  A general theory is developed to study individual based models which are
discrete in time. We begin by constructing a Markov chain model that converges
to a one-dimensional map in the infinite population limit. Stochastic
fluctuations are hence intrinsic to the system and can induce qualitative
changes to the dynamics predicted from the deterministic map. From the
Chapman-Kolmogorov equation for the discrete-time Markov process, we derive the
analogues of the Fokker-Planck equation and the Langevin equation, which are
routinely employed for continuous time processes. In particular, a stochastic
difference equation is derived which accurately reproduces the results found
from the Markov chain model. Stochastic corrections to the deterministic map
can be quantified by linearizing the fluctuations around the attractor of the
map. The proposed scheme is tested on stochastic models which have the logistic
and Ricker maps as their deterministic limits.
