### Summary

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

**1.** The relative importance of density-dependent and -independent factors on interannual variation in over-winter survival was investigated in the fluctuating population of Soay sheep on St Kilda, Scotland, over the period 1985–96.

**2.** Population density had a negative effect on survival in lambs and adult males while adult female survival showed no evidence of density dependence over the observed range of population densities.

**3.** Climatic fluctuations associated with the winter North Atlantic oscillation index (NAO) also affected survival, which decreased in winters that were relatively warm, wet and windy. The effect was most pronounced in lambs.

**4.** Survival was modelled using logistic regression analysis with and without year fitted as a random effect. The former incorporated stochastic year to year variation in survival. Results from the two modelling approaches were similar in terms of the regression coefficients estimated. However, the standard errors of the year-dependent covariates, population size and NAO, were underestimated when the random year effect was ignored, leading to incorrect inferences about the relative significance of terms being made.

**5.** Using both modelling approaches, density dependence was found to have a greater influence on survival than the effect of NAO in lambs and adult males, whereas in adult females NAO was the more important.

**6.** Once random between-year effects were taken into account, the individually varying terms such as body weight and faecal egg count were the most significant factors explaining differences in survival.

### Introduction

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

There has been a recent increase in interest in the influence of density-independent factors and environmental stochasticity on survival and population dynamics (Leirs *et al*. 1997; Sæther 1997; Gaillard, Festa-Bianchet & Yoccoz 1998; Grenfell *et al*. 1998). A growing body of evidence shows that ecological processes are affected by climatic fluctuations (Grant & Grant 1989; Forchhammer, Post & Stenseth 1998; Post & Stenseth 1998). This has implications for the debate concerning the relative importance of intrinsic (density-dependent) and extrinsic (density-independent) factors on population changes. Sæther (1997) suggested that, in the absence of predation, the population dynamics of ungulates were determined by a combination of both density-dependent and stochastic environmental effects, operating through changes in survival and fecundity rates. An example is the density-independent cohort effects of spring temperatures on birth weight of red deer (*Cervus elaphus* L.) which were found to significantly influence over-winter survival rates (Albon, Clutton-Brock & Guinness 1987; Rose, Clutton-Brock & Guinness 1998), while the effects of birth weight on survival were intensified at high densities (Clutton-Brock *et al*. 1987). Post & Stenseth (1998) have shown that growth in the moose (*Alces alces* L.) population on Isle Royale, USA, and increases in white-tailed deer (*Odocoileus virginianus* Zimmerman) abundance in Superior National Forest, USA, were both influenced by delayed density-dependent feedback, as well as global climatic fluctuations and predation by wolves (*Canis lupus* L.).

Measures of age- or sex-specific survival rates have, until recently, been based on transversal life-table methods, with little knowledge of their reliability (Gaillard *et al*. 1993). However, several long-term studies of ungulate populations with individually known animals (reviewed by Sæther 1997; Gaillard *et al*. 1998) have enabled accurate estimates of vital rates to be made, partly due to advances in the modelling of capture–mark–recapture data (Lebreton, Pradel & Clobert 1993). Jorgenson *et al*. (1997) remarked that the degree to which survival rates varied between years remained largely unknown. Since then, Gaillard *et al*. (1998) have demonstrated that across 16 species of large herbivore the coefficients of variation in survival between-years varied little in prime-aged females (from 2 to 15%) but were very variable in juveniles (from 12 to 88%). Furthermore, it has been shown that juvenile survival is more sensitive to both density dependence and, in particular, to seasonal food availability and therefore stochastic variation, than adult survival (Sinclair 1977; Gaillard *et al*. 1998). We conducted our analyses separately on lambs, adult females and adult males to investigate the different susceptibilities of these components of the population.

