Altenberg, L.
McNamara and Dall (2011) identified novel relationships between 1) the abundance of a species in different environments, 2) the temporal properties of environmental change, and 3) selection for or against dispersal. Here, the mathematics underlying these relationships in their two-environment model are investigated for arbitrary numbers of environments. A population statistic, the fitness-abundance covariance, is introduced, which quantifies the property they describe. It is the covariance between growth rates and the excess abundance of the population over what it would be without heterogeneous growth rates. Its value depends on the phase in the life cycle when the population is censused, and the pre-dispersal and post-dispersal values differ as an example of Fisher's Fundamental Theorem. The fitness-abundance covariance is shown to involve the Reduction Principle from the population genetics literature on the evolution of genetic systems and migration, which is reviewed.