In this presentation I talk about the problem of approximating the behaviour of middle-sized population models involving non-linear rates. I describe a number of systems, each with a very different non-linear behaviour, and then show that the binomial and Poisson moment closure approximations have the potential to accurately represent the expected behaviour of these models. I then compare the two approximation methods to the mean field and normal moment closure approximations in terms of applicability and accuracy, in addition to investigating their empirical distributions, to derive conclusions regarding their strengths and drawbacks.