Abstract
Long-term evolution of quantitative traits is classically and usefully described as the directional change in phenotype due to the recurrent fixation of new mutations. A formal justification for such continual evolution ultimately relies on the “invasion implies substitution” principle. This states that whenever a mutant allele causing a small phenotypic change can successfully invade a population, the ancestral (or wild-type) allele will be replaced, whereby fostering gradual phenotypic change if the process is repeated. It has been argued that this principle holds in a broad range of situations, including spatially and demographically structured populations experiencing frequency and density dependent selection under demographic and environmental fluctuations. However, prior studies have not been able to account for all aspects of population structure, leaving it unsettled in precisely which models does the “invasion implies substitution”-principle really hold. In this paper, we start by laying out a program to explore and clarify the generality of the “invasion implies substitution”-principle. Particular focus is given on finding an explicit and functionally constant representation of the selection gradient on a quantitative trait. We then show that the “invasion implies substitution”-principle generalizes to well-mixed and scalar-valued polymorphic multispecies ecological communities that are structured into finitely many demographic (or physiological) classes. We do this by setting up a continuous-time mutant-resident dynamical system for several interacting populations and species, and derive a closed expression for the selection gradient by separating the population dynamical and evolutionary timescales using geometric singular perturbation methods. We show that the selection gradient is constant in the relevant timescale and that it depends only on the resident phenotype, individual growth-rates, equilibrium population densities and reproductive values, all of which are calculated from the resident dynamics. Furthermore, we relate our results to previous work and discuss the theoretical tools required to address such problems. Our work contributes to the theoretical foundations of evolutionary ecology.