  Active force generation by actin-myosin cortex coupled to the cell-membrane
allows the cell to deform, respond to environment, and plays an important role
in cell motility and cell division. Several membrane bound activator proteins
move along the membrane and couple to its local curvature. In addition, they
can act as nucleating sites for the growth of filamentous actin. While actin
polymerization can generate outward push on the membrane, inward pull from the
contractile acto-myosin cortex can propagate to the membrane via actin
filaments. Using linear stability analysis and numerical calculations we
demonstrate the existence of three different dynamical phases depending on the
activity. The spherical membrane gets destabilized towards pattern formation,
generation of localized pulsation and traveling waves. We present our results
in terms of various phase diagrams, and evolution of the coupled fields. Our
results have implications for living cells and can be verified in experiments
on artificial cell-like constructs.
