### Introduction

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Ecologists are interested in the relative contribution of different demographic rates to variation in population growth, because knowledge of these contributions is required to inform wildlife-management strategies (Caswell, Fujiwara & Brault 1999) and to explore selection on life-history traits (Benton, Grant & Clutton-Brock 1995; Saether & Bakke 2000; van Tienderen 2000). For example, to halt a decline in population size, an obvious first step is to identify the demographic cause of the decline. Several methods to identify the relative contributions of demographic rates to variation in population growth have been developed (Varley & Gradwell 1960; Brown & Alexander 1991; Brault & Caswell 1993; Horvitz, Schemske & Caswell 1997; Sibly & Smith 1997) and have been used widely for ungulates (Brown *et al*. 1993; Coulson *et al*. 1999; Albon *et al*. 2000; Gaillard *et al*. 2000), birds (Erikstad *et al*. 1998; Saether & Bakke 2000; Wisdom, Mills & Doak 2000), reptiles (Mills, Doak & Wisdom 1999; Caswell 2001), small mammals (Dobson & Oli 2001) and marine mammals (Brault & Caswell 1993). Reviews of this literature have led to proposed generalizations (outlined later) about the associations between demographic rates and variation in population growth. These generalizations, however, are speculative because (i) comparisons of the various methods have not been made, (ii) most research has ignored the contribution of the covariation between demographic rates on variation in population growth and (iii) there are few comparisons between multiple populations of the same species living in contrasting environments.

The first method developed to decompose the variation in population growth into the relative contributions of different demographic rates was *k*-value analysis (Varley & Gradwell 1960). This involves regressing *k*-values describing state-specific per generation mortality and recruitment rates against the sum of the *k*-values. *K*-value analysis was used widely until Manly (1990) and Royama (1996) highlighted several problems with the method. At this point, ecologists began using approaches developed in demography (e.g. Tuljapurkar 1982; Lande 1988) or developed their own approaches (e.g. Sibly & Smith 1997). The most accurate of these methods is structured accounting of the variance of demographic change (SDA, Brown & Alexander 1991), which provides a complete account of causes of variation in a time series of per capita change [(*N*_{t+1} −* N*_{t})*/N*_{t} where *N*_{t} is population size in year *t*]. Because it requires perfect demographic knowledge of a population, however, the method has been rarely applied.

Another method based on matrix models (Caswell 2001) has been applied widely by researchers interested in stochastic demography. Following Caswell's (2000) definitions, we call this approach the retrospective matrix method. It provides an asymptotic first-order linear approximation of a complete decomposition of variation in the natural rate of increase (*N*_{t+1}*/N*_{t}; Lande 1988; Brault & Caswell 1993; Horvitz *et al*. 1997; Gaillard *et al*. 2000). The approximation is asymptotic because it is based on elasticities (or sensitivities) of the dominant eigenvalue (λ) to elements of a deterministic matrix: the method assumes that temporal fluctuations in the demographic structure of the population do not have a substantial impact on variation in a time-series of *N*_{t+1}*/N*_{t}. In contrast, SDA incorporates any influence of demographic fluctuations on variation in the per capita change [(*N*_{t+1 }− *N*_{t})*/N*_{t} = *N*_{t+1}*/N*_{t}− 1]. The natural rate of increase (*N*_{t+1}/*N*_{t}) and per capita change (*N*_{t+1}*/N*_{t}−1) are quantities with different means (1 and 0, respectively) but equivalent variances. Consequently, the decomposition provided by the retrospective matrix method can be compared with the decomposition provided by SDA. Having made the distinction between the two quantities for brevity we will refer to both *N*_{t+1}*/N*_{t} and *N*_{t+1}*/N*_{t}* − 1* as population growth and the asymptotic approximation of *N*_{t+1}*/N*_{t} as λ.

The retrospective matrix method differs from the prospective matrix method, which determines the functional dependence of deterministic population growth on a demographic rate. Specifically, the prospective matrix method determines how much λ would change if a change in a demographic rate were to occur, while the retrospective matrix method extends the prospective matrix method to approximate a decomposition of variation in population growth by incorporating variation in demographic rates.

Research developing the retrospective matrix method was initiated by Tuljapurkar (1982), who provided the first analytical approximation of a link between demographic rates and variation in population growth, by decomposing the dominant Lyapunov exponent of population growth [specifically ln(λ)]. Lande (1988) and Brault & Caswell (1993) introduced the now widely used decomposition of variance in population growth which was subsequently made popular by, among others, Horvitz *et al*. (1997), Saether & Bakke (2000) and Gaillard *et al*. (2000). As far as we are aware there have been no comparisons of the performance of the first-order linear approximations of the decomposition of variance in population growth provided by the retrospective matrix method with the complete decomposition provided by the SDA method.

