Pace and shape
The concepts of pace and shape of mortality distinguish two different aspects of mortality (Baudisch 2011). The pace of mortality captures how long a species is expected to live. Pace is fast in populations that die within days or weeks (e.g. Bromus tectorum; Griffith 2010), slower in populations that can live years (e.g. Cryptantha flava; Lucas, Forseth & Casper 2008) and even slower when individual plants can live for centuries (e.g. Borderea pyrenaica; García, Dahlgren & Ehrlén 2011). Pace thus sets the time-scale of the life course of individuals in a population. In contrast, the shape of mortality is independent of time. Shape captures the direction and relative magnitude of change in mortality. Capturing direction, shape describes whether the force of mortality increases, remains constant, or decreases over an individual's life course. Capturing magnitude, shape details how much mortality changes over age, that is, whether it doubles, triples or changes by several fold relative to its average level.
Pace is measured by life expectancy at maturity, denoted by e0 (Baudisch 2011), because life expectancy is the most robust measure of life span, and senescence is a phenomenon affecting the adult life course (Williams 1957; Hamilton 1966). By definition, any shape measure needs to be independent of time and to provide a threshold level that corresponds to constant mortality, that is, negligible senescence. This threshold marks the boundary between senescence and negative senescence and clearly distinguishes shape values for increasing (senescence), constant (negligible senescence) and decreasing mortality patterns (negative senescence, sensu Vaupel et al. 2004). Shape values that are close to the threshold correspond to mortality trajectories that barely change over age. They have a ‘shallow’ shape. Shape values further away from the threshold represent mortality patterns with a ‘steep’ shape. Here, we measure shape (S) by the percentage survivors up to the expected age at death, S = l(e0). If mortality is constant over age, then l(e0) = e−1, because l(x) is given by
and e0 by
therefore l(x) = exp(−μx), e0 = 1/μ, and thus l(e0) = 1/exp(1). This inverse of e, the base of natural logarithms, is approximately equal to 0.37 and marks the threshold level of shape, that is, values above 0.37 correspond to mortality patterns that increase with age, those below 0.37 point to mortality patterns that fall with age, and values at 0.37 are indicative of mortality patterns that remain constant with age (see proof in mathematical Appendix A). Appendix A shows that, for monotonically decreasing mortality, fewer than 37% of adults survive longer than one life expectancy due to the initially high level of mortality, while in the case of monotonically increasing mortality, more than 37% of adults live up to and past the expected age at death, because death starts taking its toll only at higher ages.
ComPADRe III: the plant projection matrix data base
Our comparative analyses within angiosperms rest on a data base that integrates demographic, taxonomic, biogeographic and ecological information in vascular plants and algae: Comparative Plant & Algae Demographic Research (ComPADRe III; R. Salguero-Gómez, unpubl. data). Over 800 species in ComPADRe III have world-wide provenance, spanning all terrestrial ecoregions (Olson et al. 2001) and growth forms (Raunkiær 1934; see Raunkiaer's figure 2). The life expectancies of species in the data base range over several orders of magnitude, from annuals like Arabidopsis thaliana to the extremely long-lived Sequoia sempervirens (Namkoong & Roberds 1974).
At the demographic level, the data base contains thousands of published and personally communicated population projection matrices under different year, site and treatment conditions. Each population projection matrix contains vital rate information (e.g. survival, progression/retrogression, seed bank, vegetative dormancy, clonal and sexual reproduction) that determines the population dynamics of the species under the specified year, site and treatment (Caswell 2001; Morris & Doak 2002).
To examine the natural population dynamics of the largest sample size of angiosperm species possible, we calculated the element-by-element arithmetic mean A for all periods and sites for each species excluding experimental treatments (e.g. herbivory, fire or drought). Thus, each of the resulting matrices describes the population dynamics under conditions that are ‘normal’ to those species (but see 'Discussion' for limitations on spatial and temporal replication of the studies). The matrix A was further divided into three matrices: U details changes in classes throughout the life cycle of the species conditional on survival, F describes per-capita sexual reproductive contributions, and C represents clonal reproduction via ramets.
