中文摘要：

该文讨论齐次Neumann边界条件下Gierer＋Meinhardt活化一抑制扩散模型．对于空间均匀（ODE）系统，分析了内部平衡态的渐近行为及其附近极限环的存在性和稳定性；对于空间异质（PDE）系统，给出了内部平衡态的Turing不稳定性条件，说明了Turing模式和时空周期模式的存在性．最后，通过数值算例验证了相应理论结果．

英文摘要：

The diffusive Gierer-Meinhardt activator-inhibitor model system with Neumann boundary condition is investigated. For the spatial homogeneous （ODE） system, we perform the asymptotic behavior of the interior equilibrium and the existence and stability of limit cycle surrounding the interior equilibrium. For the spatial inhomogeneous （PDE） system, we consider the Turing instability of the interior equilibrium and show the existence of Turing pattern and inhomogeneous periodic oscillatory pattern. To verify our theoretical results, some numerical simulations are also done as a complement.