Individual stochasticity in the life history strategies of animals and plants

The life histories of organisms are expressed as rates of development, reproduction, and survival. However, individuals may experience differential outcomes for the same set of rates. Such individual stochasticity generates variance around familiar mean measures of life history traits, such as life expectancy and the reproductive number R0. By writing life cycles as Markov chains, we calculate variance and other indices of variability for longevity, lifetime reproductive output (LRO), age at offspring production, and age at maturity for 83 animal and 332 plant populations from the Comadre and Compadre matrix databases. We find that the magnitude within and variability between populations in variance indices in LRO, especially, are surprisingly high. We furthermore use principal components analysis to assess how the inclusion of variance indices of different demographic outcomes affects life history constraints. We find that these indices, to a similar or greater degree than the mean, explain the variation in life history strategies among plants and animals.


Measures of variance and uncertainty
The results of the calculations of life history outcomes are typically a set of vectors whose entries give the moments of some demographic outcome (ξ) for individuals starting in each stage. Let ξ m denote the vector for the mth moments. Then the variances are given by the vector The skewness and kurtosis are most easily written in terms of the vectors of moments around the meanξ (3) ξ 4 = ξ 4 − 4(ξ 1 • ξ 3 ) + 6(ξ 1 • ξ 1 • ξ 2 ) − 3(ξ 1 • ξ 1 • ξ 1 • ξ 1 ) (4) In terms of these central moments, the vectors giving the skewness and excess kurtosis of the elements of ξ are given by The skewness quantifies the asymmetry of the distribution; positive values indicate an extended positive tail, negative values indicate the opposite. Skewness can be interpreted as a measure of inequality. Kurtosis is a measure of the extent of extreme values, measured relative to the normal distribution. Positive kurtosis indicates a distribution with heavier tails (leptokurtic) and thus more likely to exhibit extreme values, either positive or negative.

Calculation of individual stochasticity.
The calculations of individual stochasticity rely on the Markov chain transition matrix U and the fundamental matrix The (i, j) entry of N is the mean time spent in stage i, prior to absorption (i.e., death), of an individual starting in stage j. 1

Longevity
Longevity of an individual is calculated as the sum of the time spent in every transient stage, until eventual absorption (2). The first four moments of longevity are given in (3), The ith entry of η m is the mth moment of longevity for an individual starting in stage i.

Lifetime reproductive output (LRO)
Calculation of LRO requires reward matrices whose entries give the moments of the reproductive output reward associated with each transition. Let f be a vector giving the mean reproductive output of each stage. If a single type of offspring is produced f T is the first row of the fertility matrix F. For species that produce multiple types of offspring (i.e., in which F contains positive entries in more than one row), we summed all types at each age and treated the sum as the mean stage-specific reproductive output. Modeling the number of offspring as a Poisson random variable with that mean gives the reward matrices Following Theorem 1 of (4), we writeρ k for the vector, of dimension τ × 1 of the kth moments of lifetime reproduction for individuals starting in each transient stage. We define the matrix Z = I τ ×τ 0 τ ×α ; and also defineR k , the τ × τ submatrix of R k corresponding to transitions among the transient states:R In terms of these quantities, the first four moments of LRO arẽ

Age at maturity.
The age at maturity is defined as the time to first enter any stage defined as reproductive, that is any stage j for which column j of F is non-zero. The technique, detailed in Section 5.3.3 in (5) has two steps. First, the transition matrix is modified to make the reproductive stages absorbing. An individual will end in one or the other of the two absorbing states, death-before-reproduction or reproduction-before-death. Then a conditional Markov chain is constructed, conditional on reaching reproduction. The age at maturity is the time to absorption in this conditional chain. We calculated the mean, standard deviation, and coefficient of variation of this time.

Generation time.
The offspring production at age x of an individual starting in stage j is given by the vector where the entries of m (j) correspond to different types of offspring (2) . The cohort generation time, given by the mean of this distribution, is as in (2).

Extent of iteroparity
Considering previous work estimating the extent of iteroparity, which focuses on the age dispersion of reproduction (6), we used the distribution m (j) in (22) to derive a new result for the variance of the ages of mothers at the birth of offspring. The second moment of that age is The new part of this expression is x x 2 U x . We can write and then simplify this summation, = 0 + U + 2U 2 + 3U 3 + · · · +2U 2 + 6U 3 + 12U 4 + · · · (28) Solving for x x 2 U x , which appears on both sides, yields and thus µ The vector of variances is given by and from this we calculated the standard deviation and finally the coefficient of variation of the age of production of offspring. The coefficient of variation is dimensionless and is thus appropriate for comparing across life histories of different absolute lengths.

Correlations between life history outcomes
In figures 3 and 4, we show the Pearson product-moments correlations of all 16 of the demographic outcomes we include in our analyses for animals and plants, respectively. Some of these statistics are very tightly correlated; others not at all.