Global Hopf bifurcation for three-species ratio-dependent predator-prey system with two delays

In this paper, the effect of the two different delays on the dynamics of a three-species ratio-dependent predator-prey food-chain model is considered. By regarding the delay as the bifurcation parameter, the local stability of the positive equilibrium and the existence of Hopf bifurcation are investigated. Explicit formulas determining the properties of a Hopf bifurcation are obtained by using the normal form method and the center manifold theorem. Special attention is paid to the global continuation of local Hopf bifurcation when the delay τ1≠τ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau _{1}\neq\tau_{2}$\end{document}. Finally, several numerical simulations supporting the theoretical analysis are also given.