bioRxiv preprint doi: https://doi.org/10.1101/188813; this version posted September 14, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. Running Head: Thermodynamic properties of biodiversity Biological communities as interacting compartments: thermodynamic properties and diversity indices Fernando Meloni,1 Gilberto M. Nakamura,1 and Alexandre Souto Martinez1 Universidade de S˜aoPaulo, Ribeir˜aoPreto 14040-901, Brazil (Dated: today) Diversity indices provide simple and powerful metrics for assessing biological communities. Based on entropy measures, the approach considers statistical and thermodynamic inferences to deduce ecological patterns. However, concerns exist regarding the accuracy of diversity indices. Because relative quantities depend on the sorting of organisms (e.g., guilds and species) and their interactions, field observations carry inherent imprecision, thus leading to misinterpretation. Here, we present a framework that is able to appropriately achieve the thermodynamic properties in ecological systems and ensure the inference power. We demonstrate that effective abundances rather than raw abundances provide a trustful estimator of probabilities, which is evaluated through massive tests. We use empirical and synthetic data to show the advantages and reliability of this new framework under a broad range of conditions. The tests demonstrate that the replication principle is always optimized by the new estimator. Compared to other methods, this approach is simpler and reduces the importance of schemes used for sorting organisms. We highlight the robustness and the valor of effective abundances for ecological contexts: i) to assess and monitor the biodiversity, ii) to define the best sorting of organisms according to maximum entropy principles, and iii) to link local to regional diversity (α-, β-, and γ-diversity). keywords: biodiversity assessment, entropy, extensive and additive properties, estimator of probabilities, interacting compartments, predictable patterns I. INTRODUCTION from well-supported concepts, interpretations, formula- tions, and inferences from information theory, physics, The assessment of biodiversity is a primary concern statistics, and thermodynamics, and they allow deduc- 13,15,16,36,74,78 among ecologists. They are interested in monitoring ing further relationships from data . Briefly, species and ecosystems to explain how climate, soil type the calculation of diversity indices considers the number and several other environmental features affect organ- of categories and the number of organisms (abundances) 50 isms and their organization in nature15,50. Beyond ba- as primary information . The W categories are defined sic knowledge, the motivation for this interest has in- by any criteria of interest (e.g., species, genera, behavior, creased over the past decades due to the increasing hu- guild, and so forth) and are used for sorting the organ- man impacts on climate and natural ecosystems that isms, while relative quantities provide a quantitative in- place species survival and ecosystem services at risk2,26. ference. This type of basic information can be obtained The best appraisal of patterns in biological communi- for a broad range of practical conditions and reflects the ties considers details of several taxa, such as their biol- applicability of diversity indices. Furthermore, diversity ogy, genetic variability, ecology, behavior, and so forth. indices provide reliable information about global patterns However, this level of information is rarely available for of biological organization even if information about or- practical contexts, and objective measures are considered ganisms is limited. along with external inferences to deduce the community Accordingly, once the W categories are determined a patterns54,55. Taxonomic/functional compositions, the priori, the abundance Ak considers the number of organ- shape of the curves describing species-abundance distri- isms in the k-th category and determines the probability PW PW butions (SAD), and the ecological indices consist of dis- pk = Ak/A, for A = k=1 Ak and k=1 pk = 1. These tinct approaches for measuring and comparing biological probabilities are used to calculate the classic Shannon communities50. Each approach addresses levels of infor- diversity index H 50 as follows: mation, and their use shows advantages and disadvan- W 41,55 X tages for distinct contexts . Among these metrics, H = − p ln p , (1) ecological indices deserve special attention, and we ex- k k k=1 plore them in this paper. Also referred to as diversity indices, ecological indices which is equivalent to the Shannon entropy 74. In prac- are deduced from the Boltzmann-Shannon-Gibbs en- tical contexts, the H values of biological communities tropy, and they are proposed to measure order-disorder in living under distinct influences are compared by similar biological communities. These indices enjoy advantages sampling effort. In such cases, the variation ∆H is inter- bioRxiv preprint doi: https://doi.org/10.1101/188813; this version posted September 14, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 2 preted as a metric that addresses the influence of a par- gether to elaborate constraints and produce distinct a ticular driver (e.g., climate changes or human impacts) priori hypotheses, and they are compared next in accor- on the ecological framework 13,50,53. dance with MaxEnt principles 13,34,35. The arrangement In addition to the H index, there are several other that better achieves the thermodynamic properties de- ecological indices that provide complementary aspects of fines the most plausible granular scheme. However, the the biological communities. The notation approach also brings important concerns. For instance, knowledge of tens, hundreds, or sometimes thousands of W 1 X distinct organisms is a prerequisite for accurate results aH = ln pa (2) 1 − a k using guilds. Beyond the necessity of specialists, this im- k=1 plies some subjectivity in selecting the traits and respec- unifies the three most important ecological indices13,15,36: tive weights (if weights are addressed). Finally, guilds are the species richness W (number of categories), the H considered for particular groups of organisms and under index (Shannon entropy), and the Ginni-Simpson index specific situations (e.g., tree species in forests, arthropods in soil, fishes in lakes, and so on), and they cannot be D = PW p2 , where the last infers the curve asymme- k=1 k readily inferred for wide proposes. Therefore, the opin- try of ranked relative quantities. For a = 0 and R > 2, ion and context may affect the evaluation of probabilities we have 0H = ln(W ), which is the harmonic mean; for 1 pk, the values assumed by the diversity indices, and the a → 1, H = H, which is the geometric mean; and subsequent conclusions. for a = 2, 2H = 1/D, which is the arithmetic mean. These cases are special cases of the R´enyi generalized Notwithstanding the importance of the assortment of entropy and are important because they predict useful organisms, we emphasize here that the granular scheme relationships. One expects to find correlations between is only one side of the problem for the accuracy for biodi- the abundances of organisms and the values assumed by versity assessment. We claim that ecologists should ad- distinct diversity indices, such as af(A) ≡ aH, which dress further sources of data variability before defining also implies that g(0H) ≡ 1H and so on 15,36. These the best assortment. Ecological systems admit complex correlations match the concept of the replication princi- spatial-temporal dynamics (many of which are poorly un- ple in ecological contexts13,15,38, and they are true only if derstood), which finally drives the presence and quanti- the additive and extensive thermodynamic properties are ties of organisms. Beyond environmental influences, such ensured13,34,78. Assuming this circumstance, the entropy dynamics derive from intricate food webs and interaction observed for small samples can be re-scaled to predict networks, which finally frame the biological organization the entropy of the entire system. This type of inference in the ecosystems. The consequence is a mutual and would be used, e.g., to explain how the biological diver- non-trivial dependence of relative quantities. Because sity is spatially partitioned (γ-, α- and β- diversities)12,38. ecological information is generally acquired through field Although H and other diversity indices provide an inter- observations and samplings, fluctuations in relative abun- 45,51 esting approach to assess and interpret biological commu- dances may accrue relevant “noise” in empirical data , nities, tests using empirical data often show a different leading to uncertain probabilities pk for diversity indices. reality, and the topic remains open 10,12,14,24. Consequently, the reproducibility of ecological experi- 7 The inconsistency between theoretical predictions and ments remains a topic of scientific concern . empirical results is justified by the lack of a precise def- Taking these arguments into consideration, we conjec- inition of appropriate classes for ecological systems34.A ture that insights taken from non-equilibrium systems common approach for sorting organisms considers taxo- 19,40,52,80 could be adapted to a static approach for