We present a method for obtaining survival and coexistence results for a class of interacting particle systems. This class includes: a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, a model for the evolution of cooperation of Ohtsuki, Hauert, Lieberman and Nowak, and a continuous time version of a non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath and Levin. Each of these, for a range of parameter values, can viewed as a "voter model perturbation," meaning the dynamics are "close" to the dynamics of the voter model, a simple, neutral competition model. The voter model is mathematically tractable because of its dual process, a system of coalescing random walks. We show that when space and time are rescaled appropriately the particle density converges to a solution of a reaction diffusion equation. Analysis of this equation leads in some cases to asymptotically sharp survival and coexistence results, which are qualitatively different from the (pure) voter model case. This work with Rick Durrett and Ed Perkins is closely related to earlier work of Durrett and Neuhaueser on models with rapid stirring.