## Introduction

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.