Most applications of the retrospective matrix method have been based on a yearly description of the life cycle, and have reported contribution to age- or stage-specific demographic rates to variation in population growth. In contrast, most applications of the SDA approach have reported results from a seasonal description of the life cycle and have reported contributions of age-classes, or of demographic rates to variation in population growth. Both the retrospective matrix method and SDA can be applied to seasonal and non-seasonal descriptions of the life cycle, and both methods allow variance decompositions to be reported across age classes, across demographic rates or across class-specific demographic rates. We report decompositions that are comparable to previous publications using both approaches. As the retrospective matrix method has been applied mainly to non-seasonal descriptions of life cycles and has been applied more widely than SDA we compare the two approaches with a non-seasonal description of the life cycle.

Here we present the first comparison between the SDA and the retrospective matrix method using detailed individual-based life-history data from two populations of bighorn sheep (*Ovis canadensis*, Shaw) in Alberta, Canada (Jorgenson *et al*. 1997; Festa-Bianchet & Jorgenson 1998) and a population of red deer on Rum, Scotland (Clutton-Brock, Guinness & Albon 1982; Coulson *et al*. 2003b). First we present a complete decomposition of variation in population growth for the bighorn sheep populations using SDA and a seasonal description of the life cycle. These results can be compared with previous publications using SDA, including application of the approach to the red deer data (Albon *et al*. 2000). Next, we use SDA to provide results from a non-seasonal description of the life cycle that are compared with results from the retrospective matrix method. We report results that (i) support Royama's (1992) observation that results depend on the description of the life cycle used, (ii) show that for ungulate populations the retrospective matrix method provides an acceptable approximation of the complete decomposition of population growth provided by SDA, (iii) report that in all populations examined covariation between demographic rates contributed substantially to variation in population growth and (iv) suggest that cross-population generalizations concerning the association between demographic rates and variation in population growth may be unwarranted if covariation between demographic rates is strongly associated with variation in population growth. Finally we caution against the unfettered use of prospective and retrospective matrix methods for conservation.

### Discussion

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Our analyses produced five important results. First, populations of the same species living in different environments can have contrasting associations between variation in demographic rates and variation in population growth. Second, demographic rates with the highest elasticities of λ are not often those showing the largest contributions to variation in population growth. Third, there is little evidence that either of the previous generalizations (Gaillard *et al*. 2000; Coulson & Hudson 2003) is supported. Fourth, any decomposition of variation in population growth should consider covariation between demographic rates. Finallly, the retrospective matrix method provides a satisfactory decomposition under the majority of circumstances we investigated.

The proximate reason for the contrasting results between the two bighorn sheep populations is their difference in variation around demographic rates. For large mammals, adult survival will invariably have the largest potential impact (elasticity) as adult survival rates are high and the majority of animals within a population are adults (Gaillard, Festa-Bianchet & Yoccoz 1998; Gaillard *et al*. 2000). The contribution of a demographic rate to variation in population growth, however, is also dependent on its temporal variation. Because adult survival at Sheep River was high but variable it accounted for three times as much variation in population growth compared to Ram Mountain, where adult survival rates were high and constant. The ultimate reason for the contrasting results is differences in the ecological processes that generate variation in population growth [pneumonia and predation at Sheep River (Festa-Bianchet 1988b; Jorgenson *et al*. 1997; Ross *et al*. 1997) and density-dependence at Ram Mountain (Festa-Bianchet & Jorgenson 1998; Festa-Bianchet *et al*. 1998a; Festa-Bianchet, Gaillard & Côté 2003)].

The large diversity of ecological processes that affect populations of large vertebrates generates different demographic responses that potentially allow all life-history stages to play a substantial role in contributing to distributions of population growth characterized from a time series. As Gaillard *et al*. (2000) and Caswell (2000) observed, prospective and retrospective matrix methods can identify different demographic rates influencing different moments of a distribution of population growth. Although this result is not surprising − there is no a priori expectation for a demographic rate to be strongly associated with both the mean (the prospective matrix method) and the variance (the retrospective matrix method) of a time-series of population growth − some conservation managers may consider the prospective and retrospective matrix methods as addressing the same question (see Caswell 2000 for an excellent discussion of the two methods).

Our two SDA analyses also provide support for Royama's (1992) observation that the way the life history is described can influence the interpretation of an analysis of population dynamics. The different ways of reporting the SDA analyses also demonstrate there are various ways to report a decomposition of variation in population growth. Retrospective matrix methods typically report contributions of age-specific demographic rates measured on an annual scale. However, there are circumstances − for example, estimating the role of recruitment or survival across age-classes to variation in population growth − when it could also be illuminating to report the contribution of age-classes across demographic rates, or the contribution of demographic rates across age-classes. This can be achieved using the retrospective matrix method by summing contributions across rates or classes. Recently Lande, Engen & Saether (2003) showed how to decompose variation in population growth into contributions from density-dependence, environmental and demographic stochasticity. It may prove useful to combine the various decomposition methods available to estimate the contribution of different ecological processes within and between age-classes and demographic rates.

