Data and limitations.

The data used in this study are from the United States Geological Survey (USGS) datasets, Rainbow Trout Growth Data and Growth Covariate Data from Glen Canyon, Colorado River, Arizona (2012–2021) [21]. These freely-available datasets include ten years of rainbow trout growth measurements from release and recapture surveys in the lower Colorado River basin, along with seven environmental covariates known to influence growth rates. We have also posted all data and code online for replication.

The dataset originates from a tag-recapture program in which rainbow trout were physically tagged upon initial capture. However, a limitation is the absence of consistent individual identifiers across capture events. While tagging was conducted, the dataset does not include reliable codes linking multiple observations of the same fish. As a result, some individuals may appear in the dataset more than once without a definitive way to track their recapture history. Given this constraint, we treated each capture event as an independent observation with its associated covariates (e.g., age, environmental factors, location). This decision avoids imposing assumptions about individual identity but introduces a potential for pseudo-replication if some fish are included multiple times. Because we cannot distinguish repeat captures from new individuals, within-individual temporal dependence was not explicitly modeled. Future studies could improve upon this by ensuring traceable tag identifiers and applying longitudinal or hierarchical models to fully leverage the repeated-measures structure when available.

To accommodate the temporal structure of the data in a biologically meaningful way, we modeled the time between release and recapture events using each fish’s specific time-at-large, the duration (in days) between capture events. Rather than using static calendar timepoints, we leveraged interval-based alignment of biometric and environmental data. Environmental variables were recorded as monthly means between survey trips, while fish data included both release and recapture dates. We applied two integration protocols: (1) For fish recaptured within a single monthly interval, the corresponding monthly mean values were assigned to the observation. (2) For fish spanning multiple months, we calculated time-weighted averages of environmental variables over the entire time-at-large period. This approach ensured that covariates reflected the fish’s actual environmental exposure, preserving both ecological fidelity and temporal accuracy in model development.

Although the lack of individual IDs precluded the use of formal mixed-effects or autocorrelation models, our design mitigates these concerns by encoding relevant temporal dynamics through biologically informed durations and matched covariates. The final dataset included 9,798 observations of catch and release with complete environmental integration. No observations were excluded. Many of the algorithms we employed (Random Forest, XGBoost, Neural Networks) are robust to mild violations of independence assumptions and can accommodate the potential pseudo-replication without strong parametric assumptions. Bayesian methods provide a principled framework for explicitly modeling interdependence through joint probability structures, such as hierarchical models, spatial processes, and temporal autocorrelation structures.

Variables (features).

Each variable in the merged dataset was selected based on its biological relevance to rainbow trout growth and habitat dynamics. The response variable, or fork length at recapture, is a direct indicator of somatic growth, a critical metric for understanding individual fitness and population health. Time at large captures the duration over which growth can occur and may interact with seasonal and environmental conditions.

The independent variables used in the growth model fall into five biologically meaningful categories: initial condition, spatial, seasonal, temporal, and environmental. Each category is explicated in order.

Initial condition variables include fork length at release () and weight at release, These variables are crucial for modeling size-dependent growth trajectories. Smaller individuals often exhibit higher relative growth rates due to elevated metabolic demands and compensatory growth dynamics. These initial traits anchor the baseline from which subsequent growth is measured.

Spatial variation is represented by the river mile at release, which captures fixed geographic differences in habitat quality along the river corridor. These differences can influence early growth potential due to longitudinal gradients in water temperature, prey density, substrate, shading, and flow regimes. River position is especially relevant in riverine systems where upstream and downstream reaches may differ markedly in ecological conditions.

Seasonal effects are captured by both the release month and recovery month (both converted to sets of indicator variables), which represent seasonal cycles in water temperature, photoperiod, and prey availability, particularly aquatic insects. Seasonal timing also relates to high-flow events, which can alter habitat structure and displace prey, thereby influencing energy intake and expenditure.

Temporal exposure is modeled using time at large, which quantifies the duration available for growth between release and recapture. In addition, release year and recovery year (again, sets of indicator variables) account for interannual variability in unmeasured but biologically important conditions such as drought, flood pulses, or shifts in food web dynamics. These temporal covariates help capture broad-scale environmental fluctuations that may not be reflected in short-term or localized metrics.

Environmental variables include average river discharge over the time-at-large interval, water temperature, solar insolation, reactive phosphorus concentration, and rainbow trout biomass. Discharge influences habitat availability, drift-feeding opportunities, and the energetic cost of station-holding. Water temperature governs metabolic rate and food conversion efficiency, constraining growth within species-specific thermal limits. Solar insolation serves as a proxy for in-stream primary production, affecting food web productivity. Reactive phosphorus concentration is a limiting nutrient for algal growth, thereby modulating base-level productivity. Biomass provides an ecologically valid index of density-dependent pressures, such as competition for resources. Rather than relying on individual fish metrics, biomass estimates were derived from a Jolly-Seber mark-recapture model applied to size-class abundance data across the central 3 km of the study reach [21], reflecting population-level conditions that influence growth trajectories.

