Most ecological theories that aim to explain coexistence in megadiverse communities employ a set of three rules to describe the stochastic geometry of biodiversity: (i) individuals exhibit intraspecific clustering; (ii) species abundances vary according to a log-normal distribution and (iii) the spatial arrangement between species is independent. The first two rules have received strong empirical support, but the third remains largely unexplored. To address this deficiency, we evaluated the independent species arrangement rule in a species-rich shrubland and its potential drivers, that is, the levels of species richness and intraspecific clustering exhibited by a given species at different scales, and the relative abundance of such species in the community. We found that interspecific associations were rare and that independence was positively related to species richness and intraspecific clustering, but negatively related to relative species abundances. Synthesis. Our results agree ...

Trade-offs are a fundamental concept in evolutionary biology because they are thought to explain much of nature’s biological diversity, from variation in life-histories to differences in metabolism. Despite the predicted importance of trade-offs, they are notoriously difficult to detect. Here we contribute to the existing rich theoretical literature on trade-offs by examining how the shape of the distribution of resources or metabolites acquired in an allocation pathway influences the strength of trade-offs between traits. We further explore how variation in resource distribution interacts with two aspects of pathway complexity (i.e., the number of branches and hierarchical structure) affects tradeoffs. We simulate variation in the shape of the distribution of a resource by sampling 106 individuals from a beta distribution with varying parameters to alter the resource shape. In a simple “Y-model” allocation of resources to two traits, any variation in a resource leads to slopes less than -1, with left skewed and symmetrical distributions leading to negative relationships between traits, and highly right skewed distributions associated with positive relationships between traits. Adding more branches further weakens negative and positive relationships between traits, and the hierarchical structure of pathways typically weakens relationships between traits, although in some contexts hierarchical complexity can strengthen positive relationships between traits. Our results further illuminate how variation in the acquisition and allocation of resources, and particularly the shape of a resource distribution and how it interacts with pathway complexity, makes it challenging to detect trade-offs. We offer several practical suggestions on how to detect trade-offs given these challenges. Methods Overview of Flux Simulations To study the strength and direction of trade-offs within a population, we developed a simulation of flux in a simple metabolic pathway, where a precursor metabolite emerging from node A may either be converted to metabolic products B1 or B2 (Fig. 1). This conception of a pathway is similar to De Jong and Van Noordwijk’s Y-model (Van Noordwijk & De Jong, 1986; De Jong & Van Noordwijk, 1992), but we used simulation instead of analytical statistical models to allow us to consider greater complexity in the distribution of variables and pathways. For a simple pathway (Fig. 1), the total flux Jtotal (i.e., the flux at node A, denoted as JA) for each individual (N = 106) was first sampled from a predetermined beta distribution as described below. The flux at node B1 (JB1) was then randomly sampled from this distribution with max = Jtotal = JA and min = 0. The flux at the remaining node, B2, was then simply the remaining flux (JB2 = JA - JB1). Simulations of more complex pathways followed the same basic approach as described above, with increased numbers of branches and hierarchical levels added to the pathway as described below under Question 2. The metabolic pathways were simulated using Python (v. 3.8.2) (Van Rossum & Drake Jr., 2009) where we could control the underlying distribution of metabolite allocation. The output flux at nodes B1 and B2 was plotted using R (v. 4.2.1) (Team, 2022) with the resulting trade-off visualized as a linear regression using the ggplot2 R package (v. 3.4.2) (Wickham, 2016). While we have conceptualized the pathway as the flux of metabolites, it could be thought of as any resource being allocated to different traits. Question 1: How does variation in resource distribution within a population affect the strength and direction of trade-offs? We first simulated the simplest scenario where all individuals had the same total flux Jtotal = 1, in which case the phenotypic trade-off is expected to be most easily detected. We then modified this initial scenario to explore how variation in the distribution of resource acquisition (Jtotal) affected the strength and direction of trade-offs. Specifically, the resource distribution was systematically varied by sampling n = 103 total flux levels from a beta distribution, which has two parameters alpha and beta that control the size and shape of the distribution (Miller & Miller, 1999). When alpha is large and beta is small, the distribution is left skewed, whereas for small alpha and large beta, the distribution is right skewed. Likewise, for alpha = beta, the curve is symmetrical and approximately normal when the parameters are sufficiently large (>2). We can thus systematically vary the underlying resource distribution of a population by iterating through values of alpha and beta from 0.5 to 5 (in increments of 0.5), which was done using the NumPy Python package (v. 1.19.1) (Harris et al., 2020). The resulting slope of each linear regression of the flux at B1 and B2 (i.e., the two branching nodes) was then calculated using the lm function in R and plotted as a contour map using the latticeExtra Rpackage (v. 0.6-30) (Sarkar, 2008). Question 2: How does the complexity of the pathway used to produce traits affect the strength and direction of trade-offs? Metabolic pathways are typically more complex than what is described above. Most pathways consist of multiple branch points and multiple hierarchical levels. To understand how complexity affects the ability to detect