### Introduction

- Top of page
- Summary
- Introduction
- METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- References
- Appendix

Theoretical analyses have identified several factors that affect time to extinction of a population. In a deterministic model with no age-structure extinction will always occur when the specific growth rate *r* is negative, the time to extinction from initial population size *N*_{0} being ln *N*_{0} (Richter-Dyn & Goel 1972). Population fluctuations are, however, also influenced by stochastic effects. The environmental variance σ_{e}^{2} is generated by random variations affecting all individuals in the population similarly, whereas the demographic variance σ_{d}^{2} is created by random independent individual variations in births and deaths. These stochastic effects increase the risk of extinction because the population size may become small from random effects, but also because it reduces the population growth rate (Leigh 1981). The stochastic population growth rate, defined as the expected change in the logarithm of the population size between seasons, is *s* = *r* − σ_{e}^{2}/2 − σ_{d}^{2}/2*N*, where *N* is the population size and *r* is the per capita growth rate at an absolute scale (Lande 1998). The variance in the change in population size from one year to the next is σ_{e}^{2}*N*^{2} + σ_{d}^{2}*N*. In large populations the environmental variance may create large stochastic fluctuations and it reduces the growth rate to *r*_{0} = *r* − σ_{e}^{2}/2. Hence, environmental stochasticity may increase substantially the probability of extinction, even for large populations. When the population size is reduced, the impact of the demographic variance on the population growth rate increases, and may actually create a stochastically determined Allee-effect (Lande 1998). All these different effects of stochasticity imply that extinction may occur even when *r* is positive.

Reliable population projections require correct description of population growth and density dependence, as well as a realistic modelling of the stochasticity. In addition, uncertainties in the parameter estimates must be considered when predicting future population sizes or the time to extinction in a population viability analysis. Sæther *et al*. (2000a) suggested, following Dennis, Munholland & Scott (1991), that the population prediction interval (PPI) may be a useful concept in such analyses that embraces the expected dynamics, the stochasticity in the model, and the accuracy of the parameter estimates. A PPI is a stochastic interval that includes the unknown variable to be predicted with probability (1 − α). In population viability analysis, adopting the precautionary principle (IUCN 1994), one should use the upper one-sided intervals ranging from *t*_{α} to infinity, which means that the extinction time is predicted to be smaller than *t*_{α} with probability α. The interpretation of a prediction interval is the same as for a confidence interval, except that we draw inference about a stochastic quantity rather than a parameter. Uncertainty in the parameters does not change the risk of extinction of the population, but affects the confidence we have in the population predictions. In a small island population of song sparrow *Melopspiza melodia* (Wilson), neglecting uncertainties in the parameter estimates led to an 33% overestimation of *t*_{α} (Sæther *et al*. 2000a). The lack of consideration of uncertainties and biases in population parameters has often made it difficult to verify predictions from population viability models (Beissinger & Westphal 1998). As a consequence, such difficulties have been used to question population viability as a useful management tool (Ludwig 1999). However, according to the ‘precautionary principle’ (IUCN 1994), the preciseness in the predictions should be considered and included in recommendations about the management of endangered or threatened species. Large uncertainties should result in a more cautious approach than in those cases where available information permits accurate population projections.

Many bird populations in the European agricultural landscape are now declining rapidly (Pain & Pienkowski 1997; May 2000,Donald, Green & Heath 2001). Although the population sizes in several of those cases are already probably far below the carrying capacity, the population size is often still large. Extinction of such abundant species does not seem to be of immediate concern. However, Lande & Orzack (1988) suggested that time to extinction even of large populations could be quite short. Thus, application of quantitative criteria for risk assessment as suggested by Mace & Lande (1991) seems necessary even for such species.