Training, validation, and test sets.

Following data merging and feature engineering, we randomly split the dataset into training and testing subsets using a 70/30 ratio, with a fixed pseudo-random number seed to ensure reproducibility. Seventy percent of the data was allocated for training, while the remaining 30% served as a held-out test set for evaluating model generalization. To prevent information leakage, no test data was used during model development. Within the training set, we conducted hyperparameter tuning via 5-fold cross-validation. Specifically, we employed randomized search over a targeted hyperparameter space for each ML model to identify optimal configurations while managing computational cost. This strategy balances tuning efficiency with robustness and helps reduce the risk of overfitting. Because model evaluation was performed exclusively on an untouched 30% test set, completely isolated from training and hyperparameter tuning, the resulting performance metrics provide an unbiased estimate of true generalization ability.

Data scaling.

To prepare the data for model training, two scaling strategies were considered to normalize the predictor variables. In the first approach, all continuous predictors were scaled to a [0, 1] range using a Min-Max transformation. This full-scaling strategy was used for ML models, where consistent feature ranges are important for convergence and optimization. In the second approach, two biologically meaningful predictors, (initial length) and Time at Large, were excluded from scaling to preserve their native units and biological interpretability. This partial-scaling strategy was applied to models grounded in biological reasoning. All remaining continuous features in this approach were scaled identically to the full-scaling method. In both cases, scaling parameters were computed using only the training data and then applied to the test data, ensuring that no information from the test set leaked into the model development process. Importantly, the response variable () was never scaled to support interpretability. Both scaled datasets were used in parallel to assess whether preserving the original scale of biologically relevant variables improved interpretability or predictive performance.

Software.

All data preprocessing, model implementation, and statistical analyses were conducted using Python 3.13, the latest stable release of the Python programming language [22]. Python offers a robust open-source coding ecosystem ideal for ML, data wrangling, and statistical analysis, with libraries such as NumPy [23], pandas [24], scikit-learn [25], XGBoost [26], LightGBM [27], and Pytorch [28], enabling efficient and reproducible modeling workflows. Python’s widespread adoption in the scientific community, flexible syntax, and extensive package repository made it an ideal platform for this research.

Bayesian fabens model formulation.

To estimate the parameters of the Fabens VBGM using Bayesian inference, we specified the following hierarchical model:

(3)

(4)

(5)

where:

• : fork length at initial marking,
• : fork length at recapture,
• t: time at large (in years),
• : asymptotic maximum fork length,
• k: growth rate coefficient (positive, covariate-dependent),
• : matrix of scaled continuous covariates,
• : matrix of dummy variables for seasonal effects,
• : vector of regression weights for continuous covariates,
• : vector of regression weights for dummy predictors.

The prior distributions were specified as follows:

(6)

(7)

(8)

(9)

(10)

Formulation.

The joint posterior distribution of the model parameters

given the observed data

is proportional to the product of the likelihood and prior distributions:

(11)

where the predicted length at recapture is defined as:

(12)

and the individual-specific growth coefficient is modeled as:

(13)

Bayesian Gompertz model formulation.

The second biological model implemented was the [14] growth equation, a sigmoid-shaped function that describes asymptotic growth using a double-exponential form. Originally developed to model human mortality, the Gompertz function has since been widely applied in biological and ecological contexts to capture growth processes that begin rapidly but decelerate over time as they approach an upper physiological limit. The model assumes that the relative growth rate decreases exponentially with time, making it particularly well-suited for species exhibiting rapid early development followed by gradual slowing as size approaches a maximum threshold. The Gompertz equation models the predicted fork length at time t as:

(14)

To estimate the parameters of the Gompertz growth model using Bayesian inference, we specified the following hierarchical model:

(15)

(16)

(17)

where:

• : fork length at recapture,
• t: time at large (in years),
• : asymptotic maximum fork length,
• k: growth rate coefficient (positive, covariate-dependent),
• : matrix of scaled continuous covariates,
• : matrix of dummy variables for seasonal effects,
• : vector of regression weights for continuous covariates,
• : vector of regression weights for dummy predictors.

The prior distributions were specified as:

(18)

(19)

(20)

(21)

(22)

Formulation.

The joint posterior distribution of the model parameters

given the observed data

is proportional to the product of the likelihood and prior distributions:

(23)

where the predicted length at recapture is defined as:

(24)

and the individual-specific growth rate is:

(25)

Bayesian justification.

