The time distribution of reproductive value measures the pace of life


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  1. It is agreed that, for senescence to occur, the intensity of natural selection must decline with age. Measures of the change in the intensity of natural selection with age include reproductive value and sensitivity of fitness to changes in survival and fecundity.
  2. To investigate the performance of these indices in predicting the pace and duration of life, which must be inversely related for senescence to occur, we quantified the temporal distribution of these measures employing a generalised logistic distribution tailored for this purpose. This distribution has three parameters two of which measure pace (units: per time) and one which measures duration (units: time). We hypothesised that, given their influence on the shape of the distribution, the time distribution parameters would also be correlated with specific life-history attributes. We tested these hypotheses employing demographic projections for a sample of 207 perennial plant species of varied life form and ecology.
  3. The results confirmed the expected relationships for the time distribution parameters of reproductive value, but not in general for other indices. In particular, a tight inverse relationship between one of the parameters of pace and the duration parameter of the time distribution of reproductive value ordered species along a fast–slow continuum where these two attributes compensate each other. That is, reproductive value was spread over a temporal scale that was in inverse proportion to its accruement.
  4. Synthesis. The tight negative power relationship between the pace and duration of life as measured on the time distribution of reproductive value provides the strongest support so far to the idea that the pace of life determines its duration and, as a corollary, the idea that reproductive value must be directly proportional to the intensity of natural selection. Senescence is the unavoidable consequence of the devaluation of the reproductive value currency.


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.

Materials and methods

A Biologically Meaningful Time Distribution

If the time course of reproductive value or of any of the measures of selection proposed by Hamilton (1966) and Baudisch (2005) were to occur at a constant rate, it would follow an exponential distribution:

display math(eqn 1)

where g is the rate of change of the measure of selection y, the latter expressed as a fraction of its total cumulative distribution, and x is age. More generally, however, this rate is likely to change as the organism ages. The simplest situation where the rate of change g varies monotonically with time (with probability b) is described by the logit, the logarithm of the odds in a binomial process. The inverse logit converts the logarithm of the odds into a probability (e.g. Liao 1994). In addition to this, biological processes are generally subject to time delays (e.g. sexual maturity takes time to be reached). This is taken into account by the lagged form of the inverse logit function, math formula, where t is the time lag.

Thus, the rate of change of g becomes math formula and its substitution into (1) yields

display math(eqn 2)

This cumulative distribution function (cdf) allows quantification of three different aspects of the cumulative temporal distribution of the measure of selection under investigation. Parameter g determines the rate at which the cdf rises, producing ‘diverging’ trajectories when b and t are held constant (Fig. 1a). Parameter b, on the other hand, shortens the time span over which the majority of the process occurs: increasing values of b reducing the temporal spread of the process (Fig. 1b). Finally, parameter t delays the process producing ‘parallel’ cdfs (Fig. 1c). In the case of reproductive value (vx), a steep rise of its cdf would indicate sexual precocity and, thus, g would be expected to be negatively correlated with age at sexual maturity (α). On the other hand, by being a measure of concentration of the time distribution of vx, b should be inversely related to the unstandardised form of demographic entropy (S), a measure of the temporal spread of reproduction (Demetrius 1974; Demetrius later differentiated between what he then termed standardised entropy, H (eqn. 4.87 in Caswell 2001), and the numerator of this measure, which he called unstandardised entropy, S; eqn. 4.96 in Caswell 2001). Finally, because t is an overall measure of the duration or delay of the distribution, we expect it to be positively correlated with life expectancy at birth (L).

Figure 1.

The effect of parameter values of a generalised logistic proposed to quantify the time distribution of measures of selection on the shape of the distribution. In each of the three graphs, one parameter is increased (from a to c: g, b and t), in the order continuous, dashed and dotted curves, while the other two are fixed at the values shown.

