competitor-competitor-mutualist model is investigated, where all the coefficients are time-dependent and asymptotically approach periodic functions, respectively. Under certain conditions, it is shown that the limit periodic system of this asymptotically periodic model admits two positive periodic solutions $(u_{\mathrm{1}}^T,u_{\mathrm{2}T},u_{\mathrm{3}}^T),(u_{\mathrm{1}T},u_{\mathrm{2}}^T,u_{\mathrm{3}T})$ such that $u_{iT}\le u_i^T(i=\mathrm{1,2},\mathrm{3})$, and the sector $\{(u_{\mathrm{1}},u_{\mathrm{2}},u_{\mathrm{3}}):u_{iT}\le u_i\le u_i^T,i=\mathrm{1,2},\mathrm{3}\}$ is a global attractor of the asymptotically periodic model. In particular, we derive sufficient conditions that guarantee the existence of a positive periodic solution which is globally asymptotically stable.