We employed Bayesian inference for both the Fabens VBGM and Gompertz growth models due to its advantages in handling ecological data with complex structure and limited information. Bayesian methods allow for the incorporation of biologically informed prior knowledge–such as plausible bounds for asymptotic size or expected variability in growth rates–which improves parameter regularization and interpretability. These methods do require scaling of the non-indicator independent variables. This is especially critical in hierarchical models with latent variables and covariate-dependent growth rates, where frequentist estimators may yield unstable or biologically implausible results. Additionally, Bayesian models yield full posterior distributions, which enable more comprehensive uncertainty quantification and straightforward propagation of parameter uncertainty through model predictions. The Bayesian framework also supports posterior predictive checks, probabilistic forecasts, and model comparison using information criteria and cross validation tools that are not as readily applicable or interpretable under a frequentist paradigm. Taken together, these features make the Bayesian approach more robust, flexible, and ecologically appropriate for modeling rainbow trout growth under temporally and spatially structured environmental conditions.

Bayesian linear model formulation.

We implemented a Bayesian linear model to serve as a baseline for evaluating growth prediction under additive linear assumptions. The model assumes a normal likelihood with a linear predictor and constant variance:

(26)

(27)

where:

• : fork length at recapture (response variable),
• : value of the jth predictor for observation i,
• : model intercept,
• : regression coefficients,
• σ: observation noise (residual standard deviation).

We placed weakly informative priors on all regression parameters to stabilize estimation without imposing strong constraints. The regression coefficients were assigned independent normal priors:

and the standard deviation of residuals was parameterized as:

with a deterministic transformation for interpretability:

We chose a Bayesian approach for the linear model to maintain consistency across other modeling frameworks and to take advantage of posterior inference, particularly under uncertainty and moderate sample sizes. Bayesian linear regression provides full posterior distributions for all parameters, allowing direct probabilistic interpretation and better uncertainty quantification than frequentist confidence intervals. Additionally, posterior draws for regression coefficients enable direct comparison with effect estimates from the nonlinear growth models. This formulation also improves robustness to multicollinearity and overfitting in high-dimensional predictor sets through regularization induced by priors. Unlike frequentist ordinary least squares (OLS), which provides only point estimates and asymptotic standard errors, the Bayesian version yields complete joint distributions and facilitates posterior predictive checks and model averaging. Moreover, by expressing the observation error in log-space, we achieved more stable sampling and better convergence behavior in MCMC, particularly under heteroscedastic conditions.

Random Forest (RF).

An RF [33] is an ensemble learning method that constructs a collection of decorrelated decision trees and aggregates their predictions to produce a more stable and accurate model. For regression tasks, the predicted outcome is computed as the average of T individual decision tree predictions:

(28)

where:

• T is the number of trees in the forest,
• is the prediction of the t-th tree.

Each decision tree is trained on a bootstrap sample drawn with replacement from the original dataset, a process known as bagging (bootstrap aggregating). Furthermore, at each node split, RF consider only a random subset of predictors, introducing additional decorrelation between trees and reducing overfitting.

By averaging over many uncorrelated trees, RF significantly reduces the variance of individual tree estimators. While single decision trees are prone to high variance and may overfit training data, the ensemble average acts as a variance stabilizer, yielding a more generalizable model. This property is particularly advantageous in ecological datasets, which often contain complex interactions, nonlinearities, and noisy measurements. Each individual tree uses a greedy algorithm to recursively partition the feature space based on impurity measures such as Mean Squared Error (MSE) used in this study.

Importantly, RF offer a built-in mechanism for assessing feature importance. This importance is typically calculated (as it is here) based on the average decrease in impurity (or prediction error) resulting from splits on each feature, aggregated across all trees.

To optimize the performance of the Random Forest model, we employed a randomized hyperparameter search using 5-fold cross-validation. The search focused on key parameters that influence model complexity and generalization, including the number of trees, maximum depth of each tree, the minimum number of samples required to split an internal node, the minimum number of samples required to form a leaf, and the strategy used to select input features at each split. Rather than exhaustively evaluating all possible combinations, we sampled 25 random configurations from the defined hyperparameter space to balance efficiency and thoroughness. Model performance was assessed using negative mean squared error as the scoring metric, and the configuration yielding the lowest average validation error was selected as the final model.

The Random Forest model was tuned using a randomized search across the following hyperparameter space, with the selected values shown in bold:

• Number of trees (n_estimators): [100, 200, 300, 400]
• Maximum tree depth (max_depth): [15, 25, None]
• Minimum samples required to split a node (min_samples_split): [2, 5, 10]
• Minimum samples required to form a leaf node (min_samples_leaf): [1, 3, 5]
• Feature sampling method for tree construction (max_features): [sqrt, log2]

These settings were selected based on performance using randomized search with 5-fold cross-validation, optimizing for generalization and predictive accuracy.

XGBoost and LightGBM (LGBM).

Both XGBoost [26] and LGBM [27] are gradient-boosted decision tree (GBDT) frameworks that build powerful predictive models through the sequential addition of weak learners (typically shallow regression trees). The prediction function is constructed as a sum of T individual tree functions:

(29)

where is the space of regression trees, and each represents a base learner trained to predict the residuals of the current ensemble.