Assuming no density regulation, and using first-order Taylor expansion for the mean and variance, the above modelling of stochasticity is approximately equivalent to: E(Δ*X/X*) = *r* − σ_{e}^{2}/ − σ_{d}^{2}/2 exp(−*X*) and var(Δ*X/X*) = σ_{e}^{2} + σ_{d}^{2} exp(−*X*), where *X* = ln *N.* Furthermore, the change in the logarithm of population size between years can often be approximated by a normal distribution. If there is no demographic stochasticity, the mean and variance are constant and the process is equivalent to a Brownian motion recorded only at discrete time steps. Dennis *et al*. (1991) have previously used Brownian motions to estimate the risk of extinction of several populations of endangered or threatened species. They introduced the concept of population prediction interval and developed it for population size *N*. They also presented estimators of various quantities (mean, median, percentiles, cumulative distribution function) associated with the distribution of time to extinction, *T*. Engen & Sæther (2000) extended these results by deriving prediction intervals also for the time to extinction. Both Dennis *et al*. (1991) and Engen & Sæther (2000) considered only the case with no demographic stochasticity.

Here we use a similar approach, including demographic stochasticity in a model of the dynamics of a declining population of barn swallow *Hirundo rustica* L. to derive prediction intervals for the time to extinction. In this way, we can examine quantitatively how various factors affect the uncertainties and the risk of extinction of this population.

### RESULTS

- Top of page
- Summary
- Introduction
- METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- References
- Appendix

The number of fledglings produced in the first clutch has decreased significantly during the period 1984–99 (linear regression: *b* (SE) = −0·091 (0·011), *F* = 69·97, d.f. = 1, 1706, *r*^{2} = 0·040, *P* < 0·0001) which equals a predicted reduction by 1·46 offspring per brood during the period of 16 years. A decrease was also observed for the number of fledglings in the second brood (linear regression: *b* (SE) = −0·032 (0·009), *F* = 11·55, d.f. = 1, 966, *r*^{2} = 0·012, *P* = 0·0007), resulting in a predicted reduction by 0·51 per brood throughout the study period. However, the total number of offspring produced per pair and season did not decrease significantly (linear regression: *F* = 1·62, d.f. = 1, 1706, *r*^{2} = 0·001, *P* = 0·20). The reason for this absence of decrease in total success is a significant increase in the number of pairs having two broods per year (Spearman rank order correlation: *r*_{S} = 0·22, *N* = 1708, *P* < 0·0001).

Very few females were able to contribute to the next generation in this population. No females contributed more than two female offspring, whereas 79% of the females were either not recorded themselves or had any offspring a later year recruited to the population. A *R*_{i} value of 1 was found in 20·9% of the females. The estimate of the demographic variance σ_{d}^{2} was 0·180.

The maximum likelihood estimates of the stochastic population growth rate for large population sizes was *r *_{0}* = −0·076 and for the environmental variance σ_{e}^{2}* = 0·024. The confidence region was quite large (Fig. 3, environmental standard deviation ranging from 0·12 to 0·26 and *r *_{0} from −0·18–0·04).

In Fig. 4 we plot the lower bound *x*_{α} for the prediction intervals for the log population size at time *t* as a function of *t*. The lower bounds *t*_{α} for the prediction intervals for the time to extinction are where the curves for *x*_{α} cross the axis. We predict with 10% confidence that extinction of this barn swallow population will occur before 22 years (Fig. 5a). With 50% confidence extinction occurs before 53 years.

Correspondingly, Fig. 4 may also be read as prediction intervals for population sizes at different times and not only as prediction intervals for the time to extinction. For instance, the 90% two-sided prediction interval at a given time is (*x*_{0·05}, *x*_{0·95}). After 50 years this interval ranged from 0 to 64 individuals (Fig. 4a).

Ignoring uncertainties in the estimates, assuming that the parameters actually were known and equal to the maximum likelihood estimates, increased the predicted time to extinction (Fig. 4b). For instance, the lower bound of the interval for the extinction time for α = 0·10 increased with 41%, from 22 years when including uncertainty (Fig. 4a) to 31 years. Accordingly, using the ‘precautionary principle’ (IUCN 1994), we should be more conservative when the estimates are uncertain, being aware of the fact that the situation actually may be much more critical than what we can conclude directly from the maximum likelihood estimates.

