  Cell division and death can be regulated by the mechanical forces within a
tissue. We study the consequences for the stability and roughness of a
propagating interface, by analysing a model of mechanically-regulated tissue
growth in the regime of small driving forces. For an interface driven by
homeostatic pressure imbalance or leader-cell motility, long and
intermediate-wavelength instabilities arise, depending respectively on an
effective viscosity of cell number change, and on substrate friction. A further
mechanism depends on the strength of directed motility forces acting in the
bulk. We analyse the fluctuations of a stable interface subjected to cell-level
stochasticity, and find that mechanical feedback can help preserve
reproducibility at the tissue scale. Our results elucidate mechanisms that
could be important for orderly interface motion in developing tissues.