Age-from-stage matrix decomposition
We obtained deterministic age-specific survivorship trajectories (lx) and the probability of reaching a given age (x) using age-from-stage approaches (Cochran & Ellner 1992; Caswell 2001, 2006) applied to the U matrices for each species. The fact that we ignore F and C matrices implies that our report on ageing patterns disregards sexual and clonal reproduction of the study species. Briefly, the method relies on the fact that even though most of these projection matrices are based on size and/or developmental stage, surviving individuals inherently ‘age’, that is, grow a year older, if they survive one annual projection. In acquiring years, and by progressing or retrogressing into a different class, or by stasis (remaining in the same class) individuals are subject to a particular stage-specific survival probability that will decrease their lx values in an iterative, distinctive manner that is a property of the projection matrix of each species. An extensive explanation of the method is given by Caswell (Caswell 2001, 2006). The transformation from a survivorship (lx) schedule to the risk of mortality (μx) schedule is straightforward (Tatar, Carey & Vaupel 1993):
We then calculated the mean age at first reproduction (Lα), using similar methods (Cochran & Ellner 1992; Caswell 2001, 2006).
The age schedules (lx, μx and Lα) were calculated considering the first nonseed bank stage as the ‘beginning of life’. This approach avoids ambiguities about the amount of time that an individual plant might stay dormant as a propagule (Burns et al. 2010), an aspect that is difficult to quantify in plant demographic studies (Baskin & Baskin 2001).
We constructed Bayesian log-linear regression models to investigate how ecoregion, plant growth form and phylogenetic relationships influenced our measures of the pace and shape of mortality across angiosperms. The aim of the analysis was to determine the extent to which the expected values of pace or shape overlapped between different combinations of growth form and ecoregion and the extent to which phylogeny influenced these expected values. The basic model follows Freckleton, Harvey & Pagel (2002), whereby the probability density for the dependent variable, either pace or shape, is given by
where y is a vector of logged values of pace or shape for each species in the data set, β is a vector of parameters linking the covariates to y, X is the design matrix containing covariate values for each species in y, σ2 is the process model error, and V(λ) is the variance-covariance (VCV) matrix obtained from the phylogenetic tree with off-diagonal elements scaled by λ, also known as Pagel's lambda (Pagel 1999).1 This parameter is a scaling factor that is informative about the role of evolution in influencing trait values of terminal taxa; if ancestral trait values have no influence on the trait values of terminal taxa, the best model will be one where the VCV matrix has had the phylogenetic structure removed by the scaling factor (i.e. it is scaled with λ = 0, and is thus equivalent to a simpler nonphylogenetic model). On the other hand, if the ancestral trait values have a strong influence and have evolved via a Brownian motion process, the best model will be one where the phylogenetic structure is maintained in the VCV matrix (i.e. the VCV matrix is scaled by multiplying by 1, perfectly recovering the tree structure). An intermediate influence of phylogeny would result in the best model having a λ value somewhere between 0 and 1.
The covariates we tested were Raunkiær growth form (Raunkiær 1934) and ecoregion (Olson et al. 2001); Table S1, Supporting information). Raunkiær's classification is based on the location of the dormant shoot meristems of the species in relation to the ground during season of adverse conditions. In this categorization, species for which meristems are located close to the ground regenerate most or all of their aerial parts every year, thus having the potential to lose senescent aerial tissues (Salguero-Gómez & Casper 2010). The categories for this variable are the following: cryptophytes (shoot meristems are below-ground; rhizome, bulb and corm-producing plants), hemicryptophytes (shoot meristems at or near the surface; many rosette-forming plants), chamaephytes (shoot meristems < 25 cm above-ground; typically succulents and shrubs), phanerophytes (shoot meristems > 25 cm above-ground; typically shrubs and trees) and epiphytes (the height of the shoot meristems is measured with respect to their position on the host plant). To classify plants with respect to their habitat, we used a simplified version of Olson's classification of ecoregions based on information extracted from each publication containing the projection matrices in ComPADRe III. The ecoregions we used were temperate (‘Temperate Broadleaf and Mixed Forest’, ‘Temperate Coniferous Forest’ and ‘Temperate Grassland, Savanna and Shrubland’), tropical (‘Tropical and Subtropical Moist Broadleaf Forest’, ‘Tropical and Subtropical Dry Broadleaf Forest’, ‘Tropical and Subtropical Coniferous Forest’, and ‘Tropical and Subtropical Grassland, Savanna and Shrubland’), desert (‘Desert and Xeric Shrubland’), mediterranean (‘Mediterranean Forest, Woodland and Scrub’), montane (‘Montane Grassland and Shrubland’) and boreal (‘Boreal Forest/Taiga’).
