Present address: Department of Zoology, Downing Street, Cambridge CB2 3EF, UK.

# Identifying the demographic determinants of population growth rate: a case study of red-billed choughs *Pyrrhocorax pyrrhocorax*

Article first published online: 16 JUN 2004

DOI: 10.1111/j.0021-8790.2004.00854.x

Additional Information

#### How to Cite

Reid, J. M., Bignal, E. M., Bignal, S., McCracken, D. I. and Monaghan, P. (2004), Identifying the demographic determinants of population growth rate: a case study of red-billed choughs *Pyrrhocorax pyrrhocorax*. Journal of Animal Ecology, 73: 777–788. doi: 10.1111/j.0021-8790.2004.00854.x

#### Publication History

- Issue published online: 16 JUN 2004
- Article first published online: 16 JUN 2004
- Received 25 August 2003; accepted 14 January 2004

- Abstract
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### Keywords:

- age-specific reproduction;
- integrated elasticity;
- life-history covariation;
- sensitivity;
- survival

### Summary

- 1Identifying which age-specific demographic rates underlie variation in a population's growth rate (λ) is an important step towards understanding the population's dynamics. Using data from a 20-year study of marked individuals, we describe patterns of demographic variation and covariation in the Scottish red-billed chough population (
*Pyrrhocorax pyrrhocorax*), and investigate which demographic rates have the greatest projected and realized influence on λ. - 2Survival, the probability of breeding and breeding success varied with age in this population. Data were sufficient to estimate year-specific probabilities of first-year, second-year and adult (all ages over 2) survival and mean breeding success. A population trajectory modelled using these parameter estimates closely matched census data, suggesting that estimates and simplifying assumptions were sufficient to accurately describe important demographic processes.
- 3Elasticity analyses based on stage-classes for which year-specific survival was estimable suggested that λ was more elastic to variation in adult survival than first- or second-year survival or breeding success. These ranks were consistent across all 15 years for which λ could be estimated directly, although the elasticity of adult survival declined with population growth.
- 4Survival and breeding success were positively correlated across years. λ remained most sensitive to adult survival when this demographic covariation was incorporated into elasticity analyses.
- 5However, elasticities calculated from a fully age-structured model suggested that λ was more elastic to variation in first- and second-year survival than to survival at any individual older age class. These ranks were robust to realistic demographic variation, but sensitive to postulated patterns of demographic covariation. We emphasize that covariation should be measured and incorporated into elasticity analyses, and that estimated elasticities must be interpreted in the context of the way in which stage-classes are defined.
- 6Of the demographic rates in which we quantified between-year variation, first-year survival varied most, followed by second-year survival, breeding success and adult survival. These rates consequently contributed more equally to variation in λ than elasticities predicted. Overall, variation in λ was caused primarily by variation in survival rather than breeding success, and variation in prebreeding survival accounted for 56% of the total variation in λ.

### Introduction

Populations of most organisms fluctuate in size, the population growth rate being sometimes positive and sometimes negative. Such fluctuations are caused by extrinsic and intrinsic drivers, such as environmental change and density dependence, and by interactions between these drivers and population structure (Sæther 1997; Gaillard, Festa-Bianchet & Yoccoz 1998; Coulson *et al*. 2001). Such drivers exert their effect via their impact on demographic rates, such as survival and reproduction, across population members. A valuable step in understanding the mechanistic basis of population change is therefore to identify which demographic rates are primarily responsible for causing variation in population growth, and which stage- or age-classes are most affected (Sæther 1997; Gaillard *et al*. 1998; Sibly & Hone 2002). However, despite the pure and applied value of describing general links between demographic variation and population growth (Wisdom & Mills 1997; Pfister 1998; Benton & Grant 1999; de Kroon, van Groenendael & Ehrlén 2000; Heppell, Caswell & Crowder 2000), few principles have emerged (Sibly & Hone 2002). Further detailed case-studies, using species with differing life-histories, are needed to understand over-arching patterns (Gaillard *et al*. 2000; Sæther & Bakke 2000; Clutton-Brock & Coulson 2002).

Variation in different demographic rates does not affect a population's growth rate (λ) equally. For example, a small change in survival may affect λ more than a similar change in fecundity (Caswell 2001). Given a quantitative description of a population's demography, the demographic rates to which λ is most sensitive (i.e. the rates in which variation would cause the greatest change in λ) can be identified (Caswell 2001). The sensitivity of λ to different demographic rates has been examined in numerous plants, reptiles, birds and mammals (Silvertown *et al*. 1993; Gaillard *et al*. 1998, 2000; Heppell *et al*. 2000; Sæther & Bakke 2000). However, sensitivities have been estimated frequently from mean demographic rates, assuming that rates vary independently and that population growth is unconstrained. This is despite the expectation that sensitivities will be influenced by demographic change, and by the degree to which rates covary and are constrained by density dependence (van Tienderen 1995; Benton & Grant 1996; Grant & Benton 2000; Wisdom, Mills & Doak 2000). Although it is increasingly recognized that rigorous sensitivity analyses should consider effects of model structure and demographic variation and constraint (van Tienderen 1995; de Kroon *et al*. 2000; Wisdom *et al*. 2000), few studies have addressed these issues as yet (although see Wisdom *et al*. 2000; Zuidema & Franco 2001; and Albon *et al*. 2000 for an alternative approach). Particularly, although demographic rates may frequently covary (van Noordwijk & de Jong 1986; Stearns 1992), natural patterns of covariation have rarely been described (Menges 2000; Sæther & Bakke 2000; Wisdom *et al*. 2000). Effects on sensitivities are consequently ignored or assumed to be minimal (Caswell 2000; Menges 2000; Zuidema & Franco 2001). Further studies that measure demographic variation and covariation and consider effects on rate sensitivities are therefore needed to accurately predict demographic effects on population growth (van Tienderen 1995; Gaillard *et al*. 2000).

