single location analysis
Writing Nt for the population size at a given location, the dynamics are modelled by:
- (eqn 1 )
where rt is the growth rate varying stochastically between years and g(·) is an increasing function defining the local density regulation. For population fluctuations driven by demographic and environmental stochasticity we have where are the environmental and demographic variance, respectively (May 1973, 1974; Turelli 1977; Engen et al. 1998; Lande et al. 2003). The demographic variance can be estimated by a simple sum of squares from individual data on survival and reproduction (Sæther et al. 1998; Lande et al. 2003). When such data are available, one will usually have data from a large number of individuals, giving accurate estimates of the demographic variance (Sæther et al. 1998). In the following we therefore consider the demographic variance to be known. We shall use the logistic form of density regulation, g(N) =βN2. Then, for small and moderate population fluctuations a simple first-order approximation gives the model:
- (eqn 2 )
The model may alternatively be written
- (eqn 3 )
where Xt= ln Nt while Ud and Ue are independent variables with zero mean and unit variance. Our main goal is to investigate the environmental noise terms Ueσe at different locations, including the population synchrony it generates. The temporal and spatial stochastic fluctuations in this term may be generated partly by other quantities such as, for example, the local abundance of other species, the local temperature and the NAO index. We introduce such quantities generally as random effects writing:
- (eqn 4 )
where yi,t is the above random covariate number i at time t, U is another normalized variable and σ2 is the component of the environmental variance that can not be explained by fluctuations in the covariates. This leads to the relation so that the covariates together explain a fraction:
- (eqn 5 )
of the total environmental variance. This decomposition of the environmental variance is also valid if the covariates are subject to noise with temporal autocorrelation. In this case the total environmental noise term Ue may also have a temporal autocorrelation. However, this does not affect our approach, which only assumes that the residual noise term obtained after correcting for the effect of the covariates is generated by a white noise process.
Assuming that Xt+1 = lnNt+1 is normally distributed when conditioned on Nt, and writing f(x; µ, σ2) for the normal probability distribution with mean µ and variance σ2, the log likelihood function takes the form:
- (eqn 6 )
where Yt denotes the vector of covariates, v(N) = σ2 +/N, and the mean is the appropriate modification of eqn 2:
- (eqn 7 )
The sum in eqn 6 is taken over those years for which the population size in the previous year is known. This means that if there are time gaps in the data of more than one year, we ignore the information contained in the population change over this gap, which in any case will be small.
Now, assuming known demographic variance, all other unknown parameters are estimated by numerical maximization of log likelihood, and uncertainties are evaluated by parametric bootstrapping involving simulating the time-series using the initial population size and the estimated parameters.
The spatial analysis will be based on studying the residuals obtained from fitting the model to time-series observations at each location z,
- (eqn 8 )
where Ê denotes the estimated expected value. We use the normal approximation and choose a parametric form for the spatial autocorrelation of the U, of the form:
- (eqn 9 )
where h(z) decreases from 1 to 0 as z increases from 0 to infinity. We use the exponential function h(z) = e–z/l and the Gaussian form which both correspond to positive definite autocorrelation functions. Using the same type of spatial autocorrelation function for the population sizes, Lande et al. (1999) and Engen et al. (2002a,b) applied the standard deviation of the function h(z) scaled to become a distribution as a measure of spatial scaling of population synchrony. Hence, the parameter l in the above functions is actually this measure of spatial scaling defined for the residuals.
Data from different locations often are available over different but partly overlapping time periods. However, for each year we do have a set of residuals with zero means, estimated standard errors, and correlations defined by eqns 8 and 9. For locations z1 and z2 the correlation is:
- (eqn 10 )
For a given set of parameters (ρ0, ρ∞, l) defining the spatial synchrony, we have a complete description of a multivariate normal distribution each year, but possibly with different sets of locations at different years. The complete likelihood function is found by multiplying together the functions for each year or actually adding the log likelihood contributions for each year. Finally, this is maximized numerically to give estimates for (ρ0, ρ∞, l). Generally, the more the time-series overlap, the more information the likelihood function contains about the correlation parameters (ρ0, ρ∞, l). In order to obtain any information at all, we must require that at least two of the series are observed at least at one common time step. Otherwise the correlation parameters will be absent from the likelihood function and hence cannot be estimated.
The sampling properties of the estimates are found by parametric bootstrapping (Efron & Tibshirani 1993), that is, by simulating the whole set of local time-series, using the initial values of the data and the estimated parameters. The residuals must be simulated from the appropriate multinormal model defined by the autocorrelation function and the distance matrix.
The multinormal likelihood function can easily be calculated numerically using a lower triangular linear transformation, the Choleski decomposition (Ripley 1987). The same representation also gives a fast method for stochastic simulations required to perform the bootstrapping. This is explained in detail in Appendix I.
- (eqn 11 )
which turns out to be close to + M for γ < 1·15. Expressed by the mean yearly migration distance d̄, this is approximately + 0·64d̄2.