  Crohn's disease is an inflammatory bowel disease (IBD) that is not well
understood. In particular, unlike other IBDs, the inflamed parts of the
intestine compromise deep layers of the tissue and are not continuous but
separated and distributed through the whole gastrointestinal tract, displaying
a patchy inflammatory pattern. In the present paper, we introduce a toy-model
which might explain the appearance of such patterns. We consider a
reaction-diffusion system involving bacteria and phagocyte and prove that,
under certain conditions, this system might reproduce an activator-inhibitor
dynamic leading to the occurrence of Turing-type instabilities. In other words,
we prove the existence of stable stationary solutions that are spatially
periodic and do not vanish in time. We also propose a set of parameters for
which the system exhibits such phenomena and compare it with realistic
parameters found in the literature. This is the first time, as far as we know,
that a Turing pattern is investigated in inflammatory models.
