Nonrandom spatial patterns are ubiquitous in ecology, and provide important information on processes structuring communities, including dispersal, competition, density-dependence and habitat associations. However, multiple processes can contribute to spatial pattern formation, and similar spatial distributions can generally be formed by different mechanisms or different combinations of mechanisms. For example, aggregation at short distances can be found both in species that disperse short distances, and in species that disperse long distances but have strong habitat associations. ,
A first step in disentangling different processes involved in pattern formation is to define the scales at which the pattern occurs, allowing us to focus on mechanisms operating at these scales. This is best done in a framework in which the spatial variation is decomposed scale by scale, enabling a clear view of how different frequencies impact the process. Spectral analysis constitutes such an approach, and the wavelet variance provides a consistent estimator of spatial variance as a function of scale.
Results/Conclusions
We analytically derive wavelet variances for multiple types of spatially explicit, individual-based models incorporating seed dispersal and density-dependent competition. We use standard statistical properties of wavelets in combination with these analytical results to estimate process parameters from observed spatial patterns. We define a likelihood function that is consistent with the process underlying the model and makes it possible to estimate the most likely parameters and their uncertainty. We demonstrate these methods with numerical examples as well as case studies of four tropical tree species on Barro Colorado Island, Panama.
The numerical examples show that minimally biased estimates of the parameters and their uncertainty can be recovered using maximum likelihood methods. For large sample sizes, the likelihood approaches a multivariate normal distribution (MVN) with known covariance matrix. For small sample sizes the likelihood can still be approximated by a MVN, but the covariance matrix (and thus parameter uncertainties) can only be determined with stochastic simulations.
Application to a real case study of tropical species reveals that models were able to describe the basic features of the observed wavelet variance. The methods presented here constitute a powerful new tool for ecologists to investigate and derive information about spatial processes from static spatial patterns.