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Abstract A fundamental pattern in ecology is that smaller organisms are more abundant than larger organisms. This pattern is known as the individual size distribution (ISD), which is the frequency distribution of all individual body sizes in an ecosystem.The ISD is described by a power law and a major goal of size spectra analyses is to estimate the exponent of the power law,λ. However, while numerous methods have been developed to do this, they have focused almost exclusively on estimatingλfrom single samples.Here, we develop an extension of the truncated Pareto distribution within the probabilistic modelling language Stan. We use it to estimate multipleλs simultaneously in a hierarchical modelling approach.The most important result is the ability to examine hypotheses related to size spectra, including the assessment of fixed and random effects, within a single Bayesian generalized mixed model. While the example here uses size spectra, the technique can also be generalized to any data that follow a power law distribution. 

Abstract Aquatic invasive species (AIS) threaten biodiversity and ecosystem services around the world, but their management has been hampered by the lack of quantifiable control targets. The introduction of Silver Carp (Hypophthalmichthys molitrix) throughout the mid‐western United States epitomizes both the impacts of AIS and the need for quantitative control targets. Silver Carp are large‐bodied planktivores that compete with native planktivores, which can cause cascading effects throughout the food web. Our study tested the threshold of abundance beyond which Silver Carp alter fish assemblage structure. We used a community size spectra (CSS) approach to evaluate fish community size structure across temporal and spatial gradients of Silver Carp abundances. We hypothesized that Silver Carp would flatten the size spectra slope because they are large‐bodied and feed at a low trophic position. Electrofishing data were obtained for the La Grange Pool of the Illinois River (1994–2021) and for six pools of the Ohio River (2015–2020). Results supported our hypothesis, showing a 98% probability that the relative biomass of Silver Carp is positively related to the CSS slope (resulting in “flattening”). This pattern was strongest in the Illinois River, where Silver Carp made up >30% of fish assemblage biomass in recent years. The pattern was weakest in the Ohio River (78% probability of a positive relationship) where Silver Carp rarely exceeded 20% of total fish biomass. Subsequent changepoint models indicated that a Silver Carp relative biomass of ~24% represents a threshold below which negative food web impacts should be minimized. Our study demonstrates a clear shift in fish community size structure following invasion by Silver Carp and suggests that pre‐invasion CSS slopes may serve as a restoration target. It also illustrates the benefits of CSS to guide Silver Carp and other AIS management. 

The study explores the individual size distribution (ISD) pattern in ecological communities, characterized by a negative correlation between individual body size and abundance (N ∼ M^λ). The parameter λ denotes the rate of decline in relative abundance from small to large individuals. Despite known influences of temperature and resource availability on body size, their effects on λ remain diverse. Leveraging data from 2.4 million individual body sizes in continental freshwater streams, the research the hypothesis that λ varies as a function of temperature and resource supply. Surprisingly, despite varied environmental conditions and complete species turnover, minimal variation in λ (mean = −1.2, sd = 0.04) was observed, with no discernible impact from temperature or resource supply. The unexpected λ value of −1.2 suggests a higher-than-expected relative abundance of large individuals, challenging assumptions of metabolic scaling at 0.75 and implying large subsidy inputs to large predators. Simulation and mesocosm experiments support a metabolic scaling coefficient of ∼0.4 for freshwater macroinvertebrates. The findings underscore remarkable consistency of individual size distributions in freshwater streams, likely driven by shallow metabolic scaling and large subsidies to large consumers.