  This is the second of two articles on the study of a particle system model
that exhibits a Turing instability type effect. About the hydrodynamic quations
obtained in \cite{CSL17a}, we find conditions under which Turing instability
occurs around the null equilibrium solution. In this instability regime: for
long times at which the process is of infinitesimal order, we prove that the
non-equilibrium fluctuations around the hydrodynamic limit are Gaussian; for
times converging to the critical one at which the process is of finite order,
we prove that the $\pm 1$-Fourier modes are uniformly away from zero.