trade-offs when combined with variation in the distribution of total flux we systematically manipulated the number of branch points and hierarchical levels within pathways (Fig. 1). We first explored the effect of adding branches to the pathway from the same node, such that instead of only branching off to nodes B1 and B2, the pathway branched to nodes B1 through to Bn (Fig. 1B), where n is the total number of branches (maximum n = 10 branches). Flux at a node was calculated as previously described, and the remaining flux was evenly distributed amongst the remaining nodes (i.e., nodes B2 through to Bnwould each receive J2-n = (Jtotal - JB1)/(n - 1) flux). For each pathway, we simulated flux using a beta distribution of Jtotalwith alpha = 5, beta = 0.5 to simulate a left skewed distribution, alpha = beta = 5 to simulate a normal distribution, and with alpha = 0.5, beta = 5 to simulate a right skewed distribution, as well as the simplest case where all individuals have total flux Jtotal = 1. We next considered how adding hierarchical levels to a metabolic pathway affected trade-offs. We modified our initial pathway with node A branching to nodes B1 and B2, and then node B2 further branched to nodes C1 and C2 (Fig. 1C). To compute the flux at the two new nodes C1 and C2, we simply repeated the same calculation as before, but using the flux at node B2, JB2, as the total flux. That is, the flux at node C1 was obtained by randomly sampling from the distribution at B2 with max = JB and min = 0, and the flux at node C2 is the remaining flux (JC = JB2 - JC1). Much like in the previous scenario with multiple branch points, we used three beta distributions (with the same parameters as before) to represent left, normal, and right skewed resource distributions, as well as the simplest case where Jtotal = 1 for all individuals. Quantile Regressions We performed quantile regression to understand whether this approach could help to detect trade-offs. Quantile regression is a form of statistical analysis that fits a curve through upper or lower quantiles of the data to assess whether an independent variable potentially sets a lower or upper limit to a response variable (Cade et al., 1999). This type of analysis is particularly useful when it is thought that an independent variable places a constraint on a response variable, yet variation in the response variable is influenced by many additional factors that add “noise” to the data, making a simple bivariate relationship difficult to detect (Thomson et al., 1996). Quantile regression is an extension of ordinary least squares regression, which regresses the best fitting line through the 50th percentile of the data. In addition to performing ordinary least squares regression for each pairwise comparison between the four nodes (B1, B2, C1, C2), we performed a series of quantile regressions using the ggplot2 R package (v. 3.4.2), where only the qth quantile was used for the regression (q = 0.99 and 0.95 to 0.5 in increments of 0.05, see Fig. S1) (Cade et al., 1999).

Overview This dataset contains a re-analysis of data extracted from Figure 5 of Dukat et al. [1]. The original figure plots the number of months with drought conditions (organized by pentads), including the linear trend lines and regression parameters for the SPEI-1, SPI-1, SPEI-3, SPI-3, SPEI-6, and SPI-6 indices.Purpose The purpose of this re-analysis is to verify the linear trend lines and regression parameters reported in the original publication. This dataset serves as accompanying material for a post-publication review on PubPeer. Recalculations based on the plotted bar values indicate significant discrepancies; specifically, the underlying data diverges from both the visual trend lines and the published regression metrics (slope, intercept, and R²) for most indices.MethodsData ExtractionThe data points (bar heights representing the number of months) and the start/end points of the trend lines were digitally extracted from the high-resolution bitmap of Figure 5 in the original publication using the plot digitization software WebPlotDigitizer [2].Re-calculationa) Linear regression analysis was performed on the extracted bar data to calculate the actual slope, intercept, and coefficient of determination (R2). These parameters serve as reference. b) Slope and intercept of the visual trend lines were calculated with their start/end points. ComparisonThe regression statistics calculated from the extracted bar data were established as the reference. Both the parameters published in the figure text and the digitized parameters of the visual trend lines were benchmarked against this reference to identify discrepancies.Statistical Analysis To verify the trends, two statistical methods were applied to the extracted data:Mann-Kendall Test: A non-parametric test (mkt user defined function) used to detect monotonic trends. It assesses whether a consistent upward or downward trend exists without assuming a linear relationship or normal distribution.Linear Regression Significance: A parametric test (pvaluetrend user defined function) that calculates the two-tailed p-value for the linear regression slope. It utilizes a t-statistic derived from the slope and its standard error to determine if the observed linear trend is statistically significantly different from zero.File DescriptionsDukat_2022_Figure_05_v03.xlsx: The master summary table. It lists the original parameters reported by Dukat et al. ("Dukat regression"), the parameters derived from this re-analysis ("re-analysis regression"), and the calculated difference between the two. It covers indices: SPEI-1, SPI-1, SPEI-3, SPI-3, SPEI-6, and SPI-6.SPEI-1_SPI-1_pentad.csv (and similar): Raw bar heights representing the number of months extracted from the figure.*_trend.csv: Coordinates representing the start and end points of the trend lines as originally plotted.References [1] Dukat, P., Bednorz, E., Ziemblińska, K. et al. Trends in drought occurrence and severity at mid-latitude European stations (1951–2015) estimated using standardized precipitation (SPI) and precipitation and evapotranspiration (SPEI) indices. Meteorol Atmos Phys 134, 20 (2022).https://doi.org/10.1007/s00703-022-00858-w[2] Rohatgi, Ankit. WebPlotDigitizer. Version 5.2. https://automeris.io/ (accessed 2025-11-05).