Abstract: The Hopf bifurcation of a class of deterministic model, which is three species food chain with diffusion phenomenon and group defense ability subject to Neumann boundary condition, is investigated. By treating the death rate of predator as bifurcation parameter, the stability of the positive constant equilibrium solution is discussed by use of Hurwitz criterion. Then, the conditions which can raise the Hopf bifurcation are given through the theoretical analysis. And also, the normal form method and center manifold theorem are used to study the Hopf bifurcation direction and stability of bifurcating periodic solutions of the spatial nonhomogeneous.

Key words: food chain model, Hopf bifurcation, diffusion

摘要： 研究了一类在Neumann边界条件下，具有扩散现象和群体防卫能力的三种群食物链模型的Hopf分支，以捕食者的死亡率为分支参数，利用Hurwitz判据讨论了系统正常数平衡解的稳定性，并通过理论分析给出了Hopf分支产生的条件，又利用规范形理论和中心流形定理得到了空间非齐次情形下Hopf分支的方向和分支周期解的稳定性。

关键词: 食物链模型, Hopf分支, 扩散