Abstract
Most phenotypes, such as gene expression profiles, developmental trajectories, behavioural sequences or other reaction norms are function-valued traits, since they vary across an individual’s age and in response to various internal and/or external factors (state variables). In turn, many individuals live in populations subject to some limited genetic mixing and are thus likely to interact with their relatives. We here formalise selection on function-valued traits when individuals interact in a group-structured population, by deriving the marginal version of Hamilton’s rule for function-valued traits. This rule simultaneously gives a condition for the invasion of an initially rare mutant function-valued trait and its ultimate fixation in the population (invasion thus implies substitution). Hamilton’s rule thus underlies the gradual evolution of function-valued traits and gives rise to necessary first-order conditions for uninvadability (evolutionary stability). Using and extending results from optimal control theory and differential game theory, we characterise the first-order condition for time-dependent traits (dynamic traits) in terms of dynamic constraints on state variables and their marginal effects on reproductive value. Our results apply to both open-loop traits, which are function of time (or age) only, and closed-loop (state-feedback) traits, which are function of both time and state. This allows us to delineate role of state-dependence of trait expression and thus to other’s traits affects selection on function-valued trait, which pertains to both life-history and social evolution.
Competing Interest Statement
The authors have declared no competing interest.