Plant Demography Data and Population Projection

The data set used in this study comes from a world-wide data base of demographic information for plants initiated in the late 1980 by Silvertown and Franco (Franco & Silvertown 1990) and greatly expanded by Salguero-Gomez (ComPADRe III, unpublished), held at the Max Planck Institute for Demographic Research, Rostock, Germany. The data base contains demographic information for close to 900 species of plants studied in their natural environments. Matrix models have been digitised for about 400 of these species and, given the fact that we had to decide when to stop adding species to our analyses, we concentrated on 207 species. The information consisted of one matrix population model for each species under natural conditions. When more than one matrix was presented (e.g. temporal and/or spatial replication), the average or pooled matrix was chosen/calculated. The species were classified into the following categories: trees (T), palms (P), shrubs (S), forest herbs (F) and herbs from open habitats (O). For expediency, we refer to these categories, which combine life form and habitat characteristics, as life forms.

The individual species' matrices were projected in MATLAB (MathWorks 2010) employing methods described in Cochran & Ellner (1992) and Caswell (2001, chapter 5) to generate age-based survival (lx) and fecundity (mx) schedules, as well as life expectancy at birth (L), and age at sexual maturity (α). The matrices for Lonicera maakii, Myrsine guianensis, Potentilla anserina and Viola fimbriatula, which contained multiple forms of reproduction, retrogression and/or dormant stages, did not yield L and/or α, and thus, the number of degrees of freedom is not always the same across all analyses (see Appendix S1 in Supporting Information). The intrinsic rate of population growth (r) was calculated as the natural logarithm of the largest eigenvalue of each matrix (λ1). In a previous study (Silvertown, Franco & Perez-Ishiwara 2001), reproductive value (vx) for species in a similar data set was calculated employing these methods, and it was observed that, because of increasing fecundity and low projected risk of mortality at advanced ages, reproductive value could remain constant or even be increasing towards the end of life. However, because this may be an artefact of the matrix projection model, which necessarily assumes constant survival in the last stage class, we assumed that the number of individuals surviving to age L would be too small and insignificant, and set maximum longevity to this value. This meant that reproductive value (vx) calculated employing the discrete version of Fisher's formula assuming lx had effectively decreased to zero at age L was also zero at this age. The life table parameters were also used to compute entropy (S) (Demetrius 1974), which measures the variability in the age at which individuals reproduce. Finally, we calculated the several measures of change in the force of selection outlined in Table 1 of Baudisch (2005). These are all measures of the sensitivity of r to changes in either linear or logarithmic measures of survival/mortality and fecundity.

Table 1. Descriptive statistics for the parameters of the time distribution of reproductive value (g, b and t) and the life-history attributes expected to be correlated with them [age at sexual maturity (α), entropy (S) and life expectancy (L)] for a sample of 207 perennial plants classified by life form (O: herbs from open habitats, F: herbs from forest understory, S: shrubs, P: palms, T: trees). The last column summarises the results of univariate analyses of variance of the difference between life forms in each of the six parameters/attributes. Homogeneous subsets of life forms at < 0.05 (Tukey's HSD) are indicated by superscripts. Because some species did not yield some life-history attributes (N = sample sizes in each group), the denominator degrees of freedom varied between 198 and 202
Parameter or AttributeLife formF4,198–202
= 80/81= 34/35= 20/21= 18= 51/52
  1. a

    < 0.001.

  2. b

    = 0.67.


Data Analyses

To quantify the parameters of the time distribution of the different measures of selection, we obtained their cumulative distribution by successively adding up their respective terms from = 0 to L. Each cumulative distribution was then standardised by dividing the series by their total sum (the last term of the cumulative distribution). The distribution function was fitted to the cumulative distributions of the individual selection measures for each individual species with the generalised nonlinear regression option of SPSS version 19 (IBM Corporation, Armonk, New York, USA), using the Levenberg–Marquardt algorithm and least square loss function. For cdfs that proved difficult to fit, alternative fitting methods were employed, but for some indices of selection, these did not produce better results: the algorithm would either fail to converge or produce evidently absurd parameter values (e.g. values of g>>1 or negative b values that produced declining or even oscillating cdfs, usually with large standard errors).