Once the demographic rate to which population growth is theoretically most sensitive is identified, it cannot be concluded that changes in this rate actually cause observed changes in population size. This is because λ varies as a function of the extent to which each rate varies as well as its sensitivity to this variation; λ will remain constant if the demographic rates with high sensitivities do not vary, but change if rates with low sensitivities vary greatly (Benton & Grant 1996; Gaillard *et al*. 2000; Sæther & Bakke 2000; Wisdom *et al*. 2000). To identify which demographic rates drive population change, the potential influence of demographic variation on λ (the rate sensitivities), the degree to which rates actually vary and the combined consequences of these patterns must all be quantified (Gaillard *et al*. 1998, 2000; Sæther & Bakke 2000; Wisdom *et al*. 2000).

Demographic variation, however, is difficult to measure in natural populations. This is particularly true in iteroparous species with age- or stage-structured life-histories, where the mean and variance of each demographic rate should ideally be estimated for each class. Long-term studies of marked individuals are generally required to estimate rates and ensure that representative temporal variation is observed (Sæther 1997; Gaillard *et al*. 1998, 2000; Sæther & Bakke 2000). Even in well-studied populations, simplifications, such as pooling consecutive age-classes, are usually necessary (e.g. Brault & Caswell 1993; de Kroon *et al*. 2000; Heppell *et al*. 2000). Further, some demographic rates prove consistently difficult to measure. For example in birds, where juveniles typically disperse after fledging (Greenwood & Harvey 1982), prebreeding survival is rarely measured accurately (Anders *et al*. 1997; Sæther & Bakke 2000). Analysts consequently assume that prebreeding survival is equivalent or proportional to survival of other age-classes, populations or species, back-calculate rates assuming population stability or simply acknowledge that survival may be underestimated (e.g. Donovan *et al*. 1995; Hiraldo *et al*. 1996; Wisdom & Mills 1997; Budnik, Ryan & Thompson 2000; Sæther & Bakke 2000). Such assumptions are rarely validated, yet may greatly influence projected links between demography and population growth. The accuracy of simplifying assumptions and parameter estimates can in theory be checked by using these estimates to model the population trajectory, then comparing the modelled trajectory to census data. However, because populations are frequently difficult to census, this powerful means of validating population models is not always available (but see Brault & Caswell 1993; Coulson *et al*. 2001; Haydon *et al*. 1999).

Here, we investigate links between observed demographic variation and the growth rate of a natural red-billed chough (*Pyrrhocorax pyrrhocorax* Linnaeus) population. The population that we studied inhabits the island of Islay (55°N, 6°W), 25 km west of the Argyll coast in south-west Scotland. This population is resident, has been studied continuously since 1981 and was fully censused using standardized methodology in 1982, 1986, 1992, 1998 and 2002 (see Finney & Jardine 2003). The number of breeding pairs varied between 78 in 1986 and 45 in 1998 (Finney & Jardine 2003). A total of 992 fledgling choughs were individually colour-ringed up to 2000, and detailed data describing their subsequent demography have been collected (Bignal *et al*. 1987; Monaghan *et al*. 1989; Reid *et al*. 2003a,b). Islay and the adjacent island of Colonsay hold virtually the entire Scottish chough population, and surveys of surrounding areas show that choughs rarely emigrate (Monaghan *et al*. 1989; Finney & Jardine 2003). Juveniles and prebreeding adults form conspicuous foraging and roosting flocks in which colour-ringed birds can be identified readily; substantial effort, also involving a network of volunteers, has been put into these observations (Still, Monaghan & Bignal 1987; Bignal, Bignal & McCracken 1997). This system therefore provides a valuable opportunity to estimate mean and between-year variation and covariation in population-wide demographic rates (including prebreeding survival) in an avian population, to validate parameter estimates by comparing population models to census data, and thus to dissect the demographic causes of variation in population growth.

Our aims in this paper are threefold. First, we estimate mean age-specific rates of survival and reproduction. We then use a prospective elasticity analysis (Caswell 2001) to identify the age-specific rates to which the population growth rate is most elastic. Secondly, we attempt to quantify the degree to which age-specific survival and reproduction vary and covary across years, and investigate the effects of variation and covariation on estimated elasticities. As simplifying assumptions are necessary at this stage, we validate assumptions and parameter estimates by modelling the population trajectory and comparing this trajectory to census data. Thirdly, we combine elasticity and variability estimates to quantify the extent to which each demographic rate contributed to observed variation in λ, and compare projected and realized influences on λ.