Unlike bagging-based methods like RF, boosting operates sequentially, where each new tree corrects the prediction error made by the ensemble so far. Both XGBoost and LGBM use gradient descent to minimize an objective function that combines a differentiable loss function (typically squared error for regression) with a regularization term to penalize model complexity:

(30)

Regularization discourages overfitting and promotes generalization by penalizing trees with excessive depth, number of leaves, or leaf weights.

XGBoost [26] constructs trees depth-wise, growing each level of the tree simultaneously and splitting all nodes at the current depth before proceeding. This strategy produces balanced trees and generally leads to faster convergence with fewer splits.

LGBM [27], in contrast, adopts a leaf-wise growth strategy: it selects the leaf with the largest reduction in loss and continues splitting from that point. This often results in deeper, more asymmetric trees, allowing for greater flexibility and improved computational efficiency on large datasets. However, it can increase the risk of overfitting on smaller datasets if not properly regularized.

To fine-tune the XGBoost model, we conducted a randomized hyperparameter search using 5-fold cross-validation. The search targeted parameters that directly influence model flexibility, learning dynamics, and regularization. These included the learning rate, maximum tree depth, number of boosting rounds, subsampling ratio for training instances, proportion of features used per tree, and both and regularization strengths. Given the use of higher learning rates, the number of boosting rounds was kept moderate to avoid overfitting. We sampled 25 random combinations from the defined hyperparameter space to ensure efficient yet thorough exploration. Model performance was evaluated based on negative mean squared error, and the configuration yielding the lowest average validation error was selected as the final model.

The XGBoost model was tuned using randomized search with 5-fold CV over the following hyperparameter space. The selected values are shown in bold:

• Learning rate (eta): [0.05, 0.1, 0.15, 0.2]
• Maximum tree depth (max_depth): [3, 4, 5]
• Number of boosting rounds (n_estimators): [200, 400, 600, 800]
• Subsample ratio of training instances (subsample): [0.7, 0.8, 0.9]
• Proportion of features used per tree (colsample_bytree): [0.8, 0.9, 1.0]
• L1 regularization term on weights (reg_alpha): [0, 0.1]
• L2 regularization term on weights (reg_lambda): [1, 1.5]

These parameters were selected to balance generalization and computational efficiency, with model performance validated using cross-validation.

For the LightGBM model, we also implemented a randomized hyperparameter search using 5-fold cross-validation. The search was designed to explore parameters central to boosting performance and model regularization. These included the number of boosting rounds, learning rate, maximum tree depth, and the number of leaves per tree, which is particularly influential in LightGBM’s gradient-based framework. Additional parameters such as subsampling ratios for rows and columns, along with and regularization strengths, were also tuned to improve generalization. A total of 20 random configurations were sampled from the defined hyperparameter space to achieve an efficient and targeted search. Model performance was evaluated using negative mean squared error, and the configuration with the lowest average validation error was selected as the final model.

The LightGBM model was tuned using randomized search across the following hyperparameter space. Selected values are shown in bold:

• Number of boosting rounds (n_estimators): [100, 150, 200]
• Learning rate (learning_rate): [0.05, 0.1, 0.15]
• Maximum tree depth (max_depth): [15, 20, −1]
• Number of leaves per tree (num_leaves): [31, 50, 100]
• Subsample ratio of training data (subsample): [0.7, 0.8, 0.9]
• Column sampling ratio per tree (colsample_bytree): [0.8, 0.9, 1.0]
• L1 regularization term on weights (reg_alpha): [0, 0.1]
• L2 regularization term on weights (reg_lambda): [0, 0.1, 1]

Support Vector Regression (SVR).

SVR extends the foundational concepts of Support Vector Machines (SVMs) to regression tasks by constructing an optimal predictive function that balances model complexity with tolerance for small prediction errors [32]. SVR estimates a continuous response variable using a linear or nonlinear mapping:

where is a (possibly nonlinear) transformation of the input vector , w is the weight vector, and b is the bias term.

Unlike OLS, which minimizes squared error, SVR introduces an ε-insensitive loss function that ignores prediction errors within a margin of ε. The optimization objective is to find a “flat” function that approximates the data within this margin while minimizing complexity. Formally, the primal optimization problem follows.

subject to:

Here, penalizes model complexity to promote generalization. The slack variables and capture deviations beyond the ε-tube, and the regularization parameter C governs the trade-off between model flatness and tolerance to errors exceeding ε.

This convex quadratic programming formulation is typically solved in the dual using Lagrangian multipliers, which naturally identifies a subset of support vectors–data points that lie outside the ε-tube and directly influence the regression function.

