## Introduction

The definition of senescence as the deterioration of state with age (reviewed by Finch 1990 and Rose 1991) and the realisation that for senescence to occur the intensity of natural selection must decrease with age (Medawar 1952) led to the proposition of specific measures of this decline (Hamilton 1966). These theoretical measures consist of separate estimators of the sensitivity of fitness (measured as population growth rate) to changes in survival and fecundity as the organism ages. Because deterioration of state is likely to be reflected in an increase in the probability of death, a decrease in the ability to reproduce, or both, it has also been suggested that the joint pattern of age-specific survival and reproduction expressed by reproductive value must provide an appropriate measure of the changing value of selection with age (Medawar 1952; Partridge & Barton 1996). It was Fisher himself who, when developing the concept of reproductive value, suggested that ‘the direct action of natural selection must be proportional to this contribution’ (Fisher 1930, p. 27). Despite the clarity of this statement, the fact that Fisher could *easily see* (our emphasis) the mathematical form that such a measure ought to take may account for his lack of emphasis on the relevance of reproductive value in the context of senescence. Fisher had derived a formula that only later would be found to be equivalent to the left eigenvector of a population model expressed in matrix form (Leslie 1948) and thought this formula was too obvious to be worried about it. Fisher died 6 years before the publication of Hamilton's paper, and we can only speculate about what opinion he would have had about Hamilton's work. What must be acknowledged, however, is the fact that Fisher suggested reproductive value specifically as a measure proportional to the changing value of natural selection with age.

Twenty-two years after the publication of Fisher's book, Medawar (1952) established the demographic dimension of the problem of senescence. Although Medawar was clear about the relevance of the changing reproductive value of the individual with age, his emphasis on the demographic signature of the decline of physiological state with age in the shape of the mortality curve may account for the weight placed by subsequent authors on it (see Finch 1990 and Ricklefs 1998). For reasons that should become clearer later, we believe reproductive value may better capture the selection conditions determining the duration of life.

Baudisch (2005) generalised Hamilton's indices of selection and found that alternative measures of the sensitivity of fitness to changes in either survival or fecundity with age predicted increased selection before it eventually declined. Our proposition here is that because these measures are still separate estimators of the intensity of natural selection with age, reproductive value may be a better measure of it. We suggest that the criteria to evaluate the performance of each of the indices suggested by Fisher and Hamilton and Baudisch would be their ability to conform to a generalised time distribution whose parameters can be linked with specific life-history components. Applied to the different measures of selection, the parameters of this distribution represent (see next section) the following: (i) the rate at which the intensity of selection initially increases; (ii) a measure of how this initial rate decreases with age; this rate also measures the concentration of the temporal spread of selection and (iii) an overall measure of duration or temporal delay in the distribution of selection. We estimated these three parameters on a sample of 207 perennial plant species for which detailed demographic information allowed estimation of the different selection indices. We hypothesised that reproductive value would produce the more consistent estimation of these parameters. More specifically, we hypothesised that: (i) the initial rate of increase in an efficient estimator of selection would correlate directly with age at sexual maturity; (ii) its temporal concentration would correlate inversely with demographic entropy (a measure of the spread of reproduction) and (iii) its temporal delay would correlate directly with life expectancy. Furthermore, because the parameters of the time distribution constitute measures of either pace (the first two parameters have units per time) or duration (the third parameter has units time), we hypothesised a positive relationship across species between the first two parameters and negative relationships between each of these first two parameters and the third. These negative correlations would be the clearest measures of a fast–slow continuum of selection, and thus of life-history variation across species (e.g. Franco & Silvertown 1996). The trade-off implied by these negative relationships would then bear upon the issue of senescence.