Background

The starting point for analytical explorations of stochastic demography is Tuljapurkar’s (1990) approximation for the stochastic log growth rate, $$\mathrm{log}\,\lambda _{\mathrm{s}\,}$$. While Tuljapurkar’s full approximation includes the effects of between‐year correlations in matrix elements, we use the simpler version of his formula that omits this complication:

One can arrive at the general result that variability in annual population growth reduces fitness by examining equation (1): $$\mathrm{log}\,\lambda _{\mathrm{s}\,}$$ is approximately equal to the log deterministic growth rate, decreased by the variability term $$-\tau ^{2}/ 2\overline{\lambda }^{2}_{1}$$. However, the more interesting question is how variation in a particular matrix element influences fitness. The terms in τ² with $$k,\, l=m,\, n$$ are simply $$\mathrm{Var}\,( e_{k,\, l}) \overline{S}^{2}_{e_{k,\, l}}$$ so that the negative effect on $$\mathrm{log}\,\lambda _{\mathrm{s}\,}$$ of variance in $$e_{k,\, l}$$ is weighted by the square of its deterministic sensitivity value. Thus, selection against variation in matrix elements will be highest for those elements with the highest sensitivity values.

This prediction is illustrated more completely by taking the derivative of equation (1) with respect to a particular covariance or variance term and approximating the sensitivity of $$\mathrm{log}\,\lambda _{\mathrm{s}\,}$$ to changes in that variance or covariance (Caswell 2001). As with deterministic sensitivities, these stochastic sensitivities are widely interpreted as measures of the force of selection acting on elements of the life history (Caswell 2001). The vast majority of the terms in τ² do not include any particular covariance term, so the resulting stochastic sensitivity values are quite simple:

Because sensitivities to matrix elements are always positive, the estimated sensitivities of $$\mathrm{log}\,\lambda _{\mathrm{s}\,}$$ to variances and covariances are always negative (eqq. [2]). Equation (2a) is the basis for the generality that those matrix elements (and more important, the life‐history traits they represent) that have higher sensitivity values should be under the strongest selection for low variance. In support of this prediction, Pfister (1998) found a negative correlation between the variances of matrix elements and their sensitivities across a range of species (the same pattern held for coefficients of variation and elasticities, which are rescaled sensitivity values; but see Morris and Doak 2004). Equation (2b) also implies that selection should favor more negative covariances between stochastically varying matrix elements (Orzack and Tuljapurkar 1989).

A Complication in Analyzing Variance‐Importance Trade‐Offs

As we noted above, the standard stochastic sensitivity results rely on the assumption that only the terms of τ² that contain a particular variance and covariance term will influence the sensitivity value for that term. Implicitly, this is an assumption that the variance and covariance terms contained in τ² are independent of one another. In fact, this is wrong. One way to see this is to express the covariance of two rates in terms of their correlation and standard deviations: $$\mathrm{Cov}\,( e_{k,\, l},\: e_{m,\, n}) =\mathrm{Corr}\,\,( e_{k,\, l},\: e_{m,\, n}) \sigma _{e_{k,\, l}}\sigma _{e_{m,\, n}}$$. A change in the covariance of these two rates can result from either a change in their correlation, a change in the standard deviation of e_k, l, or a change in the standard deviation of e_m, n. Conversely, a change in a standard deviation or variance of a matrix element must create changes in the covariances of that rate with all other rates, unless the correlations between these rates are also to change. This means that the simple sensitivity and elasticity expressions usually shown for variances and covariances in equations (2) (e.g., eqq. [14.110] and [14.111] in Caswell 2001 and eq. [9.15] in Morris and Doak 2002) are incorrect.

To derive better estimates of these sensitivities and elasticities, it helps to recast the problem in terms of correlations and standard deviations, thereby separating variance from correlation at the onset. While we are at it, it also makes for more biologically informative results to express Tuljapurkar’s expression for τ² in terms of the vital rates (e.g., survival, growth, and fecundity values) that underlie the matrix elements rather than the matrix elements themselves:

To see how these correlations can mediate selection on variability in vital rates, it helps to consider the sensitivities of $$\mathrm{log}\,\lambda _{\mathrm{s}\,}$$ to standard deviations and correlations in vital rates:

It is important to remember that unlike the sensitivities of matrix elements, deterministic sensitivities of vital rate means are not necessarily positive. Vital rates such as the probability of shrinking into a smaller size class (for plants and some invertebrates) or transitioning to postbreeding stages (observed in many animals) will usually have negative sensitivity and elasticity values. The same is true of the probabilities of death that arise from particular causes, which, especially in management contexts, are often included in matrices as mortality rather than survival probabilities. For rates such as these, strong positive correlations with vital rates having positive deterministic sensitivities may reverse the sign of otherwise positive variance elasticity and sensitivity values, leading to selection for low variance. To illustrate the possible effects of correlations on variance sensitivity values, consider the effect of just one correlated vital rate j on the sensitivity to variance in rate i. This effect is determined by a single term in the summation in equation (5), $$\overline{S}_{v_{i}}\overline{S}_{v_{j}}\sigma _{v_{j}}\rho _{v_{i},v_{j}}$$, the sign of which will vary with the combined signs of both deterministic sensitivity values and the correlation between the two rates (Table 1).

