中文摘要：

考虑具有四个离散时滞的互惠合作模型。以四个时滞τ1，τ2，τ3，τ4的两种组合作为分支参数，基于对特征方程根的分析和规范型理论，研究两种情形下平衡点的稳定性及局部Hopf分支产生的充分条件，得出确定分支周期解稳定性及分支方向的算法及计算公式。数值模拟验证了理论分析结果，并给出了Hopf分支全局存在性的数值结果。

英文摘要：

For a class of cooperative model with four discrete time delays, with two combinations of four time delay τ1,τ2, τ3, τ4 being the bifurcation parameters, the stability of equilibrium points and the sufficient conditions of local Hopf bifurcation in two cases are studied based on the analysis of the characteristic root and the standard form theory. The algorithms and formulas for determining the stability and bifurcation direction of the branch periodic solutions are obtained. Numerical simulations are carried out to support the theoretical findings, and also a numerical result about the existence of global Hopf bifurcation is shown.