Assuming that the demographic variance was zero, the estimate of the environmental variance became approximately 10% larger (0·026), because the demographic stochasticity is then absorbed in the environmental noise term. Actually, the demographic term σ_{d}^{2} is then replaced by a constant that is confounded with the constant from σ_{e}^{2}. Ignoring the demographic variance strongly increased the predicted life expectancy of the population. For instance, *t*_{0·10} increased from 22 (Fig. 4a) to 28 years (Fig. 5a), an increase of 27%. This probably occurred because inclusion of the demographic variance shortened the length of the final stage of the process to extinction (Figs 4 and 5a) by an increase in the variance as well as a decrease in the expected growth rate by the term σ_{d}^{2}/2*N*.

Ignoring environmental stochasticity strongly reduced the range of variation of the prediction interval. For instance, after 50 years *x*_{0·95} was reduced from 64 individuals with both demographic and environmental stochasticity present (Fig. 4a) to 8·5 individuals when choosing σ_{e}^{2} = 0 (Fig. 5b). As in the case for excluding demographic variance, the predicted time to extinction again increased.

### DISCUSSION

- Top of page
- Summary
- Introduction
- METHODS
- RESULTS
- DISCUSSION
- Acknowledgements
- References
- Appendix

This Danish population of barn swallow had a mean decline of 7·6% per year (Fig. 4), resulting in that a prediction of extinction before 22 years (Fig. 4a) can be performed with 10% confidence. Accordingly, after IUCN’s (1994) criteria, based on Mace & Lande (1991), this population should be classified as vulnerable. This occurs even though the population size at the last year of study was 58 pairs, illustrating that population size may be a poor predictor of extinction risk and that even relatively abundant species may go rapidly extinct.

However, these analyses assume a closed population with small emigration or immigration. If a considerable fraction of the recruits settled outside the study area, this is likely to result in an underestimate of σ_{d}^{2} because of a reduction in the proportion of females producing no recruits. The large size of the study area and the high recapture probability (Møller 1994) make it unlikely that this bias is substantial. Furthermore, immigration from outside will reduce the risk of extinction. However, the decline of the barn swallow has been known to occur over large areas (e.g. Gregory *et al*. 1999) so immigration from source areas may not be sufficient to prevent the decline.

Many bird species in the agricultural landscape in Europe now face severe declines in population sizes (Pain & Pienkowski 1997; May 2000; Donald *et al.* 2001) and may be related to recent changes in agricultural practice (Møller 2001). These results are based mainly on an analysis of population indices collected over large areas. The rate of decline in our study population was larger than recorded previously from one less comprehensive data source collected in the same region as our study situation. There is indeed a weak but statistically significant positive correlation between the population size in the study area and the bird census indices from the Danish national survey (*r* = 0·65, *N* = 16 years, *P* = 0·006, Møller 1994). The relative weak association between local intensive and regional, extensive surveys may be related to the problem that national bird census programmes rely on amateurs who do not visit randomly selected plots, and who do not cover the same study areas each year. As amateurs study birds because they like birds, it is possible that census plots with few birds have a higher probability of being discontinued than study plots with many birds. This could generate an apparent population trend that shows a smaller degree of decline than the actual decline in randomly selected plots. Accordingly, analysis based on population indices may underestimate the rate of decline so that the situation for many populations may be more severe than previously believed.

The reproductive success of barn swallows decreased during the period 1984–99 in both first and second broods, although there was no significant decrease in total annual reproductive success per pair. This latter observation was due to a significant increase in the proportion of pairs raising two broods per season. Analyses of local recruitment in the present barn swallow population have shown that significantly more individuals recruit from first than from second broods, partly because fledglings from first broods are in better body condition than nestlings from second broods (Møller 1994). Hence, this suggests that the decrease in the mean number of fledglings per pair for the first brood affected the population growth rate negatively.

