  McNamara and Dall (2011) identified novel relationships between the abundance
of a species in different environments, the temporal properties of
environmental change, and selection for or against dispersal. Here, the
mathematics underlying these relationships in their two-environment model are
investigated for arbitrary numbers of environments. The effect they described
is quantified as the fitness-abundance covariance. The phase in the life cycle
where the population is censused is crucial for the implications of the
fitness-abundance covariance. These relationships are shown to connect to the
population genetics literature on the Reduction Principle for the evolution of
genetic systems and migration. Conditions that produce selection for increased
unconditional dispersal are found to be new instances of departures from
reduction described by the "Principle of Partial Control" proposed for the
evolution of modifier genes. According to this principle, variation that only
partially controls the processes that transform the transmitted information of
organisms may be selected to increase these processes. Mathematical methods of
Karlin, Friedland, and Elsner, Johnson, and Neumann, are central in
generalizing the analysis. Analysis of the adaptive landscape of the model
shows that the evolution of conditional dispersal is very sensitive to the
spectrum of genetic variation the population is capable of producing, and
suggests that empirical study of particular species will require an evaluation
of its variational properties.

Populations and Evolution;Spectral Theory