Population dynamics result from a combination of deterministic mechanisms (e.g. competition, predation) that drive density-dependent dynamics and stochastic forces that disrupt the neat patterns that would otherwise result. Stochastic noise is often effectively viewed as a nuisance, seen as creating uncertainty and unpredictability without contributing in interesting ways to the list of mechanisms driving dynamics. However, it is becoming increasingly clear that in some situations, stochasticity itself plays an important qualitative role in shaping overall dynamical patterns, such that the dynamics cannot be fully understood by studying the deterministic mechanisms alone. Classical approaches to studying theoretical models are not well-equipped to make insights about these situations. Alternative analytical approaches exist but are not yet widely used in ecology. In this talk, I will present some useful ways to interpret and visualize effects of stochasticity in noisy ecological models, as well as some proof-of-concept examples to show the value of these approaches.