Solving the dual problem enables the use of kernel functions, defined as , to compute inner products in high-dimensional feature spaces without explicit transformation. This kernel trick allows SVR to model nonlinear relationships efficiently. Common choices include the radial basis function (RBF), polynomial, and sigmoid kernels.

To develop an SVR with interpretable coefficients, we initially restricted the hyperparameter search to a linear kernel. This choice facilitates straightforward interpretation of the feature effects on the predicted outcome and allows comparison with traditional parametric models. While linear regression models also offer interpretable coefficients, we included the linear-kernel SVR model to assess whether a margin-based linear method with regularization could improve generalization performance without sacrificing interpretability. A randomized hyperparameter search with 5-fold cross-validation was employed to tune the penalty parameter, the epsilon-insensitive loss margin, and the kernel coefficient. Model performance was evaluated using negative root mean squared error (RMSE), and the configuration with the lowest average validation error was selected as the final model.

The linear Support Vector Regression (SVR) model was tuned (5-fold CV) using a predefined hyperparameter space. Since a linear kernel was used, the gamma parameter was not applicable. The hyperparameter search space and selected values are shown below, with selected values in bold:

• Penalty parameter (C): [0.1, 1, 10, 100]
• Epsilon in the loss function (): [0.01, 0.1, 0.5]

These parameters were selected based on performance using cross-validation, with the linear kernel fixed.

A second Support Vector Regression (SVR) model was formulated to evaluate the optimal kernel type as well as the other parameters. The investigated components mirrored those of the linear SVR, with the addition of kernel selection. Three kernels (linear, polynomial, and radial basis function or RBF) were examined using hyperparameter tuning and 5-fold cross-validation. A nonlinear Support Vector Regression (SVR) model was tuned to evaluate kernel performance, including linear, polynomial, and radial basis function (RBF) kernels. The selected configuration used an RBF kernel, with other hyperparameters optimized via randomized search and 5-fold cross-validation. The hyperparameter space and selected values are listed below, with selected values in bold:

• Kernel type (kernel): [linear, poly, rbf]
• Penalty parameter (C): [0.1, 1, 10, 100]
• Epsilon in the loss function (): [0.01, 0.1, 0.5]
• Gamma (): [scale, 0.01, 0.1]

This configuration was found to provide the best trade-off between model complexity and predictive performance on the validation folds. To compare the parameter relevance of the SVR model against other ML approaches, permutative feature importance was evaluated. This method allowed for a consistent framework to assess the relative contribution of each input variable across models, enabling a robust comparison of feature influence in the presence of nonlinearities and interactions.

Artificial Neural Network (ANN).

Artificial Neural Networks (ANNs) [34] are layered computational models inspired by the architecture of the human brain. They are capable of approximating highly nonlinear, hierarchical relationships between inputs and outputs through learned feature transformations. In their most basic form, ANNs consist of an input layer, one or more hidden layers, and an output layer, with each layer composed of nodes (neurons) that compute weighted combinations of their inputs. Although artificial neural networks (ANNs) are not always the top-performing choice for structured tabular data, especially when sample sizes are moderate, their inclusion in this study was motivated by both methodological and ecological considerations. From a methodological perspective, ANNs serve as a flexible, universal function approximator capable of capturing complex, nonlinear relationships that may elude simpler parametric models. Given the known nonlinearities in fish growth and environmental interactions, this capacity warranted their inclusion. Ecologically, neural networks allow us to test whether non-tree-based nonlinear models can uncover complementary patterns in rainbow trout growth dynamics, especially in interaction-heavy scenarios where traditional models may oversimplify relationships. Including the ANN also provided a useful baseline for comparing deep learning performance against more interpretable models such as tree ensembles and biologically grounded equations.

The general forward pass for an L-layer feedforward neural network is expressed as:

(31)

where:

• denotes the weight matrix at layer l,
• is the bias vector at layer l,
• is the nonlinear activation function at layer l, such as the ReLU, sigmoid, or tanh function.

ANNs are trained by minimizing a loss function–such as mean squared error in regression–via backpropagation and stochastic gradient descent (SGD) or adaptive optimization algorithms like Adam. During training, weights and biases are iteratively adjusted to reduce the prediction error. The flexibility of ANNs comes at the cost of increased computational demand and sensitivity to hyperparameter choices, such as learning rate, batch size, and architecture depth.

The universal approximation theorem guarantees that a sufficiently large neural network can approximate any continuous function on a compact domain to arbitrary accuracy. This makes ANNs powerful tools for modeling nonlinear ecological systems with high-dimensional interactions that may be difficult to capture with traditional parametric models.

To identify an optimal configuration for the neural network model, we performed a randomized hyperparameter search using 5-fold cross-validation. The search space included a range of architectural and training parameters, such as the number and size of hidden layers, dropout rates, learning rates, weight decay values, learning rate scheduling parameters (initial cycle length and minimum learning rate), and batch sizes. From a total of 648 possible configurations, five hyperparameter combinations were randomly sampled and evaluated. For each combination, five models were trained and validated across cross-validation folds, with mean squared error used as the evaluation metric. The average validation loss across 5-folds was used to determine the most effective configuration. This initial tuning phase provided a computationally efficient yet rigorous method for selecting a high-performing neural network architecture consistent with the evaluation procedures used for other models in the study.

