Carrying simplex in the Lotka-Volterra competition model with seasonal succession with applications

Abstract

We investigate the dynamics of the Poincar \begin{document}$ \acute{\rm e} $\end{document} -map for an \begin{document}$ n $\end{document} -dimensional Lotka-Volterra competitive model with seasonal succession. It is proved that there exists an \begin{document}$ (n-1) $\end{document} -dimensional carrying simplex \begin{document}$ \Sigma $\end{document} which attracts every nontrivial orbit in \begin{document}$ \mathbb{R}^n_+ $\end{document} . By using the theory of the carrying simplex, we simplify the approach for the complete classification of global dynamics for the two-dimensional Lotka-Volterra competitive model with seasonal succession proposed in [Hsu and Zhao, J. Math. Biology 64(2012), 109-130]. Our approach avoids the complicated estimates for the Floquet multipliers of the positive periodic solutions.