The evolution of dispersal in random environments and the principle of partial control




In a recent paper, J. M. McNamara and S. R. Dall 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. Conditions that produce selection for increased unconditional dispersal are new instances of the Principle of Partial Control proposed to explain departures from reduction in 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 the rates of these processes. The model's behavior is shown to depend on the harmonic mean of the durations of the environments, causal connection between successive environments, and the eigenvalues of the environmental change matrix. Analysis of the adaptive landscape in the model shows that the evolution of conditional dispersal is very sensitive to the spectrum of phenotypic variation produced by the population, and suggests that empirical study of a particular species will require an evaluation of its variational properties.