Diversity Partitioning Using Shannon's Entropy and its Relationship to Rarefaction

Abstract

Diversity (the variety of different types of organisms) of an ecological or paleoecological system reflects processes and history operating across a range of hierarchically related scales. For example, the diversity of a biofacies is the sum of the diversity in all the local patches composing the biofacies, the diversity of a depositional system is composed of all the biofacies composing the depositional system, and the diversity of a biotic province is composed of all the landscapes composing the province. Diversity at a larger scale (γ-diversity) incorporates both the average inventory diversity of units of the next smaller scale (α-diversity) and the compositional differences, or differentiation diversity, among the smaller units (β-diversity). Many familiar means of measuring diversity can be mathematically partitioned to determine the relative contribution of different diversity components at any hierarchical level. When using richness (the number of taxa in an ecological system) as a measurement of diversity, it is necessary to use rarefaction to correct for differences in sample size. The divergence between sample-based and individual-based rarefaction curves of a composite collection (γ-diversity) incorporating all the samples (α-diversity) contributing to a given hierarchical level reflects the degree of non-random compositional difference among the smaller scale units (β-diversity). Alternatively, Shannon's entropy can be partitioned additively: β-entropy equals γ-entropy (based on a composite sample) minus average α-entropy of the constituent samples. A useful property of entropy is that it can be converted to effective richness, the number of taxa that would result in the same entropy value if all were equally abundant. Effective richness can be thought of as a unit conversion from non-intuitive entropy units to more easily understood richness units. Effective richness derived from Shannon's entropy partitions diversity multiplicatively – i.e., β-diversity is the number of compositionally distinct smaller units that contribute to the total diversity at the higher level. Diversity partitioning is rapidly becoming adopted as a tool for directly addressing how the structure of higher-level ecological and paleoecological systems reflects interactions among lower-level units in response to environmental and evolutionary changes.