A clear understanding of the forces generating these results is important to avoid their misinterpretation. In particular, the possible benefit of increased variance in a life‐history trait shown by these sensitivity results is not due to a life‐history trade‐off, in any usual sense of the term. A trade‐off would imply that increased variability in one vital rate would lead to an increased mean or decreased variability in another rate, thereby benefiting fitness. Such allocation trade‐offs are likely in many contexts and, with enough data to estimate their parameters, can be incorporated into matrix models fairly easily (e.g., van Tienderen 1995). This type of trade‐off can be one source of the negative correlations between vital rates that we might observe and use as parameters in a stochastic demography model. However, the trade‐offs themselves are not included in the analysis we explore here, which assumes that changes in one vital rate variance can occur independently of any other variance or mean. Rather, the effects of correlations that we show result from the ability of variation in negatively correlated rates to counteract each other’s effects. This reduces τ², the overall variability in annual growth or performance, from year to year. If increased variance in one vital rate will offset the variation in τ² caused by others, this variation will be favored. While Tuljapurkar’s approximation and hence our analyses do not fully account for the complexities of real structured population growth (Tuljapurkar 1990; Caswell 2001), simulations we have conducted confirm that variability in vital rates can indeed result in higher population growth rates.

Does This Correction Make a Difference?

To show how the corrected formula for variance sensitivities (eq. [5]) can alter the conclusions of a stochastic sensitivity analysis, we used information for the desert tortoise, a widely used example of a stochastic matrix model (Doak et al. 1994; Caswell 2001; Morris and Doak 2002). In particular, we contrasted the results of equation (5) with predictions that ignore all correlated vital rate effects (i.e., setting the summation on the right‐hand side of eq. [5] to 0; this is the equivalent of eq. [2a] for covariances between mean matrix elements). The corrected sensitivity values differ sharply from the uncorrected ones, increasing by up to 97‐fold in magnitude and, in four out of 11 cases, changing from negative to positive (fig. 1A). For most variances, the contribution to the entire corrected sensitivity value from indirect effects of correlated rates was nearly as high or even higher than that of the direct effects (fig. 1B).

Examination of particular vital rates makes it clear why some sensitivities change so dramatically. When including correlation effects, the sensitivity of growth to the standard deviation in survival of class 4 tortoises, $$\sigma _{\mathrm{s}\,_{4}}$$, becomes far more negative. This is because the survivorship of class 4 tortoises, s₄, is positively correlated with all of the other survival rates except that of the largest tortoises, to which λ_s is least sensitive (fig. 2A). While growth rates show a mix of positive and negative correlations, they have much lower sensitivity values than do the survival rates to which s₄ is consistently positively correlated. Thus, the negative effect on λ_s of increasing $$\sigma _{\mathrm{s}\,_{4}}$$ would be greater than otherwise predicted because s₄ varies in concert with these other survival rates. In contrast, the sensitivity to variability in size class 2 tortoises, $$\sigma _{\mathrm{g}\,_{2}}$$, goes from near 0 to positive when considering indirect effects. This is because g₂ is negatively correlated with all of the survival rates except that of the largest size class (fig. 2B). In other words, $$\sigma _{\mathrm{g}\,_{2}}$$ helps to balance the variance in these other vital rates, thereby reducing their effects on λ_s. Other vital rates, such as g₆, have low or mixed correlations with the influential survival rates. Their variance sensitivities are therefore little changed by including correlated effects (fig. 2C). Overall, the use of our new formulation substantially changes the picture that emerges from a stochastic sensitivity analysis of variance terms and shows that predictions of selection for higher variability can emerge from real life‐history patterns.

Conclusions

Matrix models have become one of the most commonly used quantitative tools in ecology, with applications ranging from the management of populations to the search for life‐history patterns. A major use of matrix models is sensitivity analysis, which provides a way to predict the fitness consequences of changes in different vital rates and the efficacy of management actions. We show here that a broad generality of stochastic matrix models—that increased variance in vital rates is never favored—is a substantial oversimplification. In particular, negative correlations in the variance of different vital rates can drastically change the sensitivity of stochastic population growth to vital rate variation, even leading to higher population growth rates with increased variability of some vital rates.

In a similar but distinct vein, Tuljapurkar (1990, pp. 82–83) provides an example of synergistic effects of matrices on population growth. Also, Tuljapurkar et al. (2003) found cases in which increasing the variability of matrix elements predicted increased population growth rate. However, in neither case have previous researchers noted the general ability of sensitivity analysis to predict selection for variation and the importance of correlation structures in generating these results. We caution that the corrected sensitivities and elasticities we present in equations (4)–(6) inherit from Tuljapurkar’s approximation (eq. [1]) the assumptions that the environment is uncorrelated from year to year and that environmental variability is not excessive. When either assumption is seriously violated, sensitivities of the population growth rate to vital rate variability could also be computed by perturbation methods described by Tuljapurkar (1990) and Tuljapurkar et al. (2003).

Overall, our results demonstrate that vital rate variation can have stronger and more varied effects on fitness and population growth than have been previously shown, opening the door to more informative analyses and applications of demographic models. We do not expect that our revised sensitivity expressions will always result in substantial changes in estimated sensitivities. However, when individuals of different stages live in close proximity, many vital rates are expected to have substantial correlations from the effects of shared environmental drivers. These correlations, which can be positive or negative, will have significant effects in sensitivity analyses. This is especially true for the estimation of the sensitivity of population growth or fitness to variation in traits.

Because of the importance of correlations in determining sensitivity values, our results emphasize the need for empirical studies to estimate correlation patterns as accurately as possible and to consider these estimates when making demographic predictions. In many cases, correlations are not considered in demographic analyses because they can be only poorly estimated (Fieberg and Ellner 2001). Furthermore, almost all estimates of correlations are biased toward 0, so we routinely underestimate the strength of their effects (Doak et al., forthcoming). While many of the problems of accurately estimating variation and correlation in vital rates cannot be easily solved, our results show the importance of these parameters for an accurate understanding of demographic and evolutionary predictions and therefore the need to judiciously use this information in future analyses.