  In this paper, we prove the local-controllability to positive constant
trajectories of a nonlinear system of two coupled ODE equations, posed in the
one-dimensional spatial setting, with nonlocal spatial nonlinearites, and using
only one localized control with a moving support. The model we deal with is
derived from the well-known nonlinear reaction-diffusion Gray-Scott model when
the diffusion coefficient of the first chemical species $d_u$ tends to $0$ and
the diffusion coefficient of the second chemical species ${d_v}$ tends to $+
\infty$. The strategy of the proof consists in two main steps. First, we
establish the local-controllability of the reaction-diffusion ODE-PDE derived
from the Gray-Scott model taking $d_u=0$, and uniformly with respect to the
diffusion parameter ${d_v} \in (1, +\infty)$. In order to do this, we prove the
(uniform) null-controllability of the linearized system thanks to an
observability estimate obtained through adapted Carleman estimates for ODE-PDE.
To pass to the nonlinear system, we use a precise inverse mapping argument and,
secondly, we apply the shadow limit ${d_v} \rightarrow + \infty$ to reduce to
the initial system.
