STATE-DEPENDENT LIFE-HISTORY EQUATIONS

Matrix population models provide a natural tool to analyse state-depen dent life-history strategies. Reproductive value and the intrinsic rat e of natural increase under a strategy, and the optimal life-history s trategy can all be easily characterised using projection matrices. The resultant formulae, however, are not directly comparable with the cor responding formulae for age structured populations such as Lotka's equ ations and Fisher's formula for reproductive value. This is because fo rmulae involving projection matrices lose track of what happens to an individual over its lifetime anti are only concerned with expected num bers of descendants one time step in the future. In contrast the usual age-dependent formulae explicitly followed a single individual throug h from birth to death. In this paper I show how the state-dependent fo rmulae can be rewritten to be directly comparable with the standard ag e-structured formulae. Although the formulae are intuitively obvious t he decomposition into current and future reproductive success differs from that previously given and is, I suggest, a more natural definitio n. The derivation of appropriate equations for optimal life-histories relies on results from dynamic programming theory; and is much more ge neral and easier than previous derivations. The value of rewriting pro jection matrix results in terms of the lifetime of an individual organ ism is illustrated by an example in which the optimal plastic response to an environment is derived.