### Materials and methods

#### age- and year-specific reproduction and survival

Choughs on Islay first breed aged 2–4 years, nest in traditional cavity sites and lay one clutch of three to six eggs each spring (mean 4·7 ± 0·1, see also Bignal *et al*. 1987). Reproduction and survival vary between years and are correlated with weather but not population size (Reid *et al*. 2003a). Each year from 1981 to 2002, 50–75% of occupied nest-sites were visited near the end of the chick-rearing period. The number of chicks fledging (breeding success) was recorded and a sample of fledglings was marked with unique combinations of colour-rings. Subsequent monitoring of colour-ringed (thus known-age) individuals allowed the mean breeding success (m_{i}) of each age-class *i* to be estimated (Reid *et al*. 2003b).

Over 9000 resightings of colour-ringed choughs have been documented to date, distributed across Islay and across age-classes. Capture–mark–recapture models based on these resightings were used to estimate maximum likelihood age- and year-specific survival probabilities. Individuals were noted as observed or not observed during 1 May–1 July each year. Because few data were collected in 1982 or 2002, survival analyses covered 1983–2001. Survival probabilities were therefore estimated for the years 1983–84 to 1999–2000 (see Lebreton *et al*. 1992). A fully time-dependent Cormack–Jolly–Seber model was initially fitted (program mark, Lebreton *et al*. 1992; White & Burnham 1999). This model fitted the data (‘Release’ tests for heterogeneity, *P* > 0·1) and data were not greatly overdispersed (variance inflation factor ≈ 1·07, Cooch & White 1998). This model was constrained to estimate age- and year-specific survival probabilities and appropriate resighting probabilities (see also Reid *et al*. 2003a). Akaike's information criterion, adjusted for small sample sizes and overdispersion (qAICc), was used to identify the most parsimonious model that included the desired survival parameters. As previous analyses suggested that survival did not differ significantly between breeding males and females (Reid *et al*. 2003b) and the sexes of prebreeding choughs were unknown, age-specific survival probabilities were estimated across all individuals. As Islay's chough population is relatively isolated, apparent survival probabilities are likely to primarily reflect variation in mortality rather than dispersal.

As not all colour-ringed choughs were observed in all years, the proportion of each age-class that bred in each year could not be counted directly. Probabilistic models cannot be used to estimate this proportion retrospectively because previous breeding, unlike survival, cannot be inferred from subsequent sightings in a resident population. Instead, we divided the number of females observed breeding at each age by the total number of females observed at this age. As resighting probability did not vary with age (see Results), these calculations are unlikely to be biased by differential observation of breeders and nonbreeders. Individuals observed within non-territorial roosts or flocks early during the breeding season were assumed to be non-breeders. Because chough sexes cannot be determined in the field other than by breeding behaviour, we assumed an equal sex ratio in prebreeding birds.

To quantify covariation among demographic rates we calculated correlation coefficients between pairs of rates for which year-specific values were estimated. To quantify and compare the degree of temporal variability in each rate, we calculated across-year coefficients of variation (CV, a rate's standard deviation divided by its mean).

#### projection model structure and analysis

Different methods of estimating a population's growth rate make different simplifying assumptions. To assess the consequences of these assumptions, we used a range of modelling approaches. To investigate the consequences of demographic change for population growth, we primarily used a fully age-structured Leslie matrix projection model for the population, based on a prebreeding census of females and birth-pulse dynamics (Caswell 2001). Top row fecundity terms (f_{i}) described the mean number of daughters surviving to the next breeding season per female in each age class *i*. Assuming an equal sex ratio at fledging,

- ( (eqn 1))

where c_{i} is the proportion of females of age *i* that attempted to breed, m_{i} is the mean breeding success of these females and P_{1} is the mean probability of surviving from fledging to age one (first-year survival). Subdiagonal transition probabilities (*P*_{i}) described the probability of surviving between subsequent age classes. The fully age-structured model had 14 dimensions (thus allowing ages up to the maximum observed on Islay; Reid *et al*. 2003b).

We calculated the asymptotic population growth rate (λ) as the dominant eigenvalue of the projection model (Caswell 2001). We calculated elasticities as a measure of the sensitivity of λ to variation in each demographic rate. Elasticities (e_{ij}) quantify the proportional change in λ resulting from a proportional change in a matrix entry a_{ij} while other entries are held constant (Caswell 2001):

- e
_{ij}= (a_{ij}/L)(δλ/δa_{ij})( (eqn 2))

where δλ/δa_{ij} is the partial derivative of λ with respect to a_{ij} (the rate sensitivity). As a mean-standardized sensitivity, elasticities allow comparison of the relative effect on λ of variation in different matrix components (Caswell 2001). However, the probability of breeding, breeding success and first-year survival appear in prebreeding projection models only within compound fecundity terms (eqn 1, Caswell 2001). As we were specifically interested in links between λ and variability in individual demographic rates, we focused on elasticities of rates rather than matrix entries. In this model formulation, elasticities of age-specific breeding probability (e(c_{i})) and breeding success (e(m_{i})) equal their respective fecundity elasticities (e(f_{i})) and can therefore be determined directly from the matrix model, while e(P_{1}) = Σ e(f_{i}) (Heppell *et al*. 2000; Sæther & Bakke 2000; Caswell 2001). Analytical values were confirmed numerically as the gradient of a log(rate) vs. log(λ) plot over a small range of demographic variation (Caswell 2001). Elasticities of individual demographic rates, unlike elasticities of matrix entries, do not sum to one (Caswell 2001).

