Biomass–Likelihood Relationship in Ecosystems

• The biomass–likelihood relationship is defined as the statistical link between total biomass and the probability of observing a system state, serving as a key predictor in ecological modeling.
• In complex ecosystem models like the GLV system, higher biomass sharply increases the likelihood of stable attractor states by reducing effective self-inhibition.
• This relationship informs applications from exoplanet biosignature detection to fisheries stock assessments by connecting biomass to observational data and model predictions.

The biomass–likelihood relationship characterizes the statistical connection between the total biomass of a state in an ecological, biogeochemical, or resource system and the likelihood (probability) of observing, detecting, or inferring that state, given model dynamics and observation error structures. Across disciplines including complex ecosystems theory, exoplanet biosignature modeling, forest inventory estimation, and fisheries stock assessment, the relationship encodes how system-level or community-level biomass summarizations—often functioning as sufficiency statistics—govern either the prevalence of particular ecosystem states or the detectability of biogenic signals.

1. Biomass as a Macroscopic Organizing Principle in Dynamic Ecosystems

In the context of complex ecosystems modeled by the random Generalized Lotka–Volterra (GLV) equations, total biomass $B$ of a state $\{N_i^*\}$ is defined as

$B = \sum_{i=1}^S N_i^*$

with $N_i^*$ the steady-state abundance of each surviving species ($N_i^*\approx 0$ treated as extinct). Patro, Taylor & Goyal (Patro et al., 10 Nov 2025) demonstrate that in multistable GLV systems, a state's likelihood—that is, the fraction of random initial conditions converging to a given stable attractor—depends almost exclusively and sharply on $B$, rising orders of magnitude with increasing biomass.

High-biomass states correspond to low effective self-inhibition; at steady state,

$$B^*_i = 1/A_{ii} \qquad\text{or generally}\qquad A_{\text{eff}}^\alpha = 1/B_\alpha$$

where $A_{ii}$ is the self-inhibition coefficient for species $i$ and $A_{\text{eff}}^\alpha$ for state $\alpha$. This inverse law underpins the biomass–likelihood link: states with high $B$ (low $A_{\text{eff}}$) grow faster, outcompete alternatives, and thus dominate the landscape of possible system outcomes.

2. Probability of State Occurrence in Multistable Systems

The probability $p_i$ that a random initialization yields state $i$ rises steeply with biomass $B_i$. Empirical and analytic forms quantify this, typically as

$\ln p_i \approx -\frac{C}{B^* - B_i}$

for $B_i < B^*$, with constants $B^*, C$. Explicit formulae in monodominant and block-structured communities (see Eq. (9–11) Supplement, (Patro et al., 10 Nov 2025)) connect $p_i$ directly to $B_i$ via the effective self-inhibition: $$p_i = \sum_{k=1}^i \frac{\chi_k^{S-k+1} - \chi_{k-1}^{S-k+1}}{(S-k+1)\prod_{j=k}^S \chi_j}$$ where $\chi_i = D - A_{ii}$ and $B_i = 1/A_{ii}$.

In block-structured GLV systems, this generalizes to an integral over Gaussian-distributed initial abundances: $$p_\alpha = \int_{0}^\infty \frac{1}{s_\alpha} \,\varphi\Big(\frac{z - m_\alpha}{s_\alpha}\Big) \prod_{\beta \ne \alpha} \Phi\Big(\frac{z - m_\beta}{s_\beta}\Big)\,dz$$ with block-specific means and variances and $\chi_\alpha = D - 1/B_\alpha$. Across diverse $(\mu, \sigma)$ regimes, the biomass dependence robustly predicts the prevalence of states except for extremely rare attractors ($p \lesssim 10^{-5}$) where structure overlaps invalidate mean-field assumptions.

3. Coarse-Graining: Biomass and Diversity as Sufficient Statistics

The GLV studies (Patro et al., 10 Nov 2025) reveal that the full structural detail of interaction matrices can be coarse-grained: every stable state behaves as a "block" characterized by macroscopic biomass $B_\alpha$ and richness $L_\alpha$. The probability landscape $p_\alpha(B, L)$ is recoverable without knowledge of microscopic $A_{ij}$ values. In this model,

• Inter-block interactions are approximated by global means $\mu$
• Each block’s self-inhibition is set by $A_\text{eff}^\alpha = 1/B_\alpha$
• Initial abundance sums are Gaussian for $L_\alpha \gtrsim 5$ (Irwin–Hall)

The result is a powerful macroscopic organizational principle for community assembly and regime prediction, providing tractable analytic predictions for states’ likelihoods.

