  This is the first of two articles on the study of a particle system model
that exhibits a Turing instability type effect. The model is based on two
discrete lines (or toruses) with Ising spins, that evolve according to a
continuous time Markov process defined in terms of macroscopic Kac potentials
and local interactions. For fixed time, we prove that the density fields weakly
converge to the solution of a system of partial differential equations
involving convolutions. The presence of local interactions results in the lack
of propagation of chaos, reason why the hydrodynamic limit cannot be obtained
from previous results.