Because the matrices used for this analysis were based on a range of modelling approaches, the population dynamics resulting from them could vary widely from species to species. These differences could affect comparative analysis of mortality patterns between species. To account for these potential confounding effects, we added life cycle criteria, matrix dimension and clonality as covariates in our analyses. The life cycle criteria used to classify individuals in a population were size, developmental stage, age or a combination of these state variables (Caswell 2001). We categorized each species’ mean control matrix, the matrix resulting from the element-by-element arithmetic mean of the matrices that report control conditions, into one of the seven possible combinations (i.e. size, stage, age, size and stage, size and age, stage and age, as well as size, stage and age). The number of dimensions of a population projection matrix (i.e. the number of life cycle stages) is positively correlated with the rates of progression and retrogression and negatively correlated with class-specific survival (Enright, Franco & Silvertown 1995; Salguero-Gómez & Plotkin 2010) and could thus also influence age-based trajectories obtained from them (R. Salguero-Gómez, unpubl. data). Finally, we classified the matrix of each species as genet-based or ramet-based. The latter was defined as species whose C matrix had at least one element > 0. This information is critical because in our analyses the mortality trajectory only refers to the U matrix (see section 'Age-from-stage matrix decomposition' above). Matrix dimension, the criteria used to specify the life cycle of the species, and the presence of clonality were included to control for potential confounding effects in our analyses.
Finally, to incorporate the phylogenetic structure in the aforementioned VCV matrix, we constructed a fully resolved tree by first obtaining a family-level tree with PHYLOMATIC (http://phylodiversity.net/phylomatic), then manually resolving it using MESQUITE (Maddison & Maddison 2009) by employing information from the Angiosperm Phylogeny Website (Stevens 2001) and specific detailed studies of phylogenetic relationships within families (see C. Mbeau-Ache & M. Franco, unpubl. data). Phylogenetic distances were interpolated by employing the bladj function of PHYLOCOM (Webb, Ackerly & Kembel 2008), using the estimated node ages from Wikström, Savolainen & Chase (2001).
After obtaining all the covariates and the VCV matrix mentioned previously, we constructed a Bayesian log-linear regression model for both pace and shape. The full Bayesian model was
where the first element of the input (p(y|β…) is the likelihood function described in (eqn 2), and the second element of this input (p(β|β0…) corresponds to the prior densities for the parameters. We used a conjugate multivariate-normal prior density for β, with mean β0 and VCV matrix βs I, and inverse gamma for σ2, with parameters s1 and s2. For parameter λ, we used a truncated normal distribution, with lower truncation at 0 and upper truncation at 1. We used a Markov Chain Monte Carlo algorithm to find posterior parameter distributions. We ran five parallel simulations for 10 000 iterations with a burn in of 2000 iterations, and each simulation was started from different parameter values to assess convergence. We used direct sampling for parameters β and σ2, and a Metropolis algorithm for Pagel's λ (Clark 2007).
To test the influence of the covariates described earlier, we ran 16 nested models for both pace and shape as dependent variables. These models had different combinations of explanatory variables (i.e. ecoregion and growth form), phylogeny (i.e. Pagel's λ > 0 or fixed to be equal to 0) and the presence of confounding variables (i.e. life cycle criteria, matrix dimension and clonality). We evaluated model fit by calculating deviance information criterion (DIC; Spiegelhalter et al. 2002).
When the selected model(s) included any of our explanatory covariates (ecoregion or growth form), both of which are categorical, we used Kullback–Leibler discrepancies (Burnham & Anderson 2001) to determine the level of overlap between the posterior distributions of their parameters. For instance, let Pi and Pj be the posterior densities for parameters βi and βj, where i and j are specific combinations of growth form or ecoregion. The K–L value that measures the amount of overlap of Pi over Pj, namely K(Pi, Pj), is calculated as
If both distributions are identical, then K(Pi, Pj) = 0, suggesting that there is no distinction between the expected values from i and j; as the K–L values increase, the discrepancy between the two distributions becomes higher. As can be inferred from (eqn 4), the relationship is asymmetric, namely K(Pi, Pj) ≠ K(Pj, Pi). Thus, for each combination of i’s and j's, we calculated two K–L values. We used McCulloch's calibration (McCulloch 1989) to make the K–L values easier to interpret. This calibration modifies the K–L values to range between 0.5 and 1, where values close to 0.5 imply high overlap and values close to 1 correspond to minimal overlap.