The expected relationships between the distribution and life-history parameters mentioned previously were investigated by phylogenetic generalised least squares (PGLS) models employing the caper package in R (Orme et al. 2012). To conduct these analyses, a phylogeny of the species in our data set was first constructed. This was achieved by first obtaining a higher-level tree in phylomatic (, discontinued 10 Jan 2013; Phylomatic v3 now at; Webb & Donohue 2005), which was then resolved manually at species level in mesquite (Maddison & Maddison 2009) employing information from the Angiosperm Phylogeny Website (Stevens 2012) and specific detailed molecular, and a few morphological studies of phylogenetic relationships within families (see Appendix S2). Phylogenetic distances were interpolated employing the bladj function of phylocom (Webb, Ackerly & Kembel 2008), using estimated node ages from Wikström, Savolainen & Chase (2001).


Model Fit

The model produced highly significant fits to the time distribution of the cumulative reproductive value for all 207 species of perennial plants used in this study with R2 values ranging from 0.981 to > 0.999 (Fig. 2; SEM 1). However, with the exception of the sensitivity of population growth rate (r) to the logarithm of fecundity (dr/dlnmx; last equation in Table 1 of Baudisch 2005), it performed poorly for all the other measures of the force of selection acting separately on survival and fecundity, often failing to converge (as explained in the Methods section). For this reason, parameter estimation could only be achieved for a handful of species in most measures of the sensitivity of r to survival/mortality and fecundity. This is understandable because many of these functions tend to decline monotonically, yielding cdfs that are convex, rather than sigmoidal, in shape. For dr/dlnmx, the fits were also highly significant but slightly lower and more variable than those for vx (mean R2 ± SD for the time distribution fits of vx and dr/dlnmx: 0.997 ± 0.006 and 0.996 ± 0.011, respectively; SEM 1). Interestingly, there was no correspondence (correlation) between the values of each of the three parameters for the distributions of vx and dr/dlnmx (R2 = 0.0051, > 0.20; R2 = 0.0135; > 0.08; R2 = 0.0076, > 0.15; one-tailed tests for the correlations between g, b and t of vx versus dr/dlnmx, respectively).

Figure 2.

Illustration of the fit of the time distribution to the vx data of five species, one from each of the life forms defined in the text. From left to right, and in the approximate order of life expectancy that one might expect the life forms to occur: Aquilegia sp. (O), Guarianthe aurantica (F), Lindera benzoin (S), Euterpe edulis (P) and Garcinia lucida (T). Grey line: vx projected from the matrix model; black line: model fit.

Expected Relationships between Life-History Attributes and the Parameters of the Time Distributions of vx AND dr/dlnmx

Before describing these relationships, it is important to note that, although with plenty of overlap, there were significant differences in age at sexual maturity (α), entropy (S) and life expectancy (L) among the five life forms (Table 1). The degree of difference among these life forms in the parameters of the distribution function for reproductive value, however, was lower than that for life-history attributes. Parameters g and t showed significant differences between life forms, but b did not (Table 1).

As hypothesised for the time distribution of vx, g was inversely related to α, b was inversely related to S and t was directly related to L, but the proportion of variance accounted for by each of these three relationships was small (Fig. 3, Table 2). The equivalent relationships for dr/dlnmx had lower proportion of variance accounted for (Table 2; figures not shown). Life form had no influence in any of these relationships (> 0.10 in all cases). In the case of t of vx versus L, regression through the origin produced a slope equal to 0.96, indicating a close one to one match despite substantial variation from species to species (the identical scale on both axes of Fig. 3c should allow the reader to mentally draw a line of equality). In these three PGLS models, phylogenetic signal (Pagel's λ) was equal to zero, and thus, the results are essentially undistinguishable from ordinary least squares.

Table 2. Phylogenetic generalised least squares models of the relationships between the parameters of the time distribution of reproductive value and life-history attributes, and those between the parameters of the time distribution of Baudisch's dr/dlnmx and the same life-history attributes in a sample of 207 species of perennial plants. α: age at sexual maturity, S: demographic entropy, L: life expectancy. Life form was not significant in any of these relationships and is therefore not included in the models
Dependent variableEffectSlope (SE) F 2,201–205 P R 2
g of vx α −0.0035 (0.0007)27.88< 0.0010.117
b of vx S −0.0608 (0.0110)30.39< 0.0010.125
t of vx L 0.568 (0.0513)122.5< 0.0010.372
g of dr/dlnmx α −0.0014 (0.0007)4.420.010.017
b of dr/dlnmx S −0.440 (0.1729)6.470.0020.026
t of dr/dlnmx L 0.585 (0.0674)75.38< 0.0010.266
Figure 3.

