Darwinian fitness, its directional derivative, and Hamilton's rule for limited dispersal with class structure under within and between generation environmental stochasticity

Darwinian fitness is equated here with invasion fitness and defined as the quantity determining the fate--certain extinction or possible spread--of a single mutant type. We derive it, together with its phenotypic derivative, for evolution in group-structured populations under limited genetic mixing, where the demography of the focal species and its environment is modeled as a discrete-time stochastic process. Reproduction, physiological development, dispersal, and survival are influenced by interactions within and between groups and by environmental fluctuations within and across generations. Using multitype branching processes in random environments, we show that invasion fitness is predicted by a stochastic growth rate that can be represented biologically in two meaningful genealogical ways. First, as the long-term geometric mean of the expected per-capita number of mutant copies produced per time step by a representative member of the mutant lineage. Second, as the the long-term geometric mean of the expected reproductive-value-weighted per-capita number of mutant copies produced by such an individual. This latter representation is useful for computing the phenotypic directional derivative of invasion fitness. Moreover, this derivative can be written as an actor-centered inclusive-fitness effect derived from properties of the resident population process. This effect depends on class-specific fitness differentials, relatedness, reproductive values, and class frequencies. However, unless generation- and class-specific fitness defines a stochastic matrix, the derivative does not separate stochastic reproductive values from relatedness and class frequencies, and must be evaluated by simulations. In summary, we formalize invasion fitness biologically quite generally and show how Hamiltons marginal rule is deduced from it.