The best-performing artificial neural network (ANN) configuration was selected based on average validation loss across five cross-validation folds. The selected hyperparameters follow.

• Hidden layer sizes (hidden_sizes): [256, 128, 64]
• Dropout rates per layer (dropout_rates): [0.4, 0.3, 0.2]
• Learning rate (learning_rate): 0.0005
• Weight decay (weight_decay): 5e-5
• Learning rate restart interval (lr_restart_interval): 20
• Minimum learning rate (min_lr): 1e-6
• Batch size (batch_size): 32

Fabens VBGM.

The VBGM assumes that growth decelerates asymptotically as an organism approaches its theoretical maximum size . The key strength of this model is that it provides posterior distributions over all unknowns, including individual-level growth rates, allowing for full uncertainty quantification. Unlike traditional VBGM approaches that assume constant growth rates, this model incorporates individual heterogeneity through environmental and temporal covariates, which is critical in ecological modeling where growth is often affected by dynamic habitat conditions.

The performance metrics for the VBGM suggest a solid, biologically informed baseline. The model yielded an RMSE of 16.82 and an MAE of 11.60, indicating that the average deviation of predicted fish lengths from observed values was around 11–17 mm. From a biological standpoint, this level of error is relatively modest given the natural variability in growth patterns among fish populations, and it suggests that the model is effectively capturing the underlying growth dynamics. The value of 0.9620 and adjusted of 0.9619 imply that over 96% of the variability in observed lengths is explained by the model, reinforcing the biological plausibility of VBGM in modeling somatic growth processes. Additionally, the AIC (24,945.04), AICc (24,945.05), and BIC (24,962.99) metrics indicate strong model parsimony, supporting the idea that the growth curve is both statistically and biologically well-specified without unnecessary complexity. (While a Widely-Applicable Information Criterion (WAIC) is generally preferred, use of the AIC, AICc, and BIC supported comparison.) Overall, these results affirm the VBGM’s capacity to capture length-at-age patterns with high fidelity, aligning with theoretical expectations of metabolic scaling and resource allocation in fish development.

Gompertz model.

The Gompertz model also demonstrate reasonable but somewhat inferior predictive performance relative to the VBGM. The performance on the test set follows: RMSE = 16.924, MAE = 11.674, =0.962, Adjusted =0.962, AIC = 24982.28, AICc = 24982.29, BIC = 25000.24.

While the Gompertz model outperformed the VGBM in the baseline, no-covariate setting likely due to its flexible curvature and early growth deceleration, its performance declined relative to VBGM once covariates were introduced in a Bayesian framework. One reason for this reversal is structural: the Gompertz model includes a single global rate parameter, which can limit its ability to accommodate feature-dependent variation in growth. In contrast, the VBGM, though initially more rigid, benefits more from the inclusion of biologically relevant covariates that can adjust parameters such as the asymptotic size or growth rate across individuals or conditions. This flexibility, combined with the regularizing effects of Bayesian estimation, enables the VBGM to better capture context-specific growth dynamics and reduces overfitting. The Gompertz model, by contrast, may become overparameterized or less stable when extended with covariates, diminishing its initial advantage.

Bayesian linear model.

We fit a Bayesian linear regression model using NumPyro to estimate the relationship between trout recapture length () and a combination of continuous and categorical predictors. All continuous predictors were standardized prior to model fitting, while the response variable () remained in its original unit of millimeters to preserve ecological interpretability.

The model used a Normal prior for the coefficients (), a LogNormal prior on the standard deviation (), and was sampled using the NUTS with 4 chains, each producing 1,500 posterior samples after a 500-sample tuning phase.

The Bayesian linear model demonstrated strong predictive accuracy compared to baseline methods. It achieved an of , with an adjusted of , indicating that the model captured the majority of the variance in the response variable. The RMSE was 25.72 mm, and the MAE was 18.54 mm, both reflecting substantial improvements over prior versions and competitive performance relative to more complex models.

The information criteria supported the model’s improved fit: AIC = 27,533.81, AICc = 27,535.44, and BIC = 27,821.15. These values are substantially lower than those observed in earlier specifications and within a competitive range compared to tree-based or boosting models, suggesting that the revised Bayesian linear model more effectively captured the underlying structure of the data.

While the Bayesian framework offers clear benefits in interpretability and uncertainty quantification, its application here did not result in a competitive model. Future improvements may involve incorporating hierarchical structures, interaction terms, or more flexible priors to better accommodate ecological complexity in trout growth modeling.

RF.

