On the role of spatial stochastic models in understanding landscape indices in ecology

Spatial stochastic models play an important role in understanding and predicting the behaviour of complex systems. Such models may be implemented with explicit knowledge of only a limited number of parameters relating to spatial relationships among locations. Consequently, they are often used instead of deterministic-mechanistic models, which may potentially require an unrealistically large number of parameters. Currently, in contrast to spatial stochastic models, the parameterization of the joint spatial distribution of objects in landscape models is more often implicit than explicit. Here, we investigate the similarities and differences between bona fide spatial stochastic models and landscape models by focusing mostly on the relationships between processes, their realizations (patterns), representation and measurement, and their use in exploratory as well as confirmatory data analysis. One of the most important outcomes of recognizing the importance of stochastic processes is the acknowledgement that the spatial pattern observed in a landscape is only one realization of that process. Hence, while ecologists have been using landscape pattern indices (LPIs) to characterize landscape heterogeneity and/or make inferences about processes shaping the landscape, no stochastic modelling framework has been developed for their proper statistical elucidation. Consequently, several (mis)uses of LPIs draw conclusions about landscapes which are suspect. We show that several reports about sensitivities of LPIs to measurements have common roots that can be made explicitly manageable by adopting stochastic models of spatial structure. The key parameters of these stochastic models are composition and configuration, which, in general, cannot be estimated independently from each other. We outline how to develop the stochastic framework to interpret observations and make some recommendations to practitioners about everyday usage. The conceptual linkages between patterns and processes are particularly important in light of recent efforts to bridge the static-structural and the dynamic-analytic traditions of ecology.