  In many adult tissues, stem cells and differentiated cells are not
homogeneously distributed : stem cells are arranged in periodic "niches", and
differentiated cells are constantly produced and migrate out of these niches.
In this article, we provide a general theoretical framework to study mixtures
of dividing and actively migrating particles, which we apply to biological
tissues. We show in particular that the interplay between the stresses arising
from active cell migration and stem cell division give rise to robust stem cell
patterns. The instability of the tissue leads to spatial patterns which are
either steady or oscillating in time. The wavelength of the instability has an
order of magnitude consistent with the biological observations. We also discuss
the implications of these results for future in vitro and in vivo experiments.
