Copulas reveal complex and informative dependencies propagating throughout ecology

All branches of ecology study relationships among environmental and biological variables. However, ubiquitously used approaches to studying such relationships, based on correlation and regression, provide only a small slice of the complex information contained in the relationships. Other statistical approaches exist that provide a complete description of relationships between variables, based on the concept of the copula; they are applied in finance, neuroscience and other fields, but not in ecology. We here explore the concepts that underpin copulas and examine the potential for those concepts to improve our understanding of ecology. We find that informative copula structure in dependencies between variables is exceedingly common across all the environmental, species-trait, phenological, population, community, and ecosystem functioning datasets we considered. Many datasets exhibited asymmetric tail dependence, whereby two variables are more strongly related in their left compared to right tails, or vice versa. We describe mechanisms by which observed copula structure and tail dependence can arise in ecological data, including a Moran-like effect whereby dependence structures between environmental variables are inherited by ecological variables; and asymmetric or nonlinear influences of environments on ecological variables, such as under Liebig9s law of the minimum. Copula structure and tail dependence can also reveal causal relationships between variables. We describe consequences of copula structure, including impacts on extinction risk and the stability through time of ecosystem services. By documenting the importance of a complete description of dependence, advancing conceptual frameworks, and demonstrating a powerful approach, we aim to elevate copula reasoning to widespread use in ecology.