Temporal variance in population density V is very generally related to mean density x by the power function, V = ax(b), with the value of b typically ranging from 1 to above 2 in different sets of data. We describe three simple Population dynamic models which are capable of generating variance-mean relationships with the observed range of b values. In the first model, the level of density dependence is correlated with mean density. This model is rejected because we find no relationship between the incidence of significant density dependence and mean density in data for moths and aphids. In the two other models, the level of environmental stochasticity in population change is correlated with mean density, affecting either the intrinsic (density-independent) growth rate or the equilibrium density (density dependence). Only the former model is capable of generating both positive and negative relationships between temporal variability in population size and density dependence. A positive relationship is predicted for species with high intrinsic growth rate, which may generate complex dynamics and thereby add to population variability. Aphids have higher intrinsic growth rates than moths, and as predicted, the relationship between population variability and density dependence is positive in aphids but negative in moths.