The estimate of the demographic variance in this population of barn swallow was far lower than recorded previously in the passerines song sparrow (σ_{d}^{2} = 0·66), great tit *Parus major* L. (σ_{d}^{2} = 0·57) and dipper *Cinclus cinclus* L. (σ_{d}^{2} = 0·27) (Tufto *et al*. 2000). This low estimate is mainly because only a small proportion of the females is able to produce offspring that are later recruited into the population. Although we do not yet know the mechanism generating the decline of this population, degradation of quality of the breeding habitats, due probably to altered agricultural practices (Møller 2001), is a likely explanation. Whether a low demographic variance is a general feature of declining populations still remains to be determined.

Although the demographic variance was low, it especially affected the final stage of the process to extinction (Figs 4 and 5a), and strongly influenced the prediction of the time to extinction. In fact, the effect of the demographic stochasticity on the time to extinction was comparable to the effects of the environmental stochasticity (Fig. 5). In populations with larger demographic variance the demographic stochasticity will have an even stronger effect. This is in contrast to the results of a previous analysis of the viability of a great tit population, where the environmental variance affected the time to extinction more strongly than the demographic variance (Sæther *et al*. 1998). This is probably because our barn swallow population was smaller and much closer to extinction than the great tit population, which fluctuated around a carrying capacity of 216 pairs. In such a large population the effects of the demographic stochasticity on the variance in the population growth rate will be small (Lande 1998). To illustrate, we predicted the time to extinction of the barn swallow population, choosing the demographic variance previously estimated (Sæther *et al*. 1998) in the great tit population (σ_{e}^{2} = 0·569, assuming no density dependence). This strongly shortened the predicted time to extinction, *t*_{0·10} decreasing with 27% from 22 (Fig. 4a) to 16 years. Thus, reliable estimates of the demographic variance are crucial for a correct prediction of the time to extinction of small, declining populations. Accordingly, estimates of time to extinction (e.g. Dennis *et al*. 1991; Gaston & Nicholls 1995), ignoring demographic stochasticity, are likely to underestimate the risk of extinction of small populations. The lower 10% quantile of the estimated inverse Gaussian distribution of the time to extinction (Dennis *et**al*. 1991), ignoring uncertainties in the estimates, was 34 years. However, when estimating the time to quasi-extinction (Ginzburg *et al*. 1982) at a population size considerably above zero, this effect is likely to be small.

The environmental stochasticity strongly influences the dynamics (Sæther *et al*. 2000b) and the time to extinction of small passerine populations (Sæther *et al*. 1998). Although our estimate of the environmental variance was lower than in three other passerine species (Tufto *et al*. 2000), assuming no environmental variance still increased the time to extinction (Fig. 5b). The value of *t*_{0·10} was 35 years, an increase of 59% from the case where both demographic and environmental stochasticity were present (Fig. 4a). Furthermore, the width of the prediction interval strongly decreased when no environmental stochasticity was assumed to be present (Fig. 5b). This illustrates the necessity of correctly estimating and modelling the stochasticity in the population dynamics when predicting population trajectories.

Several approaches can be chosen to include uncertainty in parameter estimates in the construction of the prediction interval. One approach is the Bayesian, in which a prior distribution, for example a non-informative one, is converted into a posterior distribution when conditioned on the observations (Berger 1985). This is a conceptually simple way to get rid of the nuisance parameters by integrating over their posterior distribution, finally claiming that one actually knows the probabilities of any event of interest. Several computer programs (e.g. Spiegelhalter *et al*. 1996) are now available to perform the analysis that will also make a Bayesian approach feasible for complicated models. The major problem with the Bayesian methods in population viability analysis is that they are not based on the frequency interpretation of probability (Dennis 1996). Accordingly, one will calculate the *t*_{α} and use the Bayesian interpretation directly, claiming that the probability that the population goes extinct after the observed *t*_{α} conditioned on the data is actually 1 − α. The actual frequentistic probability, however, interpreting *t*_{α} as a stochastic variable, may be somewhat different. The process of correcting the coverage probability so that the coverage is approximately correct with respect to the frequentistic definition of probability would usually be outside the range of what a Bayesian statistician would accept. When using Bayesian methods in population viability analysis, one should be aware of this problem and make it a general rule that coverage probabilities, regardless of the chosen method, should be checked carefully by intensive stochastic simulations to reveal the frequentistic properties of the method.