## Introduction

All natural populations experience some random environmental variation. Mounting evidence suggests that the frequency and severity of environmental variation is changing, at local and global scales. Consequently, stochastic modelling approaches have become increasingly common in conservation studies concerned with predicting population persistence (Morris & Doak, 2002; Lande *et al*.2003; Boyce *et al*. 2006). Environmental variation is also important when studying life history evolution, where the degree of variation is expected to contribute to the evolution of lifespan and reproductive traits (Tuljapurkar & Horvitz 2006; Morris *et al*. 2008).

Stochastic matrix models have become a standard tool for investigating population growth in variable environments. These models describe populations in terms of temporally and/or spatially variable vital rates that quantify transition rates between stages that may be age classes, ontogenetic stages, spatial regions, or other characteristics. The long-run growth rate of such a population is the stochastic growth rate *a*. This growth rate is widely used in biology as a fitness measure in evolutionary problemsproblems (see Tuljapurkar *et al*. (2009) for a discussion), and as a descriptor of growth or persistence in ecological and PVA analyses (e.g. Morris & Doak 2002).

As important as the growth rate itself is its sensitivity to changes in model parameters. Deterministic sensitivities measure the effect on growth rate of a small change in one or several matrix elements, assuming all other rates remain constant (i.e. its derivative) (Caswell 2001). These changes are called perturbations, and they can be chosen to evaluate sensitivities with distinct biological meanings. (One can measure sensitivity to changing a single vital rate, or to changing all of them, for example.) A closely related quantity, elasticity, measures the response of growth rate to proportional, rather than absolute, perturbations. Tuljapurkar (1990) found an exact method for calculating sensitivities of the stochastic growth rate. Tuljapurkar & Horvitz (Tuljapurkar *et al*. 2003) demonstrated how to decompose the proportional sensitivity of the growth rate into changes in the means and to changes in the variances of life history parameters. It is also possible to assess sensitivity and elasticity to perturbations within a single environment or subset of environments within an overall range of specified environmental states (Caswell 2005; Ezard *et al*. 2008; Aberg *et al*. 2009). Habitat-specific sensitivities can be important for populations experiencing frequent disturbance, or when environmental variation is defined by specific variables like rainfall or temperature. In the stochastic case, one can estimate stochastic sensitivities using Tuljapurkar's approximation (Tuljapurkar 1982) if environmental variation is small (see Caswell 2001 for an exposition).

However, in stochastic environments both a mean growth rate and a variance are required to fully describe population dynamics (Tuljapurkar & Orzack 1980). Consider a population composed of *N* individuals at time *t*. Define the total population growth over time *t* as Λ(*t*) = *N*(*t*)/*N*(0). In the limit of large *t*, log Λ(*t*) is asymptotically normally distributed (Tuljapurkar & Orzack 1980). Suppose we have many sample paths of the stochastic environmental process, then for large *t* we can estimate the mean stochastic growth rate as and its variance across sample paths as . This variance (*v*) is used in studying population extinction (Beissinger & McCullough, 2002), analysing time series of population data (Engen *et al*. 2005; Lande *et al*.2006; Saether *et al*. 2007), estimation of effective population size (Engen *et al*. 2010), and making stochastic population forecasts (Lee & Tuljapurkar 1994). For a complete picture of stochastic population dynamics we need to understand the properties of *v*, not just of *a*. Particularly in situations where *v* is large, a distribution-focused approach may be more appropriate than existing mean-focused sensitivity analyses.

As illustrated in Fig. 1, there are several ways the distribution of population growth could respond to perturbation (Tuljapurkar & Orzack 1980). We aim to examine two questions : (1) how does one formulate a joint-interpretation of sensitivities of the distribution of population growth that accounts for changes to both the mean and the variance? (2) Under what circumstances is it appropriate to use only a mean-focused sensitivity analysis, and when should biologists use our more distribution-focused framework?

Here we present a new exact method for calculating the sensitivity of the variance of stochastic population growth (*v*). Our formulas apply to general kinds of stochastic variation (large or small, serially correlated or not) in models with a general age-stage structure. The general sensitivity calculation allows us to estimate sensitivities and elasticities of the variance to specific changes in life history parameters both across habitats and in specific habitat states. The variance sensitivity also allows the calculation of the sensitivity of cumulative extinction risk, which is useful in studying conservation.

We apply the new sensitivity calculations to empirical data from an at-risk population of polar bears (*Ursus maritimus*)(Hunter *et al*. 2007, 2010) and discuss the results in the context of conservation management. Increases in stochastic variation necessarily increase the uncertainty of population projections and decrease predicted persistence times. It is clear that management efforts will benefit in several ways from an understanding of which vital rates most strongly affect *v*. We are especially interested in cases where *a* and *v* respond differently to the same change in a particular vital rate. Our results raise important questions about how we should interpret a change that increases both the mean and the variance of growth. Can we manage for both *a* and *v*?