A continuum of information-based temporal stability measures and their decomposition across hierarchical levels

When biodiversity-stability relationships are assessed from temporal patterns of species (or species assemblages) biomass or other key variables, ecological stability is commonly quantified as the inverse of the coefficient of variation (CV) or its square. However, just as biodiversity cannot be fully characterized by a single value, the complexity of temporal stability/invariability cannot be completely captured by one metric, especially since CV is disproportionally sensitive to large data values. Ecologists now recognize that species diversity can be fully characterized by a continuum of Hill-number-based measures parametrized by a diversity order q[≥]0, which determines the sensitivity of the measure to species abundance. Building on the intuitive concept that temporal stability can be quantified by the closeness between the data vector and the ideally maximally stable vector, we propose a continuum of information-based measures of temporal stability/invariability parameterized by an order q>0. This continuous parameter q determines the sensitivity of the measure to the magnitude of biomass or other ecosystem functions. By varying q, researchers can differentially weight small, medium, or large values in a time series, thereby disentangling their respective contributions to stability. Our framework unifies and generalizes classical measures: the case q=1 links to Shannon entropy, reflecting MacArthurs 1955 stability concept; q=2 connects to the conventional CV-based measure. Unlike these traditional metrics, our approach explicitly accounts for the number of data values (i.e., time points), adjusting for time-series-length effects to enable fair and meaningful comparisons across datasets of varying lengths. We extend the framework to hierarchical structures by developing additive and multiplicative decompositions of stability in a metacommunity (or metapopulation) into alpha and beta components. The beta component can be further used to obtain measures that quantify (a)synchrony among communities (or populations). For q=2, the resulting (a)synchrony measure provides a mathematically rigorous, CV-based metric. The proposed measures are illustrated using 22-year biomass time series data from the Jena Experiment in Germany. We developed the R package iSTAY (information-based stability measures) and online tools for computation and visualization. Our measures are adaptable to other functions beyond biomass and are applicable in both temporal and spatial contexts. Open Research StatementThe original biomass data of the Jena Experiment, covering the years 2003 to 2023, can be retrieved from https://jexis.idiv.de/ddm/Data/ShowData/624, while the 2024 data are available at https://jexis.idiv.de/ddm/Data/ShowData/695. These datasets, along with the R code used in this study, are currently accessible on Github at https://github.com/AnneChao/MS_iSTAY for review purposes and will be archived on Zenodo upon journal acceptance.