Diversity is a concept central to ecology, and its measurement is essential for any study of ecosystem health. But summarizing this complex and multidimensional concept in a single measure is problematic. Dozens of mathematical indices have been proposed for this purpose, but these can provide contradictory results leading to misleading or incorrect conclusions about a community’s diversity. In this review, we summarize the key conceptual issues underlying the measurement of ecological diversity, survey the indices most commonly used in ecology, and discuss their relative suitability. We advocate for indices that: (i) satisfy key mathematical axioms; (ii) can be expressed as so-called effective numbers; (iii) can be extended to account for disparity between types; (iv) can be parameterized to obtain diversity profiles; and (v) for which an estimator (preferably unbiased) can be found so that the index is useful for practical applications.

Dietary diversity is an established public health principle, and its measurement is essential for studies of diet quality and food security. However, conventional between food group scores fail to capture the nutritional variability and ecosystem services delivered by dietary richness and dissimilarity within food groups, or the relative distribution (i.e., evenness or moderation) of e.g., species or varieties across whole diets. Summarizing food biodiversity in an all-encompassing index is problematic. Therefore, various diversity indices have been proposed in ecology, yet these require methodological adaption for integration in dietary assessments. In this narrative review, we summarize the key conceptual issues underlying the measurement of food biodiversity at an edible species level, assess the ecological diversity indices previously applied to food consumption and food supply data, discuss their relative suitability, and potential amendments for use in (quantitative) dietary intake studies. Ecological diversity indices are often used without justification through the lens of nutrition. To illustrate: (i) dietary species richness fails to account for the distribution of foods across the diet or their functional traits; (ii) evenness indices, such as the Gini-Simpson index, require widely accepted relative abundance units (e.g., kcal, g, cups) and evidence-based moderation weighting factors; and (iii) functional dissimilarity indices are constructed based on an arbitrary selection of distance measures, cutoff criteria, and number of phylogenetic, nutritional, and morphological traits. Disregard for these limitations can lead to counterintuitive results and ambiguous or incorrect conclusions about the food biodiversity within diets or food systems. To ensure comparability and robustness of future research, we advocate food biodiversity indices that: (i) satisfy key axioms; (ii) can be extended to account for disparity between edible species; and (iii) are used in combination, rather than in isolation.

In order to describe the sensitivity of a cellular automaton (CA) to a small change in its initial configuration, one can attempt to extend the notion of Lyapunov exponents as defined for continuous dynamical systems to a CA. So far, such attempts have been limited to a CA with two states. This poses a significant limitation on their applicability, as many CA-based models rely on three or more states. In this paper, we generalize the existing approach to an arbitrary $N$ -dimensional $k$ -state CA with either a deterministic or probabilistic update rule. Our proposed extension establishes a distinction between different kinds of defects that can propagate, as well as the direction in which they propagate. Furthermore, in order to arrive at a comprehensive insight into CA’s stability, we introduce additional concepts, such as the average Lyapunov exponent and the correlation coefficient of the difference pattern growth. We illustrate our approach for some interesting three-state and four-state rules, as well as a CA-based forest-fire model. In addition to making the existing methods generally applicable, our extension makes it possible to identify some behavioral features that allow us to distinguish a Class IV CA from a Class III CA (according to Wolfram’s classification), which has been proven to be difficult.