The relationships between model parameters and life-history attributes: (a) g vs. α, (b) b versus S, and (c) t versus L in a sample of 207 species of perennial plants classified by life form: O: herbs from open habitats, F: herbs from forest understory, S: shrubs, P: palms, T: trees.

Notice that the relationships depicted in Fig. 3a and b are in reality ‘wedges’ in which any value is allowed within their confines. Thus, a wide range of values of g and b are possible at low values of α and S, respectively, but both g and b become restricted to smaller values as, respectively, α and S increase. We will come back to interpret the wedge shape of these relationships in the discussion.

Among the three parameters of the time distribution of reproductive value (log transformed), t was negatively correlated with g (Pearson = −0.308) and b (= −0.891), and the latter two were consequently positively correlated with each other (= 0.310). As it would be expected from the positive correlation between t and L described previously, the relationship between g and t was negative and also formed a wedge (figure not shown). The most consistent relationship was that between b and t, as this did not form a wedge, but was characterised by a tight power relationship (R2 = 0.80) with a PGLS slope equal to −0.96 and whose 95% confidence interval includes −1, suggesting almost perfect compensation (Fig. 4; Table 3). The equivalent relationship for b and t of dr/dlnmx accounted for a smaller proportion of variance (R2 = 0.312) and was shallower (slope = −0.51) (Table 3). Interestingly, the relationships between b and t showed evidence of phylogenetic signal for both vx and dr/dlnmx (Pagel's λ = 0.365 and 0.389, respectively).

Table 3. Phylogenetic generalised least squares models of the relationships between parameters t and b (log transformed) of the distributions of reproductive value and the distribution of Baudisch's dr/dlnmx for 207 species of perennial plants
Dependent variableEffectIntercept (SE)Slope (SE) F 2,205 P R 2
t of vxlog(b)0.583 (0.082)−0.955 (0.034)802.5< 0.0010.796
t of dr/dlnmxlog(b)1.002 (0.212)−0.510 (0.053)94.3< 0.0010.315
Figure 4.

Power relationship between parameters b and t of the distribution of reproductive value and of the distribution of dr/dlnmx in a sample of 207 species of perennial plants. Symbols as in Fig. 3.


Although many studies have investigated the possible relevance of reproductive value to the process of senescence (e.g. Vahl 1981; Thompson 1984; Moller & De Lope 1999; Newton & Rothery 1997; Brown & Roth 2009; Bouwhuis et al. 2012), ours is the first to quantify three different aspects of its temporal distribution across a number of species of varied ecology. This distribution allowed us to quantify two measures of pace (g, related to how rapidly sexual maturity is reached, and b, related to the temporal concentration of reproduction throughout the life cycle) and one measure of duration (t, correlated with life expectancy) similar in spirit to the characterisation of pace and shape suggested by Baudisch (2011). All three parameters measured on vx correlated with the hypothesised life-history attributes better than the parameters estimated on the only measure of selection that fitted the time distribution model, dr/dlnmx. Life-history theory predicts that the rate with which individual species mature must be negatively correlated with lifespan (Williams 1966; Tinkle, Wilbur & Tilley 1970; Stearns & Crandall 1981; Charnov 1990) and, among other studies, earlier age at first reproduction has been associated with an earlier onset of reproductive senescence in red deer (Cervus elaphus) (Nussey et al. 2006), male blue-footed boobies (Sula nebouxii) (Kim et al. 2011) and 20 other mammal and bird species (Jones 2008). Similarly, an increase in reproductive effort early in life has been associated with accelerated senescence in fertility of collared flycatchers (Ficedula albicollis) (Gustafsson & Part 1990). The literature in this respect is large and consistent with the idea of trade-offs between survival and reproduction, and our comparative results agree with these observations.