We used Monte Carlo simulations to estimate confidence limits for λ and investigate effects of demographic variation and uncertainty on elasticity estimates (Alvarez-Buylla & Slatkin 1991; Wisdom *et al*. 2000). Five hundered projection matrices were parameterized independently by drawing demographic rates randomly from lognormal (breeding success) and beta (survival and breeding probability) distributions with mean and variance matching observed values. Elasticity values and λ were calculated from each matrix, and their distributions were described (as Wisdom *et al*. 2000; Fieberg & Ellner 2001). As observed demographic variance comprises both biological and sampling variance, this approach may overestimate variance in elasticities and λ. To check whether this affected our conclusions, simulations were repeated using half the observed variance (Wisdom *et al*. 2000). Matrix models and simulations were run in Excel, using the PopTools add-in.

To assess effects of demographic covariation on estimated elasticities, we calculated integrated elasticities. Integrated elasticities (IE) quantify the total effect of variability in a specific demographic rate on λ, including both direct effects and indirect effects via covariation with other rates (van Tienderen 1995). For a demographic rate *x*,

- IE
_{x}= Σr_{y}_{xy}e_{y}CV_{y}/CV( (eqn 3))_{x}

where r_{xy} is the correlation between rates *x* and *y*, and e_{y} and CV_{y} are the standard elasticity and coefficient of variation of *y*, respectively (van Tienderen 1995; Sæther & Bakke 2000). While elasticities identify the demographic rates to which λ is most sensitive, the contribution of each rate to variation in λ depends also on the extent to which the rate varies. To estimate the actual contribution of each demographic rate to population change we multiplied the rate's integrated elasticity by its coefficient of variation to give a variance-standardized elasticity (IE_{x}CV_{x}, van Tienderen 1995). The total variance in λ can be estimated as the square of this quantity summed across all variable demographic rates:

- ( (eqn 4))

(Horvitz, Schemske & Caswell 1997; Gaillard *et al*. 2000; Caswell 2001). We therefore estimated the proportional contribution of each rate *x* to total variation in λ (χ_{x}) as:

- ( (eqn 5))

Data were sufficient to estimate year-specific values of age-specific demographic rates only after pooling consecutive age-classes. To validate necessary assumptions and resulting parameter estimates, we used stage-structured projection models based on pooled age-classes to estimate year-specific values of λ and thus model the population trajectory. We sequentially multiplied an initial population census by consecutive annual estimates of λ, then compared the resulting trajectory to other censuses. We defined the breeding population in each census year as the total number of confirmed (eggs or young present) and probable (nest-building observed, but breeding not confirmed due to site inaccessibility) breeding pairs, and excluded the few territorial pairs where breeding was thought not to have been attempted (see Monaghan *et al*. 1989; Finney & Jardine 2003).

When demographic rates vary between years due to environmental stochasticity, the asymptotic growth rate (λ) estimated from standard matrix models may over-estimate the long-term population growth rate. The stochastic parameter (λ_{st}) may be a more accurate metric (Tuljapurkar 1990; Caswell 2001). As λ_{st} cannot be calculated analytically for complex life-histories (Caswell 2001), we used (Tuljapurkar 1990) small noise approximation:

- ( (eqn 6))

where τ^{2} = Σ* _{x}*(δλ/δ

*x*)

^{2}σ

^{x2}+ Σ

*(δλ/δ*

_{x}*x*)(δλ/δ

*x*)cov(

*x*,

*y*).

Cov(*x*,*y*) is the covariance between demographic rates *x* and y, is the variance of rate *x*, and λ_{0} and the sensitivities δλ/δ*x* are evaluated using mean demographic rates. This approximation assumes no between-year autocorrelation in environmental conditions and therefore demography (as observed on Islay, Reid *et al*. 2003a), and is robust to reasonable demographic variation (CV ≈ 0·40, Tuljapurkar 1990; Nations & Boyce 1997).

### Results

#### age-specific reproduction and survival

Figure 1 shows the mean breeding success (m_{i}) and apparent survival probability (*P*_{i}) of each female age-class, estimated across all years of data. Both breeding success and survival increased and then declined with age (see Reid *et al*. 2003b for further discussion). The most parsimonious survival model included year-specific but not age-specific resighting probability (Table 1a). Resighting probability averaged 0·72 ± 0·04 across years (range 0·41–0·93). Even after sexes were pooled, data were insufficient to estimate survival probabilities for all older ages individually. Combined probabilities were consequently estimated for 11- and 12-year-olds, and for choughs aged 13 or over: survival was assumed to be homogeneous within these groups. The estimated proportion of females that attempted to breed (c_{i}) increased from 0% of 1-year-olds to 28% of 2-year-olds, 81% of 3-year-olds, 97% of 4-year-olds and effectively 100% of older females.

