  We study the existence and stability of small-amplitude periodic waves
emerging from fold-Hopf equilibria in a system of one reaction-diffusion
equation coupled with one ordinary differential equation. This coupled system
includes the FitzHugh-Nagumo system, caricature calcium models,
consumer-resource models and other models in the real-world applications. Based
on the recent results on the averaging theory, we solve periodic solutions in
related three-dimensional systems and then prove the existence of periodic
waves arising from fold-Hopf bifurcations. Numerical computation in [J. Tsai,
W. Zhang, V. Kirk, and J. Sneyd, SIAM J. Appl. Dyn. Syst. 11 (2012),
1149--1199] once suggested that the periodic waves from fold-Hopf bifurcations
in a caricature calcium model are spectrally unstable, yet without a proof.
After analyzing the linearization about periodic waves by the relatively
bounded perturbation, we prove the instability of small-amplitude periodic
waves through a perturbation of the unstable spectra for the linearizations
about the fold-Hopf equilibria. As an application, we prove the existence and
stability of small-amplitude periodic waves from fold-Hopf bifurcations in the
FitzHugh-Nagumo system with an applied current.
