Concentration phenomena on Y-shaped metric graph for the Gierer–Meinhardt model with heterogeneity

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.