Model | qAICc | ΔqAICc | qAICc weight | Parameters | Deviance |
---|---|---|---|---|---|

(a) Age-specific survival | |||||

ϕ(a)p | 3547·4 | 102·3 | 0·000 | 16 | 1043·6 |

ϕ(a)p(a) | 3563·5 | 118·4 | 0·000 | 30 | 1031·0 |

ϕ(a)p(y) | 3445·1 | 0·0 | 0·999 | 33 | 906·3 |

ϕ(a)p(1y,ady) | 3458·2 | 13·1 | 0·001 | 50 | 883·8 |

ϕ(a)p(1y & 2y,ady) | 3478·8 | 33·8 | 0·000 | 49 | 906·6 |

(b) Between-year variation in age-specific survival | |||||

ϕ(a)p(y) | 3445·1 | 9·5 | 0·004 | 33 | 906·3 |

ϕ(1y,2y,ady)p(y) | 3435·5 | 0·0 | 0·455 | 68 | 822·8 |

ϕ(1y,2y,ady,13 +)p(y) | 3435·9 | 0·4 | 0·382 | 69 | 820·9 |

ϕ(1y,ady)p(y) | 3437·7 | 2·2 | 0·152 | 52 | 859·2 |

ϕ(1y,2y,3y,ady)p(y) | 3444·0 | 8·5 | 0·007 | 83 | 798·6 |

ϕp(y) | 3640·2 | 204·6 | 0·000 | 19 | 1130·2 |

ϕ(y)p(y) | 3648·0 | 212·5 | 0·000 | 35 | 1105·2 |

#### fully age-structured projection model

Based on mean demographic rates for each age, the fully age-structured projection model gave λ = 0·99 (95% CI 0·83–1·18, or 0·93–1·09 with 50% variance). Thus λ did not differ significantly from unity. Figure 2 shows the estimated elasticities of breeding success (and breeding probability, as e(c_{i}) = e(m_{i})) and survival at each age. This analysis suggested that λ was most sensitive to variation in first-year (fledging to age 1) and second-year (age 1–2) survival, followed consecutively by survival through subsequent years. Simulations showed that these ranks were robust to estimated variance in demographic rates; using the full degree of observed demographic variation, λ was most elastic to first- and second-year survival in 100% of simulations, followed by third- then fourth-, fifth-, sixth- and seventh-year survival in 99, 99, 96, 83 and 67% of simulations, respectively.

#### demographic variation and covariation

Variation in λ depends on the extent to which demographic rates vary as well as the sensitivity of λ to this variation, and overall sensitivities are affected by covariation among rates. We therefore attempted to quantify between-year variation and covariation in demographic rates. Data were insufficient to estimate the probability of survival for every individual age in every year. We therefore used maximum likelihood models to identify pooled age-classes (stage-classes) that were statistically distinct with respect to survival while retaining sample sizes sufficient to estimate year-specific values. The most parsimonious model included first-year, second-year and adult (all ages over 2) classes (Table 1b). Consistent with the suggestion that survival declines in old age in choughs (Fig. 1, Reid *et al*. 2003b), there was some support for including a separate class for choughs aged 13 or over (Table 1b). However, because too few choughs reached this age to quantify between-year variation in survival, we retained a single adult category for all ages over 2. Year-specific estimates of adult survival did not differ greatly when the few old birds were included or excluded. Table 2 shows the year-specific survival estimates, with overall means and coefficients of variation. Survival varied markedly between years in first-years (likelihood ratio, = 46·0, *P* < 0·001) and marginally so in second-years ( = 25·9, *P* = 0·04) and adults ( = 24·0, *P* = 0·05). Mean breeding success also varied between years (one-way anova*F*_{21,675} = 1·6, *P* = 0·04, Table 2) but there were insufficient data to quantify between-year variation in age-specific breeding success or the proportion of females that attempted to breed. In summary, year-specific values of first-year, second-year and adult survival and mean breeding success were estimable. These demographic rates were all to some degree positively correlated across years; a good breeding year tended to be a good survival year across all ages (Table 3).

Year | m | P_{1} | P_{2} | P_{ad} | λ | e(P_{ad}) | e(m), e(P_{1}), e(P_{2}) |
---|---|---|---|---|---|---|---|

Mean | 1·99 ± 0·15 | 0·43 ± 0·02 | 0·64 ± 0·03 | 0·80 ± 0·01 | |||

CV | 0·16 | 0·36 | 0·29 | 0·11 | |||

1981 | 2·00 ± 0·33 | ||||||

1982 | 1·66 ± 0·26 | ||||||

1983 | 2·36 ± 0·28 | 0·58 ± 0·08 | |||||

1984 | 2·44 ± 0·29 | 0·74 ± 0·08 | 0·78 ± 0·10 | ||||

1985 | 2·38 ± 0·25 | 0·52 ± 0·05 | 0·91 ± 0·06 | 0·95 ± 0·09 | 1·27 | 0·67 | 0·17 |

1986 | 1·85 ± 0·22 | 0·38 ± 0·06 | 0·63 ± 0·08 | 0·75 ± 0·07 | 0·93 | 0·72 | 0·14 |

1987 | 2·16 ± 0·25 | 0·41 ± 0·06 | 0·67 ± 0·10 | 0·80 ± 0·06 | 1·02 | 0·70 | 0·15 |

1988 | 1·91 ± 0·25 | 0·39 ± 0·07 | 0·55 ± 0·10 | 0·84 ± 0·06 | 1·00 | 0·76 | 0·12 |

1989 | 1·81 ± 0·27 | 0·20 ± 0·07 | 0·47 ± 0·13 | 0·69 ± 0·07 | 0·78 | 0·81 | 0·10 |

1990 | 1·96 ± 0·28 | 0·48 ± 0·09 | 0·61 ± 0·22 | 0·86 ± 0·07 | 1·07 | 0·72 | 0·14 |

1991 | 2·12 ± 0·20 | 0·36 ± 0·06 | 0·49 ± 0·12 | 0·87 ± 0·07 | 1·02 | 0·78 | 0·11 |