It has previously been demonstrated that over-winter survival of the Soay sheep (*Ovis aries* L.) on St Kilda was strongly density-dependent (Clutton-Brock *et al*. 1991; Grenfell *et al*. 1992), but in recent high population years anticipated crashes have not occurred, especially within the adult population. It has become apparent that at high population sizes the system is particularly sensitive to a combination of density-dependent and -independent factors. Consequently populations above a certain threshold can increase, decrease or remain constant in size depending on the extrinsic environmental conditions (Grenfell *et al*. 1998). It would generally be expected that the effects of weather on population dynamics should become more evident as a system approaches the ecological carrying capacity (Sinclair 1989). In support of this, it has been shown that population growth rates or survival were more variable at high density, when density-independent effects were stronger, in populations of red deer on Rum (Benton, Grant & Clutton-Brock 1995), bighorn sheep (*Ovis canadensis* Shaw) in the Canadian Rocky mountains (Portier *et al*. 1998) and in Dall's sheep (*Ovis dalli* Nelson) in Alaska (Bowyer, Leslie & Rachlow, in press).

Grenfell *et al*. (1998) found that the effects of March gales and April temperatures had a greater influence on Soay sheep survival above a threshold density than at low density. Here, we have investigated the influence of a stochastic environmental variable, climatic fluctuation, on over-winter survival. In northern Europe, fluctuations in winter climate are strongly correlated with interannual variations in the atmospheric circulation over the North Atlantic (Wilby, O’Hare & Barnsley 1997). An annual index of this North Atlantic oscillation (NAO) can be measured by the difference in normalized sea level pressures between Lisbon, Portugal and Stykkisholmur, Iceland, between December and March (Hurrell 1995). In the British Isles, high positive values are associated with warm, wet winters with strong westerly winds, whereas low negative values indicate cold, dry winters (Wilby *et al*. 1997). Variations in the abundance of zooplankton species have already been linked with fluctuations in NAO, with the implication that the NAO may play a comparable role to the El Niño southern oscillation in pelagic ecosystems (Fromentin & Planque 1996). In terrestrial systems, breeding phenologies of a number of species of birds and amphibians have been shown to be well correlated with fluctuations in NAO (Forchhammer *et al*. 1998a). Furthermore, direct and delayed effects of the NAO on population dynamics of red deer have been found in Norwegian populations (Post *et al*. 1997; Forchhammer *et al*. 1998b), and in moose and white-tailed deer populations in the United States (Post & Stenseth 1998).

Most of the previous analyses of survival of Soay sheep on St Kilda have used logistic regression analysis (Clutton-Brock *et al*. 1992; Bancroft *et al*. 1995; Illius *et al*. 1995; Clutton-Brock *et al*. 1996; Moorcroft *et al*. 1996). This has advantages over other techniques such as the analysis of one-way contingency tables in that several factors can be controlled for simultaneously. However, conventional logistic regression, as applied through generalized linear modelling, does not allow a distinction to be drawn between fixed and random effects and cannot take account of more than one source of variation in the data. Here we investigate the effects of density-dependent and stochastic year-to-year variation in the survival of the Soay sheep, using an extension of logistic regression from the more statistically sophisticated class of generalized linear mixed models (GLMMs). This allowed random year effects to be fitted within the framework of conventional logistic regression. We assessed the appropriateness of conventional logistic regression, without random effects, for analysing survival data across years, by comparing the results with those obtained using mixed modelling.

### Discussion

- Top of page
- Summary
- Introduction
- Materials and methods
- Results
- Discussion
- Acknowledgements
- References

Our analyses have shown that both density-dependent and density-independent factors influenced survival of different components of the population. Lambs were strongly affected by both factors, with density dependence being the stronger of the two. In adults, survival of males was also density-dependent but to a lesser extent than in lambs, whereas over the range of population densities observed, female survival was not significantly affected by density dependence. Adult survival was also less strongly affected by density-independent factors than lamb survival. Juvenile survival is generally lower and more variable than adult survival and tends to be more sensitive to resource availability and changes in weather (Sinclair 1977; Fowler 1987; Owen-Smith 1990; Gaillard *et al*. 1998). Our results therefore corroborate these earlier findings.

Previous analyses of survival on St Kilda have found that both adult male and female survival were density-dependent (e.g. Clutton-Brock *et al*. 1991; Clutton-Brock *et al*. 1997). While there was no statistical evidence for a difference in density dependence between the sexes, there was also no significant effect of density on adult female survival over the range of population densities encountered during the course of this study, once stochastic between-year variation was accounted for. The difference between this and previous results could be explained in terms of the different statistical techniques used. However, the result has probably also been exaggerated by the longer run of data now available, in particular including the winter of 1996/97 when, despite the largest ever population density, adult female survival was high. This therefore provides some new evidence that on St Kilda adult female survival may, after all, be in keeping with the general principle of adult survival being buffered against density effects. Of nine long-term ungulate population studies reviewed by Sæther (1997) and a further four reviewed by Gaillard *et al*. (1998), the Soay sheep and red deer on Rum were the only populations to show a density-dependent decline in adult survival. Both are island populations, living close to their carrying capacities and are frequently resource-limited.

