This general definition of density dependence (eqn [1]) facilitates comparison of population dynamics across species with different types of life history. Here the author illustrates this approach by comparing the stochastic density-dependent population dynamics of different bird species. We assume that the expected adult annual

variance ¿2 in relation to (a) clutch size, (b) adult survival rate s, survival and fecundity rates are independent of age, that density dependence is exerted by the adult fraction of the population on any combination of juvenile and adult vital rates, and, finally, that deviations of the adult population at time t from equilibrium x(t) = N(t) - K are expected to be small or moderate. Based on these assumptions, we obtain a linearized autoregressive model with time delays from 1 to a years:

where bj is the autoregressive coefficient for time lag i, u>(t) is a noise term with mean zero and variance o^, describing environmental stochasticity, including transient fluctuations in age structure and autocorrelations due to long-term fluctuations in the biotic or abiotic environment.

The autoregression coefficients bi for species with age at maturity a > 2 decreased with time lag (Figure 4), indicating that the effects of the previous years' population sizes on current population size decreased with time. These autoregression coefficients bi do not directly reveal the strength of delayed density dependence because they depend on life-history parameters as well as density dependence in the vital rates. In a species with a > 1 with no density dependence in subadult or adult survival rates, b1 equals the adult annual survival rate, s. Similarly, in a species that matures at 1 year, if b1 = 0 the population autocorrelations for all time lags will be zero, corresponding to a white noise process for the population size N(t) that indicates strong density dependence.

Again using the general definition of density dependence (eqn [1]), it is evident that there is a relationship between total density dependence in the life history and the autoregression coefficients a

Figure 4 Mean values (±SD) of the autoregression coefficients bi for different lags in the population dynamics of birds in relation to variation in age at maturity of birds. Circles represent species that mature at 1 year, triangles age at maturity at 2 years, squares age at maturity at 3 years, and reversed triangles species that mature at 4 years or older.

In our data set from birds, the density-dependent effects at annual scale were independent of life history, including generation time (Figure 5a) as well as clutch size (Figure 5b) and adult survival rate (Figure 5c). As a consequence, the stationary variance in the time series a2N was also independent of life history.

We have previously shown that many avian demographic traits such as clutch size and age at maturity scale closely with adult lifespan. Accordingly, we find that several features of population dynamics measured on a timescale of generations can be predicted from life-history characteristics. The strength of total density dependence in the life history D increased with generation time T (Figure 5d) and adult survival rate (Figure 5f) but decreased with clutch size (Figure 5e). This implies that the effect on the population growth rate per generation of a change in population size was larger for long-lived than for short-lived species. Consequently, the rate of return to equilibrium measured in generations decreases with generation time T (correlation coefficient of log10-transformed values = —0.73, P<0.001, n = 23).

To compare the residual variation in the population process, we must account for interspecific variation in age at maturity that will cause differences in the lag structure of the population dynamics. We first estimate the variance in the stationary distribution of the population sizes a2N in our model (eqn [2]) and then calculate the variance in the noise of a first-order process with a single time lag of 1 year, a", that will give the same stationary

Figure 4 Mean values (±SD) of the autoregression coefficients bi for different lags in the population dynamics of birds in relation to variation in age at maturity of birds. Circles represent species that mature at 1 year, triangles age at maturity at 2 years, squares age at maturity at 3 years, and reversed triangles species that mature at 4 years or older.

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