1992 | 1·87 ± 0·30 | 0·30 ± 0·07 | 0·44 ± 0·11 | 0·63 ± 0·07 | 0·76 | 0·75 | 0·13 |

1993 | 1·67 ± 0·24 | 0·26 ± 0·07 | 0·44 ± 0·14 | 0·68 ± 0·09 | 0·78 | 0·80 | 0·10 |

1994 | 1·27 ± 0·27 | 0·24 ± 0·08 | 0·43 ± 0·17 | 0·89 ± 0·09 | 0·95 | 0·88 | 0·06 |

1995 | 1·80 ± 0·28 | 0·41 ± 0·09 | 0·87 ± 0·23 | 0·94 ± 0·09 | 1·16 | 0·73 | 0·13 |

1996 | 2·33 ± 0·35 | 0·26 ± 0·08 | 0·71 ± 0·16 | 0·83 ± 0·10 | 1·00 | 0·75 | 0·13 |

1997 | 1·64 ± 0·29 | 0·35 ± 0·09 | 0·80 ± 0·19 | 0·84 ± 0·08 | 1·02 | 0·74 | 0·13 |

1998 | 2·45 ± 0·28 | 0·60 ± 0·09 | 0·66 ± 0·16 | 0·83 ± 0·07 | 1·13 | 0·66 | 0·17 |

1999 | 2·11 ± 0·23 | 0·48 ± 0·10 | 0·57 ± 0·13 | 0·77 ± 0·09 | 1·02 | 0·71 | 0·15 |

2000 | 2·13 ± 0·22 | ||||||

2001 | 2·00 ± 0·33 | ||||||

2002 | 1·69 ± 0·19 |

P_{1} | P_{2} | P_{ad} | |
---|---|---|---|

m | 0·71*** | 0·42* | 0·17 |

P_{1} | 0·55** | 0·44* | |

P_{2} | 0·64** |

#### covariation and trait elasticities

As we could not quantify between-year variation in all age-specific demographic rates, we could not quantify covariation between all components of the fully age-structured life history. We therefore used simplified stage-structured models to investigate effects of demographic covariation on rate elasticities, based on demographic rates for which between-year variation and covariation could be measured. We first validated stage-structured models and parameter estimates by modelling the population trajectory and comparing this trajectory to census data. The stage-structured model included the second-year and adult age-classes for which year-specific survival was estimated (Fig. 3). We also included a distinct third-year class because not all 3-year-olds bred. We assumed that third-year survival equalled adult survival in each year: this was supported by survival models (Table 1b). On average, 2- and 3-year-old females fledged approximately 60% and 75% of the offspring of middle-aged adults, respectively (Fig. 1). As 2 and 3-year-olds comprised a minority of observed breeders, we assumed that these age-classes, respectively, achieved 60% and 75% of the mean breeding success in each year, and that a constant proportion of each age-class bred each year.

Using mean rates of stage-specific survival and reproduction estimated across all years, the stage-structured model gave λ = 1·00 (95% CI 0·78–1·14, or 0·88–1·08 with 50% variance). This did not differ from the estimate of 0·99 from the fully age-structured model or from unity. However, across the 15 years (1985–99) for which stage-structured models were fully parameterized, λ ranged from 0·76 to 1·27 (Table 2). These year-specific estimates of λ were used to model the population trajectory for 1985–2000. Although truncation of the survival analysis meant that λ could not be estimated directly for 1982–84, 2000 or 2001 (Table 2), it was valuable to estimate λ for these years to extend the modelled trajectory to the 1982 and 2002 censuses. Eleven of 22 fledglings colour-ringed in 2000 were observed during 2001, giving a minimum juvenile survival rate of 0·5 for this cohort. We estimated other unknown pre-adult survival probabilities from the mean breeding success observed in each year, using empirical correlations between reproduction and survival (Table 3). As we had little power to estimate adult survival from mean breeding success (Table 3), we set unknown values equal to the overall mean of 0·80 (Table 2). The modelled population trajectory matched the census data remarkably well (Fig. 4); predicted population sizes differed from censuses by less than 4% on average.

Analysis of the mean stage-structured model suggested that λ was more sensitive to variation in adult survival than to variation in breeding success or first-year or second-year survival (Table 4). These ranks were consistent across all 15 years for which all demographic rates were directly estimated (Table 2), although elasticities varied significantly with λ; the elasticity of adult survival decreased while the fecundity elasticities increased as λ increased (Fig. 5). We calculated integrated elasticities to assess the effect of demographic covariation on the sensitivity of λ to variation in each demographic rate. λ remained most elastic to integrated variation in adult survival, although the relative importance of this term was reduced slightly relative to first- and second-year survival and breeding success (Table 4).