The tuned Random Forest model was then applied to the pristine test set data, achieving an RMSE of 18.40 mm and a MAE of 12.20 mm. The model explained approximately 95.45% of the variance in the outcome (, adjusted ), indicating excellent predictive accuracy. The information criteria values (AIC = 25,563.22, AICc = 25,564.78, and BIC = 25,844.57) further confirm the model’s strong fit relative to other approaches. The three most important predictors identified by the Random Forest were (length at release), time at large, and weight at release. These are intuitive and biologically plausible variables, reinforcing their relevance in predicting growth outcomes.

XGBoost.

XGBoost exhibited the lowest predictive error with an RMSE of 16.14 mm, slightly ahead of Random Forest (16.46 mm), VBGM (16.82 mm), and the Gompertz model (17.07 mm). Although the differences in RMSE are consistent across repeated validation folds, the absolute gap between the best-performing ML models and the Bayesian VBGM is under 1 mm. This suggests that the improvements in predictive accuracy, while statistically detectable, may not translate to biologically meaningful differences in growth estimation for management purposes.

LightGBM.

LightGBM achieved an RMSE of 16.25 mm and an MAE of 10.80 mm, delivering predictive performance nearly equivalent to XGBoost. With an of 0.9645 and an adjusted of 0.9640, the model explained a similarly high proportion of variance in the outcome. Although slightly higher than XGBoost, the information criteria (AIC = 24,830.76, AICc = 24,832.32, and BIC = 25,112.11) still reflect a strong model fit and reinforce LightGBM’s competitiveness among gradient boosting approaches. The top three most influential features were (length at release), time at large, and the release river map, indicating the model’s ability to integrate both biological and spatial predictors effectively.

Linear SVR.

The linear SVR model showed noticeably weaker performance compared to the top ensemble methods in this study. It yielded an RMSE of 25.35 mm and a MAE of 15.40 mm–substantially higher than those of XGBoost (RMSE = 16.14) and LightGBM (RMSE = 16.25), and even lagging behind the Random Forest model (RMSE = 18.40). With an of 0.9137 and adjusted of 0.9122, the SVR explained just over 91% of the variance in recapture length–respectable, but clearly lower than the approximately 96% explained by the best-performing models.

From a model selection perspective, SVR also fared less favorably. Its AIC = 27,446.61 and BIC = 27,727.96 were substantially higher than those for XGBoost (AIC = 24,790.38, BIC = 25,071.73) and LightGBM (AIC = 24,830.76, BIC = 25,112.11), indicating poorer balance between model fit and complexity. It also trailed the Random Forest in information criteria, suggesting that SVR was less efficient overall in modeling the observed data.

While the linear Support Vector Regression (SVR) model did not achieve top-tier predictive performance compared to nonlinear models, it offered a distinct advantage in interpretability. Specifically, linear SVR provides direct coefficient estimates, enabling a clear understanding of the direction and relative magnitude of each input variable’s contribution to the predicted outcome. This directional insight is valuable for interpreting ecological or biological mechanisms in fish growth modeling. However, this transparency comes at the cost of flexibility, as the linear kernel assumes strictly additive and linear relationships between predictors and the response variable. Consequently, the model is unable to capture complex interactions or nonlinear effects, which likely explains its higher error metrics relative to kernel-based and ensemble models.

Kernel-optimized SVR.

The RBF-kernel SVR model demonstrated strong predictive performance, outperforming several baseline and traditional models. It achieved a RMSE of 18.81, an MAE of 11.77, and an value of 0.952. The adjusted was similarly high at 0.951, indicating robust model fit even after accounting for model complexity. In terms of information-theoretic criteria, the RBF SVR produced an AIC of 25692.93, AICC of 25694.49, and BIC of 25974.28. While not the top-performing model across all metrics, it offered a balance between accuracy and generalization. Notably, it outperformed the linear SVR and Bayesian linear models across every metric, demonstrating the benefit of modeling nonlinear relationships in fish growth data through kernel-based methods.

To evaluate the relative influence of input features in the RBF SVR model, we employed permutation feature importance, a model-agnostic approach that quantifies importance based on changes in prediction error. This method involves systematically shuffling the values of each feature in the test set and measuring the resulting increase in the model’s prediction error, thereby capturing how much each feature contributes to the model’s performance. Results are incorporated to the feature importances analyses.

Metrics.

Table 2 summarizes the performance of all models using RMSE, MAE, , adjusted , and information criteria (AIC, AICC, BIC). Importantly, the leading models are tightly clustered in absolute error (typically within mm RMSE of one another), so the practical distinction among top performers is not accuracy alone, but whether the method provides biological interpretability and uncertainty quantification alongside predictive power.

