Population dynamics result from a combination of deterministic mechanisms (e.g. competition, predation) that drive nonlinear dynamics and stochastic forces that disrupt the neat patterns that would otherwise result. We often think of deterministic factors as being the most important, with their effects blurred secondarily by stochastic noise. In some particularly fascinating situations, however, it is unhelpful to thus emphasize deterministic drivers because stochasticity itself plays a role in shaping the overall pattern in the dynamics. In this way, stochasticity has a qualitative effect on the dynamics, such that dynamical patterns look quite different from what would result from the underlying deterministic factors alone. In this talk, I will use models to demonstrate how qualitative effects of stochasticity arise in population dynamics, and to explore why they occur. This work is part of an ongoing effort to better understand the role of stochasticity in ecological dynamics, eventually strengthening our ability to link patterns and processes.