Trait | Standard elasticity (e_{x}) | e_{x} relative to e(P_{ad}) | Integrated elasticity (IE_{x}) | IE_{x} relative to IE(P_{ad}) | CV | IE_{x}CV_{x} | χ_{x} |
---|---|---|---|---|---|---|---|

P_{1} | 0·14 | 0·20 | 0·34 | 0·29 | 0·36 | 0·12 | 26·6 |

P_{2} | 0·14 | 0·20 | 0·45 | 0·38 | 0·29 | 0·13 | 29·2 |

P_{ad} | 0·71 | 1·00 | 1·19 | 1·00 | 0·11 | 0·13 | 29·7 |

m | 0·14 | 0·20 | 0·57 | 0·48 | 0·16 | 0·09 | 14·5 |

Although we could not quantify the degree of covariation between all age-specific demographic rates, we attempted to check whether elasticities estimated from the fully age-structured model were robust to plausible patterns of demographic covariation. We calculated integrated elasticities for age-specific demographic rates assuming that observed correlations between stage-specific survival and breeding success held across all ages within the adult category. This postulated pattern of covariation greatly altered estimated elasticities; λ was most sensitive to variation in survival of breeding adults (Fig. 6) rather than to survival of prebreeding age-classes (Fig. 2).

#### contributions of demographic rates to variation in λ

Based on the stage-structured model, the contribution of each demographic rate to the total variance in λ (calculated from variance-standardized elasticities, eqns 4 and 5) was more equal than elasticity values predicted (Table 4). Variation in first-year, second-year and adult survival contributed approximately equally to observed variance in population growth, while variation in breeding success contributed slightly less (Fig. 7).

Based on mean year-specific estimates of demographic rates, we estimated a stochastic population growth rate (λ_{st}) of 0·974.

### Discussion

Understanding links between demographic variation and the population growth rate (λ) is an important step towards understanding population dynamics, and requires that patterns of demographic variation are described accurately (Gaillard *et al*. 2000). In choughs on Islay, mean breeding success and survival varied with age and varied and covaried across years (see also Reid *et al*. 2003a,b). However, as is almost inevitable in relatively long-lived species, year-specific values of demographic rates could be estimated only after pooling consecutive ages. Despite these simplifications, the population trajectory modelled using parameter estimates matched census data closely. This suggests that we quantified sufficient demographic variation to account for observed variation in population growth, and that simplifying assumptions and parameter and covariance estimates were sufficiently accurate to describe important demographic processes.

In Islay's choughs, demographic rates covary across different stages of cohort members’ life-histories (Reid *et al*. 2003a). Such ‘cohort effects’ can affect population dynamics (Lindström & Kokko 2002). Because these effects are best explored using individual-based models, a separate investigation into their influence on chough population dynamics is in progress (Scottish Chough Study Group, unpublished data). However, the congruence between the models presented here and census data suggests that the direct population dynamic consequences of delayed life-history covariation may be relatively small, at least compared to effects of current demographic variation.

The basic stage-structured projection model suggested that λ was more elastic to variation in adult survival than to variation in breeding success or first- or second-year survival. These ranks were consistent across all 15 years for which survival probabilities were directly estimated, although λ was less elastic to adult survival in years of population growth. However, standard elasticities may predict the population consequences of demographic variation inaccurately if rates covary or are constrained by density dependence (van Tienderen 1995; Grant & Benton 2000). The extent to which chough demography is constrained by population density on Islay is currently unclear. However, breeding success and survival did not vary with population size over the study period (Reid *et al*. 2003a), suggesting that overall effects of density may not be strong. However, these traits were positively correlated, possibly reflecting common effects of environmental conditions on different life-history stages (Reid *et al*. 2003a). After incorporating this demographic covariation into integrated elasticities, λ remained most elastic to variation in adult survival, although this elasticity was slightly reduced relative to that of breeding success and preadult survival. In contrast, using similar stage-structured models for eight other bird populations, Sæther & Bakke (2000) found that covariation increased the elasticity of adult survival in seven of the eight cases; these differences may reflect variation in patterns of demographic covariation between populations.

The suggestion that population growth in choughs is most elastic to variation in adult survival is consistent with previous studies of relatively long-lived vertebrates (e.g. Brault & Caswell 1993; Gaillard *et al*. 1998, 2000; Pfister 1998; Sæther & Bakke 2000). This consensus has been widely proposed as a basis for management action (e.g. Lande 1988; Escos, Alados & Emlen 1994; Crooks, Sanjayan & Doak 1998) and suggested to drive the evolution of demographic variability (Pfister 1998). However, because demographic rates can rarely be measured for all ages independently, ‘adult’ is usually defined to include multiple postrecruitment ages. The elasticity of λ to a change in survival or reproduction across all ages within the adult category is then quantified (e.g. Escos *et al*. 1994; Sæther & Bakke 2000; Wisdom *et al*. 2000). However, the total elasticity of λ to variation across any group of life-history stages equals the sum of the elasticities of each individual stage and increases with the contribution that each class makes to the total population size (Gaillard *et al*. 1998; Caswell 2001). The elasticity of λ to variation in adult survival is therefore bound to increase with the number of age-classes included within the adult category; a high total elasticity does not necessarily imply that λ is most elastic to survival at any individual adult age. Indeed, analysis of the fully age-structured model for choughs suggested that, if demographic rates vary independently, λ is in fact most elastic to variation in prebreeding survival. These ranks were robust to realistic levels of demographic variation, and are consistent with other finely age-structured elasticity analyses for iteroparous vertebrates (e.g. Wisdom & Mills 1997; Krüger & Lindström 2001). The similarity between our results and age-specific elasticities for red deer *Cervus elaphus* L. is notable (Benton, Grant & Clutton-Brock 1995). Such analyses, however, assume that demographic rates vary independently. Describing the full covariance structure of chough demography, as necessary to incorporate covariation into fully age-structured sensitivity analyses, is not possible given current data. However, incorporating the positive demographic covariation that available data suggest is plausible substantially altered elasticity estimates; elasticities of breeding success and adult survival were greatly increased. Similarly, Benton *et al*. (1995) found that demographic covariation (in this case due to reproductive trade-offs) greatly affected elasticity estimates in *C. elaphus*. Our analyses were solely a first step towards investigating how robust elasticity estimates may be to postulated patterns of demographic covariation. Further detailed measurements of covariation in natural populations, and investigation of its effects on the magnitude and rank order of elasticities, are clearly required to describe and understand general patterns and effects. Sensitivity analyses that ignore demographic covariation should meanwhile be interpreted with caution.

