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.
