Optimal Life Histories, Optimal Notation, and the Value of Reproductive Value

Reproductive value is the central construct in life history optimization. In the optimal life history, the reproductive value at every age represents the solution to a constrained maximization problem. This holds true regardless of trade-offs across age classes. The recent literature arguing to the contrary has misunderstood the principle and has neglected the constraints in the maximization. In reviewing the matter, this paper considers maximization over segments of the life history, maximization over segments taken in reverse sequence, maximization of terms of a decomposition into instantaneous contributions at a given age, and maximization of reproductive value over alternative states such as the sexes. To clarify the relation between the maximization procedure and the constraints, a simple iterative approach to achieving maximization and simultaneously satisfying the constraints is described. To arrive at the instantaneous decomposition, it is shown that reproductive value arises as an adjoint variable in an optimal control formulation, where effects across age classes are explicitly incorporated in a variable representing physiological state. In considering alternative states, this paper defines a male reproductive value and concludes that at equilibrium sex composition, the reproductive values of each sex will be equal at any age where a sex change is possible. Coincidentally, some common errors in the discrete time formulation of reproductive value and Lotka's equation are corrected.