In this paper, we consider the Gierer–Meinhardt model with the heterogeneity in both the activator and the inhibitor on the $Y$-shaped compact metric graph. Using the Lyapunov–Schmidt reduction method, we construct a one-peak stationary solution, which concentrates at a suitable point. In particular, we reveal that the location of concentration point is determined by the interaction of the heterogeneity function for the activator with the geometry of the domain, represented by the associated Green’s function. Moreover, based on our main result, we determine the precise location of concentration point for non-heterogeneity case. Furthermore, we also present the effect of heterogeneity by using a concrete example.

在本文中，我们考虑了Gierer–Meinhardt模型，该模型在激活剂和抑制剂上均具有异质性。 $ÿ$形紧凑度量图。使用Lyapunov–Schmidt归约法，我们构造了一个单峰固定解，集中在一个合适的点上。特别地，我们揭示了集中点的位置是由激活剂的异质性函数与域的几何形状（由关联的格林函数表示）之间的相互作用所决定的。此外，根据我们的主要结果，我们确定了非异质性情况下集中点的精确位置。此外，我们还通过一个具体的例子来介绍异质性的影响。