Understandable Multifunctionality Measures Using Hill Numbers

Abstract In ecology, multifunctionality measures the simultaneous performance of multiple ecosystem functions. If species diversity describes the variety of species that together build the ecosystem, multifunctionality attempts to describe the variety of functions these species perform. A range of methods have been proposed to quantify multifunctionality, successively attempting to alleviate problems that have been identified with the previous methods. This has led to a proliferation of more-or-less closely related metrics which, however, lack an overarching theoretical framework. Here we borrow from the comprehensive framework of species diversity to derive a new metric of multifunctionality. Analogously to the effective number of species used to quantify species diversity, the metric we propose is influenced both by the number of functions as well as, crucially, the evenness of performance levels across functions. In addition, the effective multifunctionality also considers the average level at which the functions are performed. The result is a measure of the cumulative performance of the system were all functions provided equally. The framework allows for the inclusion of the correlation structure among functions, thus allowing it to account for non-independence between functions. We show that the average metric is a special case of the newly proposed metric when all functions are uncorrelated and performed at equal levels. We hope that by providing a new metric of multifunctionality anchored in the rigorous framework of species diversity based on effective numbers, we will overcome the considerable skepticism that the larger community of ecologists has built against indices of multifunctionality. We thereby hope to help popularize this important concept which, as biological diversity, describes a fundamental property of ecosystems and thus lies at the heart of ecology.