While parameters b and t may be more clearly related to the ageing process, parameter g is also relevant to this issue because it determines the speed with which the peak of the reproductive value is reached and thus, if a trade-off between reproduction and survival exists, the ensuing decline in the intensity of natural selection. The negative influence of the spread of reproductive value (b, conceptually related to demographic entropy) on its duration (t, conceptually linked with L), however, does not seem to have received attention previously. On a broad scale, the relationships between the parameters quantifying the distribution of reproductive value and life-history attributes related to the ageing process go in the directions hypothesised and reinforce our confidence in reproductive value as a measure of the changing value of selection. We were initially enthusiastic about the idea of using the parameters of the time distribution of reproductive value to quantify the aspects of pace and shape of senescence suggested by Baudisch (2011). However, the units in which each of these parameters is measured made us reconsider these terms because they specifically quantify two aspects of pace (g and b units are per time) and one of duration (t units are time). If anything, the literature on sigmoid functions would refer to parameter b as a measure of shape, not t or its life-history proxy, L (see Baudisch et al. 2013). Significantly, our method identifies the pace and duration of life by reference to a single measure of expected future contribution to fitness, the reproductive value.

Specifically, despite confirmation of the direction of the relationships hypothesised, it was interesting to observe that only b and t showed a tight (presumably ‘functional’) relationship and that this relationship was characterised by almost perfect compensation. This suggests that the attributes of pace and duration of life are antagonistic and that the former constrains the latter. This relationship also suggests a compensatory symmetry resulting in a life-history invariant, that is, a power relationship with slope close to unity (Charnov 1993). This was not the case for all the other parameters whose relationships suggested limiting maximum combinations of their values, not theoretically bijective functional relationships. The fact that life-history traits only approximate expected general bivariate trends implies that they are not selected independently of each other. By incorporating all relevant demographic traits (mortality and reproductive schedules, but also age at sexual maturity, degree of iteroparity and life expectancy), the time distribution of reproductive value more accurately quantifies the variation in an individual's expected future contribution to fitness. If the relationship between b and t describes a general rule, the answer to the question of whether all plants senesce would have to be affirmative: different species senesce at different rates, and this determines their maximum life expectancy. Although with varying degrees of overlap in the attributes investigated across the five life forms here defined (Table 1), there is a continuum of species along the pace–duration relationship (Fig. 4). Incorporating life form into the model produces a slightly better fit (R2 = 0.83), with larger intercept for woody plants than herbaceous ones for vx. A similar, but weaker signal was obtained for dr/dlnmx.

Given the varied relationships that life-history attributes have (Charlesworth 1980; Roff 1992; McNamara & Houston 1996), it is perhaps remarkable that the pace and duration of life can be summarised by reference to the time distribution of a single parameter combining the age-specific schedules of survival and reproduction (Williams 1957; Partridge & Barton 1996). The opposition to the use of reproductive value as a measure of the intensity of selection seems to have arisen because of a misunderstanding of the significance of early mortality and the belief that the intensity of natural selection must necessarily decline with age (Hamilton 1966). The wasteful production of gametes and offspring, particularly in organisms with limited parental care, is a consequence of endless selection for the acquisition of more and more ‘lottery tickets’ with little chance of success. Under these conditions, selection favours the profligate spending on more and more offspring. This situation severely limits the value of the individual in early life (and consequently, their selection must be weak), but increases it as the individual develops and its chances of reproduction increase. Thus, for life to ever evolve beyond bacterial form, the intensity of natural selection for attributes that confer higher chances of survival and reproduction must necessarily increase with age and size (see Caswell & Salguero-Gómez 2013). As Baudisch (2005) and Caswell & Salguero-Gómez (2013) have shown, this depends on the measure of selection adopted. Our results advocate a re-evaluation of the significance of reproductive value as a measure of selection.


We thank Rob Salguero-Gómez for allowing us access to the ComPADRe III database. We are also indebted to the editors of Journal of Ecology, Mike Hutchings, Rob Salguero-Gómez and Richard Shefferson, as well as to an anonymous reviewer, for their suggestions for clarity and succinctness, which improved the manuscript considerably.