  A general formalism is developed to construct a Markov chain model that
converges to a one-dimensional map in the infinite population limit. Stochastic
fluctuations are therefore internal to the system and not externally specified.
For finite populations an approximate Gaussian scheme is devised to describe
the stochastic fluctuations in the non-chaotic regime. More generally, the
stochastic dynamics can be captured using a stochastic difference equation,
derived through an approximation to the Markov chain. The scheme is
demonstrated using the logistic map as a case study.