Irrespective of the degree and consequences of covariation, our results emphasize that estimated elasticities of λ to variation in demographic rates depend on how these rates are defined. This dependence has implications for the application of elasticities to population management. In long-lived species where survival in young adults is already high (e.g. Loison *et al*. 1999), management aimed at increasing mean adult survival may succeed only by enhancing survival in older individuals. The high elasticity of λ to adult survival predicted by simplified, stage-structured models suggests that this action should greatly increase λ. However, the low elasticity of λ to survival in old age indicated by more finely age-structured models, suggests that this change may have (depending on patterns of demographic covariation) little effect on population growth. Further, as selection is likely to shape individual age-specific life-history traits rather than traits averaged across multiple ages, it is not clear how closely pooled elasticities might predict the strength of selection and thus expected patterns of demographic variability (e.g. Pfister 1998).

The degree to which any demographic rate causes population change depends on the rate's variability as well as the sensitivity of λ to this variation (Gaillard *et al*. 1998, 2000; Sæther & Bakke 2000). Of the rates in which between-year variation could be measured in choughs, first-year survival varied most, followed by second-year survival, breeding success and adult survival. Consequently, these rates contributed more equally to variation in λ than (integrated) elasticities predicted. This pattern is consistent with data from a range of species (e.g. Ehrlén & van Groenendael 1998; Gaillard *et al*. 1998, 2000); a negative correlation between rate elasticity and variability may be a general phenomenon (Pfister 1998). The most elastic demographic rate may generally not cause the majority of variation in population growth.

Previous studies investigating the demographic causes of population change in birds have generally compared the importance of reproduction with that of adult survival, and focused less on prebreeding survival (e.g. Pfister 1998; Sæther & Bakke 2000; Crone 2001). This may be because temporal variation in prebreeding survival is often difficult to measure in avian populations. Our results suggest a primary importance of variation in survival rather than breeding success in driving population change in choughs, with first- and second-year survival accounting for approximately 56% of the total variation in λ (Table 4, Fig. 7). This contrasts with the paradigm that prebreeding survival explains little variation in population growth in long-lived vertebrates (Pfister 1998) but is consistent with studies that have investigated explicitly the population consequences of variation in first-year survival in birds and mammals (e.g. Arcese *et al*. 1992; Thomson, Baillie & Peach 1997; Gaillard *et al*. 1998, 2000; Cooch, Rockwell & Brault 2001). Researchers and population managers should focus more on the prebreeding life-history stages that are often under-studied in natural populations.

The chough is currently of conservation concern in Britain (Gibbons *et al*. 1996) and management intervention has been proposed on Islay. Based on mean demographic rates estimated over the 20-year study period, the asymptotic growth rate (λ) of the Islay population did not differ from 1. However, we estimated a stochastic growth rate (λ_{st}) of 0·974. The relatively small discrepancy between λ and λ_{st} suggests that the chough's life-history to some extent buffers the fitness consequences of environmentally induced demographic variation. This may be generally true when adult survival is high (Lande 1988; Benton *et al*. 1995), and the chough's relatively young age of first breeding and high fecundity compared to other long-lived birds may reflect life-history adaptation to a variable environment (Sæther, Ringsby & Roskaft 1996). However, an average annual decline in the breeding population of 2–3% is predicted given continuation of current demographic patterns. While the most suitable management actions may be determined by practical constraints more than theoretical possibilities (Benton & Grant 1999; de Kroon *et al*. 2000; Beissinger & McCullough 2002), our analyses suggest that the population growth rate is more sensitive to, and primarily affected by, variation in survival, particularly at younger ages, than to variation in breeding success. Conservation efforts may need to be directed at enhancing survival, particularly at prebreeding stages.

### Acknowledgements

Earthwatch, Merial Animal Health, Nature Conservancy Council, Royal Society for the Protection of Birds, Scottish Executive Environment and Rural Affairs Department, Scottish Natural Heritage and WWF-UK provided financial support during the study, and RSPB and SNH funded data compilation and preliminary analyses. JMR was additionally supported by Killam and Green College Postdoctoral Fellowships at UBC. Islay farmers kindly allowed access to nest sites. Numerous people assisted with data collection, including Martin and Robin Bignal, John and Pamela Clarke, David Jardine, Clive McKay, Neil Metcalfe, Allen Moore, Malcolm Ogilvie, Elizabeth Still, Judy Stroud, Paul Thomson and RSPB, SNH and Islay Natural History Trust staff. Jan Lindström advised on analyses, Tricia Bradley and Roddy Fairley provided helpful discussions, and Jeremy Wilson commented on a manuscript draft. Nest visits were licensed by NCC and SNH.

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