关键词: 互惠模型, 周期演化区域, 主特征值, 正周期解

Abstract: By studying  a class of three-species cooperating models on periodically evolving domain, we discussed  the effect of periodic evolution of domain on the persistence and  extinction of species. The existence and stability of the positive periodic solution of the model were studied by using the upper and lower solutions method, the comparison principle, the theory of quasi-monotone system and the priori estimates of parabolic equations. We recorded  ρ as the domain evolution rate. The results show that the effect of periodically  evolving domain on the persistence of species is negative when ρ^-2>1, it is positive when ρ^-2<1, and  there is no effect when ρ^-2=1.

Key words: cooperating model, periodically evolving domain, principal eigenvalue, positive periodic solution