  We investigate the dynamics of a single breathing localized structure in a
three-component reaction-diffusion system subjected to the time-delayed
feedback. We show that variation of the delay time and the feedback strength
can lead either to stabilization of the breathing or to delay-induced periodic
or quasiperiodic oscillations of the localized structure. We provide a
bifurcation analysis of the system in question and derive an order parameter
equation, which describes the dynamics of the localized structure in the
vicinity of the Hopf bifurcation. With the aid of this equation, boundaries of
the stabilization domains as well as the dependence of the oscillation radius
on delay parameters can be explicitly derived, providing a robust mechanism to
control the behavior of the breathing localized structure in a straightforward
manner.
