We investigate the existence and stability of stationary solutions in reaction–diffusion–ODE systems, which consist of a single reaction–diffusion equation coupled with ordinary differential equations. Within a bounded domain subject to Neumann boundary conditions, such systems may exhibit two types of stationary solutions: regular (i.e., at least continuous) and discontinuous. It is important to emphasize the distinctions between the dynamics of reaction–diffusion–ODE systems and those of classical reaction–diffusion systems. Regular stationary solutions include classical smooth stationary states. We show that all regular stationary solutions in reaction–diffusion–ODE systems are unstable. This result implies that these systems cannot sustain stable continuous spatial patterns, and any possible stable stationary solutions must be singular or discontinuous. In this study, we show sufficient conditions for the existence and stability of discontinuous stationary solutions.

Pulse dynamics arising in a three-component reaction–diffusion system with FitzHugh–Nagumo-type nonlinearity are investigated. Numerical investigations reveal that the pulse exhibits three distinct types of behavior as a control parameter is varied. To analytically examine these behaviors, finite-dimensional ordinary differential equations (ODEs) are derived to describe the individual motions of the pulse interfaces. The reduced ODE system successfully reproduces the pulse dynamics observed in the original reaction–diffusion system and clarifies the global bifurcation structure underlying the dynamics. The reduction method is further extended to multiple-interface solutions, offering insight into the mechanisms responsible for the numerically observed transient behaviors.