4. Biomass–Likelihood in Biosignature and Resource Detection

The biomass–likelihood concept generalizes to resource quantification and biosignature detection:

a) Biosignature Gas Detectability (Exoplanetary Atmospheres)

In exoplanet photochemical–climate models (Gebauer et al., 2021, Seager et al., 2013), biological surface fluxes (“biomass emissions”) of key gases directly set lower-atmospheric mixing ratios and, by steady-state loss–production equations, govern the depth of transmission spectral features. Detectability metrics (e.g., effective transmittance height $H_\text{eff}$ and required number of transit observations) scale with the biomass-dependent source flux. Table A (Gebauer et al., 2021) summarizes the collapse of spectral signatures as biomass is reduced, with detectable methane, N$_2$O, and CH$_3$Cl bands decaying by several kilometers of $H_\text{eff}$ from “modern Earth” to “dead Earth.”

b) Forest Biomass Estimation (Spatial Bayesian Models)

Bayesian spatial factor models (Doser et al., 10 Mar 2025) employ plot-level biomass observations linked to species presence, climate, and latent spatial factors. Here, plot or area-level biomass ($y_j(\mathbf{s}_i)$) is the target parameter, with likelihood for each species constructed from binary occupancy and a log-normal distribution on positive presence. Posterior sample propagation of biomass directly quantifies the probability of observed or predicted states; estimation precision is vastly improved over design-based or non-parametric estimators.

c) Microbial Biomass Yield Inference

Probabilistic macrochemical modeling (Paiva et al., 2020) for microbial systems relates observed metabolite and cell count changes to biomass yield $Y$ via stoichiometric and statistical models. The likelihood for $Y$ and cell weight $w$ reflects experimental noise and prior uncertainty; credible intervals on $Y$ embody the probability of biomass production conditional on experimental data.

5. Observational Likelihood Choice in Stock Assessment and Biomass Estimation

Observational error models for stock assessment (Albertsen et al., 2016) demonstrate that modeling choices for the distributional form of catch or survey data—including log-normal, gamma, multivariate log-normal, logistic-normal, Dirichlet, and Student’s t—substantially affect estimates of spawning stock biomass (SSB). The mean or median parameterizations inherent to each likelihood family propagate to expected or inferred SSB values:

• Log-normal equates the median of catch to the Baranov catch prediction, while gamma equates the mean
• Heavier-tailed models afford smoother trajectories but broader credible intervals
• Multivariate/correlated likelihoods yield tighter confidence bounds, especially at terminal years

Empirical studies demonstrate that shifting the likelihood can move SSB and average fishing mortality ($\overline{F}$) estimates by 10–23%. Since biomass enters the likelihood as the central parameter linking model outputs to observed data, the structure of the observational likelihood function is pivotal for biomass inference and subsequently for management decisions.

6. Limitations, False-Positive Risks, and Model Assumptions

Across all contexts, the biomass–likelihood relationship is subject to mechanistic, statistical, and epistemic constraints:

• In GLV models, approximations hinge on mean-field replacement of complex interactions and Gaussian initial-condition assumptions; accuracy degrades for rare, overlapping states (Patro et al., 10 Nov 2025).
• Biosignature models rely on Earth-like maintenance energy and photochemical regimes; uncertainties in Type I (energy-extracting) biosignature gases include high false-positive risk, while Type III (specialized byproducts) are constrained by low production rates and plausibility of biomass investment (Seager et al., 2013).
• Small-area estimation frameworks presuppose hierarchical spatial autocorrelation and hurdle-type mixture distributions; misspecification can propagate estimation bias or inflate uncertainty.
• In stock assessment, likelihood misspecification can invert management advice due to shifts in the estimated total biomass.

A plausible implication is that objective, discipline-specific evaluation and selection of likelihood models—paired with explicit consideration of biomass-driven dynamics—are essential for robust inference and policy guidance.

7. Synthesis and Cross-Disciplinary Significance

The biomass–likelihood relationship functions as a unifying macroscopic summary across domains where community structure, resource detectability, or the probability of ecological states is central. In high-dimensional, multistable systems, total biomass near-exclusively ranks the attractors by probability. In planetary and biosignature science, source fluxes set both expected atmospheric compositions and observability. In resource statistics, probabilistic and observational error models require careful calibration to ensure biomass is accurately and precisely inferred.

Patro, Taylor & Goyal (Patro et al., 10 Nov 2025) formalize this organizing principle for complex ecosystems, demonstrating that microscopic interaction matrices can be abstracted to two state-level statistics—biomass and diversity—that suffice to nearly predict the entire attractor landscape. This reaffirms biomass as both a dynamical and inferential sufficiency statistic in modern ecological theory and resource science.