With detailed data on three populations we can only make a limited number of comparisons. Our results, however, do not completely support either of two previously suggested generalizations. First, although the dominant demographic rate contributing to variation in population growth in increasing populations of both red deer and bighorn sheep was adult recruitment as proposed by Coulson & Hudson (2003), results from Ram Mountain in the ‘no trend’ period do not support their argument that prime-aged and older adult survival are the dominant demographic rates in static populations; in this population during the ‘no trend’ period, recruitment of yearlings by prime-aged and older adults was the dominant demographic rate. Second, results from Sheep River and Rum (see also Albon *et al*. 2000) demonstrate that the contribution of variation in juvenile survival to variation in population growth is not necessarily dominant as reported by Gaillard *et al*. (2000). The generalizations proposed by Gaillard *et al*. (2000) and Coulson & Hudson (2003) were based on studies that ignored covariation between demographic rates. Before any new generalizations can be proposed further studies examining the association between demographic rate covariation and variation in population growth are required. The probable reason for the differences between the results reported here and the results reported by Gaillard *et al*. (2000) for the same populations is that Gaillard *et al*. (2000) did not incorporate demographic rate covariation into their analyses.

The result that covariation between demographic rates provides a substantial contribution to variation in population growth is unsurprising from a life-history perspective. This is because life-history variation − including cohort effects − generates variation in demographic rates that can be correlated (Benton & Grant 1999b, 2000; Beckerman *et al*. 2002; Beckerman *et al*. 2003). Genetic constraints (Lande 1982) and environmental variation (Coulson *et al*. 2004) can generate both life-history variation and covariation in demographic rates. For example, an environmentally favourable year may lead to elevated survival and fecundity leading to no detectable survival costs to reproduction. In contrast, an environmentally challenging year may generate a survival cost of reproduction (Marrow *et al*. 1996; Tavecchia *et al*. 2005). Across a series of good and bad years this will generate covariation between survival and fecundity rates that could impact on variation in population growth. Given that environments and limiting factors differ between populations of the same species it is also unsurprising that different populations exhibit different associations between the covariation in demographic rates and variation in population growth. A critical question for ecologists concerns how these differences manifest themselves between populations and species.

Failure to ignore the contribution of covariation between demographic rates on variation in population growth can lead to misleading conclusions, with important implications for applied ecology. Conservation biologists, for example, are often interested in devising strategies that target demographic rates that contribute most to variation in population growth. Considering only the direct effects each demographic rate has on variation in population growth could lead to flawed conservation strategies if demographic rates covary and impact on population dynamics. Although previous analyses using the SDA method reported substantial contributions of demographic rate covariation (Brown *et al*. 1993; Coulson *et al*. 1999; Albon *et al*. 2000) few retrospective analyses considered such contributions (but see Dobson & Oli 2001). Part of the reason could be that few demographic studies report demographic rate covariation (Wisdom *et al*. 2000). We found that demographic rate covariation can explain between one-third and one-half of the variation in population growth. When using the retrospective matrix method we estimated only the contribution of second order covariances. We did not consider higher-order covariances, as the SDA method has never found these to be important for large mammals (Brown *et al*. 1993; Coulson *et al*. 1999; Albon *et al*. 2000). Although we are now hesitant to propose generalities, it is noticeable that the contribution of demographic rate covariation to variation in population growth is greatest in those populations that show no temporal trend. As successful population management and conservation can depend on the accurate decomposition of variation in population growth, we recommend that future application of the retrospective matrix method should consider demographic rate covariation.

The retrospective matrix method is a first-order linear approximation of the decomposition of variation in population growth (Horvitz *et al*. 1997; Caswell 2001). Although the approximation ignores fluctuations in age structure, our analyses suggest that the retrospective matrix method generally performs well for ungulate life histories. Given that the retrospective matrix method does not require the complete demographic information necessary for a complete decomposition of variation in population growth, this result is reassuring. There are, however, cases when the retrospective matrix method does not perform particularly well. For example, in the red deer population at high density the *r*^{2} value between the SDA contributions and the retrospective contributions is only 0·46. We do not know under what circumstances the performance of the retrospective approximation becomes unsatisfactory − although it should be noted that the red deer population experienced a long-term age-structure transient following release from culling, which generated marked age-structure effects on the population dynamics (Coulson *et al*. 2004). The non-parametric correlations are considerably higher than the parametric correlations for red deer, suggesting that although the retrospective matrix method does not always generate particularly accurate estimates of demographic rates contributions, it does tend to rank contributions in the correct order. In general, we consider the retrospective matrix method a very useful method for decomposing variation in population growth.

The many applications of understanding the association between demographic rates and population growth range from management to understanding evolutionary processes (Grant 1997; Benton & Grant 1999a; Caswell 2001), and any generalizations could prove useful. Our results, however, suggest that the relationship between variation in a demographic rate and variation in population growth is complex. Because both the mean value (and hence its potential effect on changes in population growth) and the temporal variation in a demographic rate determine its contribution to fluctuations in population size it appears that the source of environmental variation in demographic rates is as, or perhaps even more, important than population trajectory or distance from carrying capacity. An understanding of variation in population growth is therefore likely to require an understanding of the sources of variation in multiple demographic rates. We now require a detailed investigation of the effects of life-history variation and environmental variation on variation in population growth across taxa.