  Ecologists often interpret variation in the spatial distribution of
populations in terms of responses to environmental features, but disentangling
the effects of individual variables can be difficult if latent effects and
spatial and temporal correlations are not accounted for properly. Here, we use
hierarchical models based on a Poisson log-normal mixture to understand the
spatial variation in relative abundance (counts per standardized unit of
effort) of yellow perch, Perca flavescens, the most abundant fish species in
Lake Saint Pierre, Quebec, Canada. The mixture incorporates spatially varying
environmental covariates that represent local habitat characteristics, and
random temporal and spatial effects that capture the effects of unobserved
ecological processes. The sampling design covers the margins but not the
central region of the lake. We fit spatial generalized linear mixed models
based on three different prior covariance structures for the local latent
effects: a single Gaussian process (GP) over the lake, a GP over a circle, and
independent GP for each shore. The models allow for independence, isotropy, or
nonstationary spatial effects. Nonstationarity is dealt with using two
different approaches, geometric anisotropy, and the inclusion of covariates in
the correlation structure of the latent spatial process. The proposed
approaches for specification of spatial domain and choice of Gaussian process
priors may prove useful in other applications that involve spatial correlation
along an irregular contour or in discontinous spatial domains.
