Based on individual demographic data, the demographic variance cd can be estimated from data on individual variation among females in their fitness contributions to the following generations. The total contribution of a female i in year t (R ) is the 'number of female offspring born during the year that survive for at least 1 year' plus 1 if the female survives to the next year. The demographic variance can then be estimated as the weighted mean across years of

where R is the mean contribution of the females, a is the number of recorded contributions in year t, and E denotes the expectation. We can further partition a\ into components due to variation in fecundity, survival, and interaction between fecundity and survival. Writing B for the number of offspring produced, I = 1 if the mother survives and I = 0 if she dies, the demographic variance is the mean over years of the within-year variance of R = B +1. This can be split into its components var(R) = var(B) + var(I) + 2cov(B, I) that can be estimated separately by simple sum of squares. In long-lived species,

stochastic variation in age structure constitutes an important component of demographic stochasticity and for species with a mean age of maturity older than 3 years we must base our estimates on contributions (Bj, j) for the different age classes i in year t. We then calculate the demographic stochasticity from the projection matrix and separate this into components that are generated by demographic stochasticity in each vital rate.

It is obvious that such detailed demographic data are rarely available. However, in birds we can compute demographic variance of several species and relate the estimates to the position of the species along the 'slow-fast' continuum of life-history variation. According to one hypothesis, ad is expected to increase with adult survival rate (and hence to decrease with clutch size) because very few offspring recruit in short-lived species with a high first-year mortality. Alternatively, ad can be expected to decrease with adult survival rate because life-history constraints (small reproductive rates, high life expectancy) generate small variability in fitness among individuals in long-lived species.

In birds, the two components of the demographic variance due to stochastic variation in fecundity and survival were positively correlated. As expected from this relationship, interspecific differences in demographic variance were closely related to the size of both the fecundity component and the survival component. Interspecific differences in demographic stochasticity were well explained by life-history variation. Larger values of ad were found in species at the fast end of the avian life-history continuum, that is, in species with large clutch sizes (Figure 3a), short life expectancy (Figure 3b), early age at maturity, (Figure 3 c) and short generation times (Figure 3d). This supports the hypothesis that the level of demographic stochasticity in avian population dynamics is subject to life-history constraints on the possible range of variation in fecundity or survival, resulting in small values of ad in long-lived species with small reproductive rates.

A General Definition of Density Dependence of Age-Structured Populations

To analyze the effects of density dependence on the population dynamics, let us first consider a simplified life history in which individuals mature at age 1 year with adults having age-independent fecundity and survival. We can then write the dynamics N(t) = A[N(t- 1)]N(t- 1),

Figure 3 The mean value across bird species of the demographic (c) age at maturity, and (d) generation time T.

Age at maturity

Figure 3 The mean value across bird species of the demographic (c) age at maturity, and (d) generation time T.

where N(t) is the population size in year t and A[N(t- 1)] is the density-dependent finite rate of increase. The strength of density dependence 7 in such a model can be defined as the negative elasticity of the population growth rate A with respect to changes in population size N, evaluated at the carrying capacity K: 7 = - (0lnA/0lnN)K. This approach can be extended to an age-structured density-dependent life history in which the total density dependence in the life history, D, should be defined as the negative elasticity of the population growth rate per generation, AT, with respect to the change in the size of the adult population when fluctuating around the carrying capacity, so that

where T is the generation time. Thus, the annual rate of return to equilibrium then becomes 7 = D/T.

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