The influence of the North Atlantic oscillation on the survival of Soay sheep was that warm, wet and windy winters were negatively associated with survival in lambs and adult females. The lack of influence of NAO on adult male survival was surprising. If, however, the analysis of adult male survival was restricted to the period 1986–96, the effect of NAO became highly significant (χ^{2} = 21·3, d.f. = 1, *P* < 0·001; Fig. 2f). Furthermore, as in the case of adult female survival, NAO was then more important than population size (χ^{2} = 19·6, d.f. = 1, *P* < 0·001). This switch appeared to be due to the exclusion of the high density-dependent winter mortality of 1985/86, despite a NAO index suitable for moderately good survival (Fig. 2f).

Although it might initially be surprising that lower survival was not associated with cold winters, the negative effect of mild, wet and windy weather could be accounted for by a decrease in time spent foraging while animals sheltered from gales and hailstorms (Stevenson 1994). In red deer populations in Norway, winters with high positive NAO indices were also associated with decreases in apparent abundance (Forchhammer *et al*. 1998b). However, as well as the direct negative effect on survival observed, Forchhammer *et al*. suggested that NAO also had a delayed positive effect on the population size 2 years later, operating through female fecundity which was enhanced by improved plant growth and female body condition. The influence of NAO on plant productivity on St Kilda has yet to be examined but it seems likely that the growing season may start earlier in high NAO winters and, while improved forage may come too late for individuals already at the end of their energy reserves, may benefit survivors. A fuller analysis of the interactions between climate, plant productivity and population dynamics is now being carried out.

We might have expected to find an interaction between population size and NAO, since environmental fluctuations tend to have greater effects at high population density (Bowyer *et al*., in press). However, no significant interaction was found. This was also the case for over-winter survival in lambs of bighorn sheep, which was affected both by density and weather the previous spring, but the effects of weather were not mediated by changes in population density (Portier *et al*. 1998). By contrast, in the same study, the effects of winter and spring temperature on neonatal survival were found to interact with density, being more important in high density years. The analysis conducted by Portier *et al*. used conventional logistic regression and took no account of stochastic year-to-year variation in survival. This raises the question of whether density and weather effects would remain significant if a random year effect was included.

The individually varying parameters, body weight and faecal egg count were found to be the most important factors affecting survival. This was perhaps not surprising, since there was a lot of individual variability in attributes between years. Body weight was the primary factor influencing survival in all three age–sex classes studied. Although its influence on survival has long been acknowledged (Peters 1983; Calder 1984), especially in juveniles (Clutton-Brock *et al*. 1987; Clutton-Brock *et al*. 1992; Sedinger, Flint, & Lindberg 1995), relatively little attention has been paid to the changing effects of body weight on survival of mature animals (Festa-Bianchet *et al*. 1997). In bighorn sheep the probability of survival was found to increase with body weight in lambs but among adults significant effects were found only in old females (Festa-Bianchet *et al*. 1997). However, it should be noted that mortality of bighorns was largely due to predation, accident or disease and unlike the Soay sheep population, no evidence was found of starvation. Consequently, we would not expect body weight to play such an important role in survival. By contrast, under the conditions of extreme mortality due to resource shortages in some years on St Kilda (Grubb 1974; Clutton-Brock *et al*. 1991), heavier individuals with larger energy reserves were at an advantage. We would expect this effect to be most pronounced at high densities when competition for resource is at its greatest (Lindstedt & Boyce 1985). As anticipated, a significant interaction between population size and body weight was found in lamb survival, although not in the adult population.

