The instabilities observed in the atomic cloud of a magneto-optical trap are experimentally studied through the dynamics of the center of mass location and the cloud population. Two dynamical components are identified: a slow, stochastic one affects both variables, and a fast, deterministic one affects only the center of mass location. A one-dimensional stochastic model taking into account the shadow effect is developed from these observations and reproduces the experimental behavior. It is shown that instabilities are driven by noise and present stochastic resonancelike characteristics.