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Table 2. Comparison of model performance metrics (sorted by RMSE). Baseline for percent reduction is the baseline von Bertalanffy growth model (VBGM). Abbreviations: VBGM = von Bertalanffy growth model; BMA = Bayesian Model Averaging; ANN = Artificial Neural Network; SVR = Support Vector Regression; RF = Random Forest; LGBM = Light Gradient Boosting Machine; XGB = XGBoost; RMSE = root mean squared error.

https://doi.org/10.1371/journal.pone.0336890.t002

The best overall performance was achieved by the Stacked Ensemble, which outperformed all individual models with the lowest RMSE (15.96 mm), lowest MAE (10.62 mm), and highest (0.9658). This highlights the effectiveness of aggregating predictive strengths from both mechanistic and ML paradigms.

Among individual learners, XGBoost and LightGBM performed exceptionally well, achieving RMSE values of 16.14 mm and 16.25 mm, respectively, with high values and favorable information criteria. These results reaffirm the ability of gradient boosting algorithms to model complex nonlinear processes in ecological systems. Bayesian biological models, including the VBGM and the Gompertz equation, also ranked competitively, demonstrating that biologically grounded models remain viable, especially when well-calibrated, informed by data, and inclusive of covariates.

In contrast, the Baseline VBGM and Baseline Gompertz models—fitted without covariates—performed poorly, with RMSE values of 86.29 mm and 61.52 mm, respectively, and substantially lower values. These results emphasize the limitations of using biologically motivated models without accounting for relevant environmental and contextual variables, reinforcing the importance of covariate inclusion for predictive accuracy.

Outside of the poor-performing baseline models, the Bayesian Linear Model and Linear SVR showed the weakest performance among fully trained models, with RMSEs exceeding 25 mm and comparatively lower values, indicating a poor fit to the data. However, the Radial Basis Function (RBF) SVR improved substantially upon the linear variant, achieving an RMSE of 18.81 mm and , illustrating the benefit of nonlinear kernel methods in capturing more complex relationships. The Random Forest and Neural Network models also produced values above 0.95, but incurred higher information criteria, reflecting greater model complexity and possibly reduced generalization.

These findings support a natural next step: applying Bayesian Model Averaging (BMA) to integrate top-performing models across paradigms. The BMA approach achieved similar accuracy to XGBoost while explicitly incorporating model uncertainty through posterior-weighted predictions. This technique may be particularly advantageous in ecological forecasting contexts where interpretability, uncertainty quantification, and generalizability are critical. Future research could assess whether BMA further enhances model robustness while preserving the domain-specific strengths of both mechanistic and ML-based approaches.

Coefficients.

Fig 3 illustrates the top nine continuous features identified through average normalized importance scores across all models. The feature importance scores were normalized using min-max scaling to a [0,1] range to enable direct comparison across different models and metrics. The variable , representing asymptotic length, emerged as the most influential predictor overall. Its importance was consistently high across nearly all modeling approaches, with particularly strong influence in the Bayesian Gompertz, Bayesian VBGM, and SVR RBF models. This is expected, as is a core parameter in both the VBGM and the Gompertz models, where it serves as a theoretical maximum size. These mechanistic models explicitly rely on this variable to define growth trajectories, which explains its critical importance.

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Fig 3. Feature importance heatmap showing normalized importance values (0-1) across nine models.

Models: RF = Random Forest, XGB = XGBoost, LGBM = LightGBM, SVR-L = Support Vector Regression (Linear), SVR-R = Support Vector Regression (RBF), NN-IG = Neural Network with Integrated Gradients, VBGM = Von Bertalanffy Growth Model (Bayesian), G-Bayes = Gompertz (Bayesian), Avg = Average across all models. Features: L1 = Initial length, Time Large = Time at large, Weight Release = Weight at release, RT Biomass = Rainbow trout biomass, Solar Insol = Solar insolation, Release RM = Release river mile, SRP Conc = Soluble reactive phosphorous concentration, Water Temp = Water temperature. Color intensity and numerical values indicate relative feature importance within each model.

https://doi.org/10.1371/journal.pone.0336890.g003

The variable Time at Large, another essential parameter in traditional growth models, also ranked highly in terms of predictive contribution. In both the VBGM and Gompertz frameworks, Time at Large maps directly to the growth curve’s temporal component, linking observed size back to age or exposure duration. Its strong influence in ML models (including LightGBM and Neural Networks) underscores its general predictive relevance, even outside mechanistic contexts.

Additional high-ranking features included Weight at Release, Rainbow Trout Biomass, and Soluble Reactive Phosphorous Concentration. These environmental or biological covariates were less central in traditional growth models but proved informative in ensemble methods like XGBoost and Random Forest. This suggests that data-driven models may capture interaction effects and context-dependent relationships that lie outside the scope of classical equations.

The consistency of and Time at Large across both mechanistic and ML approaches validates their foundational role in growth modeling. Meanwhile, the variable-specific strengths of different algorithms highlight the complementary nature of ensemble and domain-specific methods when predicting ecological phenomena.