Want to study validity of rate equation and fluctuations of n for large system. Mean system size parameter. Describe this situation as non-fluctuating variable. When n fluctuates by an amount , x will fluctuate by . Master equation Need to substitute into master equation and convert to equation for , sorting terms according to order in .
Population approaches carrying capacity .
Substitute the expansions for E, E -1 . Solution Solution depends only on net birth rate . The master equation was expanded in powers of . Malthus-Verhulst problem used as an example. was found to satisfy a macroscopic deterministic rate equation. Probability distribution of x satisfied a Fokker-Planck equation.