The author deals with a reaction-diffusion system in the dumbbell shaped domain. From the condition that the coefficients are very large, he concludes that the solution approaches a locally constant function except for the very thin region of the dumbbell domain. From the dynamical system point of view, this means that some specific finite dimensional set attracts all solutions eventually. This set is sometimes called an inertial manifold. Such a research has been done first by Y. Morita [J. Dyn. Differ. Equations 2, No.1, 69-115 (1990; Zbl 0702.35129)]. The present author obtains the reduced equation on the inertial manifold more precisely. To do this, he obtains a very elaborate characterization of the second eigenvalue of the dumbbell shaped domain (Neumann B. C.). He also gives an application to the competition system and obtains a result corresponding to the Hale and Vegas’s famous work on a single equation [see J. K. Hale and J. Vegas, Arch. Ration. Mech. Anal. 86, 99-123 (1984; Zbl 0569.35048)].