The dynamics of cell edges are closely related to those of F-actin. Recent fluorescence imaging studies have shown that F-actin can spontaneously form traveling waves or moving patches at low actin concentrations. I investigate possible mechanisms for such phenomena by numerically simulating the “dendritic nucleation” model of actin network growth. The simulations treat actin network growth on a 3 by 3 micron piece of membrane. They store information about actin filaments subunit by subunit, giving an explicit three-dimensional picture of the actin network. The calculations include filament growth, capping, branching, severing, and random thermal motion. The dynamics of nucleation-promoting factors (NPFs) in the membrane are also included: they diffuse in the membrane, and detach/inactivate in the presence of F-actin. The simulations show three types of behavior which are “tuned” by the actin concentration: 1) traveling waves, 2) coherently moving patches, and 3) random fluctuations with occasional moving patches. Wave formation is favored by a long recovery time for NPFs which have been inactivated, and by weakness of the attractive interaction between filaments and the membrane. Low G-actin concentrations cause waves to break up into patches which, however, move coherently. Similar effects are seen with increasing capping-protein concentration. Diffusion of NPFs slows the waves, and, if fast enough, stops them completely, resulting in the formation of a static spot.