Our analyses have shown good agreement between parameter estimates made using logistic regression models of over-winter survival, with and without random effects. Since, from the population dynamics perspective, the magnitude of the coefficients and range of variation of the covariates are the important results, broadly similar conclusions would be reached by both modelling approaches. However, the standard errors of year-dependent covariates were substantially underestimated by logistic regression without random effects. This concurs with findings from other areas of statistics (Glasbey 1988) that standard errors are more sensitive than regression coefficients to any change in assumptions about correlation between observations. Conventional logistic regression, which treated the observations as independent, had considerably smaller standard errors for the covariates population size and NAO than GLMM, which treated the sample size for these covariates as the number of years and inflated their standard errors according to the unexplained between-year variation.

Fitting year as a fixed effect after the year-dependent covariates, and in the absence of a random effect, allowed estimation of a mean deviance, attributable to the differences between years that were unexplained by the covariates. This was applied as a scaling parameter to adjust the standard errors for the year-dependent covariates. The adjustment worked reasonably well for population size and NAO. However, when the standard errors for catch date were treated in the same manner they became double those of the GLMM value. This indicated that much of the information about catch date came from within-year rather than between-year comparisons and highlighted the inability of conventional logistic regression to deal with such terms in a satisfactory manner.

Conventional logistic regression tended to find year-dependent covariates more highly significant than GLMM leading to incorrect inferences about the relative importance of year-dependent covariates and individual variables. This enhanced significance was due to the pseudo-replication effect of failing to treat year as a random effect (Diggle 1990). Logically it seems reasonable that mortality will vary substantially from year to year due to unrecorded factors, that the influences will be similar on all animals in the population and that, consequently, our year-dependent covariates should be measured against unexplained year-to-year variation.

We found little evidence in any of our modelling to suggest additional unexplained differences in survival between individuals once the fixed and random year effects had been accounted for. In part this may be a question of power, in that there was little information in the binary survival data and therefore a lack of heterogeneity between individuals. This was exacerbated by a relatively small proportion of individuals being recaught in subsequent years and having more than one data record (169 of 303 adult females and 65 of 262 adult males). Consequently, unexplained differences between individuals were probably quite small and the use of individual as a random effect within GLMM was unnecessary. The trends can therefore be interpreted as being across animals within years, rather than within animals across years (Diggle *et al*. 1994).

Of the two methods used, the mixed model approach was preferred to conventional logistic regression analysis, in that it correctly incorporated unexplained variation between years. However, logistic regression without random effects was much faster to perform so was used for model selection. The drawback of conventional logistic regression is that it can be non-conservative, but final fitting of models using GLMM overcame this problem. It also made good use of the robustness of the iteratively reweighted least squares algorithm for fitting logistic regression, in contrast to the Schall algorithm (Schall 1991) for fitting GLMMs, whose implementation in Genstat we found to suffer from convergence problems.

An alternative approach to analysing the Soay sheep data is to build a probability model for the sighting history of each individual, leading to the integrated mark–recapture–recovery analysis (mrra) method of Catchpole *et al*. (1998). One justification for using the mrra is that crude mortality rates estimated from animals found either dead or alive in each spring will be affected by the recovery and resighting probabilities (Catchpole *et al.*, in press, a). However, our analyses are ‘conditional’: animals not seen alive or found dead are assumed to have died in the winter after the final sighting. This assumption is rather inelegant but provided the recapture rates are high, which they are, it should not introduce substantial errors into the analysis, even though there will be some wrong assignments in the year of death. The advantage of undertaking our GLMM analysis is that the data can then be analysed in standard statistical packages, with the inclusion of time-varying, individual-specific covariates, such as weight and year, as a random effect. It should come as no surprise that, within the constraints outlined above, the mrra gives broadly similar results to those presented in this paper (Catchpole *et al*. in press, b). Further methodological developments are now required to encompass the merits of both approaches.

This study has demonstrated that the considerable between-year variation in over-winter survival of Soay sheep arose from the effects of density dependence, density-independent climatic fluctuations and other unaccounted-for stochastic variation. While the influence of these factors on the population dynamics at the whole island level has already been explored (Grenfell *et al*. 1998), the implications for the dynamics of the population from the individual perspective should now be examined in greater depth. Our model provides a framework for estimating the magnitude of random year-to-year variation, which would be of interest for developing dynamic stochastic population models.

Our analysis has also emphasized the importance of including random annual effects in survival models and of taking into consideration the variation in demographic parameters expected between age and sex classes. In addition, our results add to the growing evidence of the influence of large-scale climatic fluctuations on the structure and demographic trends of ungulates at northern latitudes.