The process model
We assume a Gompertz kernel for the underlying population dynamics. The Gompertz model has been widely used in modelling population and community dynamics (e.g. Saitoh, Stenseth & Bjonstad 1997; Jacobson et al. 2004; Dennis et al. 2006; Mutshinda, O’Hara & Woiwod 2009), and has the advantage of being linear on a logarithmic scale. The model includes intra- and interspecific interactions, as well as linear terms measuring the dynamical effects of climatic variables on the growth rates of the study populations, and is designed to accommodate species covariations in response to latent environmental factors.
More specifically, let denote the actual number of individuals of species i in the community in year t (S species in total), and let Ft and Tt designate respectively the averaged winter (December–February) rainfall (in mm) and winter temperature (in degrees Celsius) in year t, standardized as indicated above. The number of individuals of species i at time t in the community is described by
where and ki are the intrinsic growth rate and the natural logarithm of the carrying capacity of species i respectively; is the interaction coefficient quantifying the effect of species j on the growth of species i (interspecific interaction), with all coefficients of intraspecific interactions, , set to 1 (Loreau & de Mazancourt 2008; Mutshinda, O’Hara & Woiwod 2009); βi,1 and βi,2 quantify the effects of winter rainfall and winter temperature on the growth rate of species i, respectively. The random shocks, , representing the variability resulting from demographic stochasticity and un-modelled (latent) environmental factors are assumed to be serially independent and normally distributed with mean zero, but are allowed to covary across species at a specific time as discussed below. The normality assumption allows us to separately model the mean and covariance structures (Ripa & Ives 2003; Mutshinda, O’Hara & Woiwod 2009). On the natural logarithmic scale, equation 1 becomes
where denotes the natural logarithm of . Equation 2 can be compactly written in matrix form as
where is the S-dimensional vector of log-transformed abundances of the S species at time t, R is a S-by-S diagonal matrix with , and 1S is the S-dimensional vector with all elements equal to 1; , , B is a n × 2 matrix with (βi1, βi2) as ith row, and is the vector of process disturbances affecting the community dynamics at time t, with one element by species. The serially independent vectors are assumed to be multivariate normally distributed around the zero-vector, with a covariance matrix denoted by i.e. . The covariance matrix is further decomposed into its environmental and demographic components as
The covariance matrix C represents the variability not explained by intrinsic dynamics or by the included environmental covariates, including the effect of interactions with un-modelled species at the same trophic level as well as species at other trophic levels. The matrix C is henceforth referred to as the environmental covariance matrix. Species are also allowed to covary in their response to latent (un-modelled) environmental factors by assuming that the elements, , ;]>on the main diagonal of C and the off-diagonal elements, Ci,j (i ≠ j), represent species-specific and joint responses to latent environmental factors, respectively. , where denotes the (population-level) demographic variance affecting the dynamics of species i from time t−1 to t, which is scaled inversely with the population size (Saether et al. 2000; Bjørnstad & Grinfell 2001; Lande, Engen & Saether 2003). It is in fact the dependence of demographic variance on the population size that makes the demographic and environmental components of the process variance involved in equation 4 statistically identifiable.
Following Saether et al. (2000), the total environmental variance, Ei, affecting the dynamics of species i can be split into a component, , attributable to the included environmental variables, and a residual environmental variance, Ci,i, quantifying the variability not accounted for by the included variables. That is,
which is simply with standardized covariates. Moreover, if and the effects of the two covariates on the dynamics of species i and j turn out to be of identical signs, then the proportion of environmental covariance between species i and j that is explained by the covariates is given by
We used the Bayesian variable selection method known as stochastic search variable selection (SSVS) (George & McCulloch 1993; Mutshinda, O’Hara & Woiwod 2009) to constrain the coefficients of spurious inter-species interactions to be close to zero so that they do not affect the model results. The rationale of SSVS is to embed a multiple regression set-up in a hierarchical normal mixture model, and use latent indicators to identify promising sets of predictors. For each coefficient of interspecific interaction, αi,j (i ≠ j), we introduced an auxiliary indicator , ;]>0 < pi,j < 1, such that γi,j = 1 when species j is included in the dynamics of species i, and γi,j = 0 otherwise. Conditionally on γi,j, we defined the prior distribution of αi,j as a mixture of two Gaussians i.e. . The positive constant c1 was selected to be small and c2 to be large. This prior specification constrains αi,j to be concentrated around zero when γi,j = 0 since the ensuing prior corresponds to the spike part of the Gaussian mixture prior placed on αi,j, which is confined around zero. On the other hand, αi,j is freely estimated from the data when γi,j = 1 since the corresponding prior, the slab part of the Gaussian mixture priors placed on αi,j, is diffuse (flat).
A Gibbs sampling methodology is used to generate samples from the joint posterior of all unknowns, including the inclusion indicators γi,j. The relevance of a single interaction effect, αi,j, is evaluated through the Bayes factor , which quantifies the amount by which the prior odds of including vs. not including αi,j into the model are changed into posterior odds by the data. If Bi,j is larger than 1, we say that the data provide more support in favour of including αi,j into the model than assumed a priori, and vice-versa. Bayes factors for comparing two hypotheses (or models) H1 and H2 are usually interpreted on the following scale due to Jeffreys (1961). B1,2 < 1: ‘Negative support for H1 (i.e. support for H2)’; 1 ≤ B1,2 < 3: ‘Barely worth mentioning evidence in favour of H1’; 3 ≤ B1,2 < 10: ‘Substantial support for H1’; 10 ≤ B1,2 < 100: ‘Strong support for H1’; B1,2 > 100: ‘Decisive support for H1’. For us here H1 and H2 represent the inclusion and exclusion of individual interaction coefficients into the model, respectively.
The observation model
We took advantage of the replicated feature of our data (time series from two light-traps: Geescroft I and Geescroft II on the same site) to explicitly accommodate potential discrepancies in capture efficiency across traps for different species. Our observation model was also specified with Gaussian errors.
More specifically, let Yi,t,k denote the observed number of individuals of species i at time t from trap k, and let . We assume that
where the random variable , intended to correct for differences in capture efficiency between species across traps, is set to zero for one of the traps (Geescroft I) to force identifiability. So we only estimate bi,2, and consider negative values of it as implying lower capture efficiency for Geescroft II compared to Geescroft I and vice-versa.