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.