Deep quantile regression for growth and maturation reaction norms

Understanding species' growth and maturation responses to anthropogenic and environmental pressures is crucial for tracking demographic shifts, phenotypic change and ensuring population sustainability. Traditional regression methods often focus on modelling the conditional mean of life-history traits, potentially overlooking heterogeneity in ecological data. In this study, we developed quantile and binary quantile regression models using deep neural networks to investigate the growth and maturation schedules across multiple quantiles. By capturing different parts of the conditional distribution, our approach accommodates heterogeneity and improves predictive accuracy and robustness compared to conventional methods. We further propose novel quantile maturation reaction norms (QMRNs) to assess changes in age and size at maturation. QMRNs outperformed widely used probabilistic maturation reaction norms in simulated datasets, demonstrating greater reliability and robustness. Applying this framework to largehead hairtail (Trichiurus japonicus) revealed distinct growth and maturation patterns between the cooler northern and warmer southern coasts of Taiwan Island, achieving lower prediction errors and providing more detailed demographic insights than traditional approaches. Overall, our quantile framework enables a more nuanced characterization of growth and maturation heterogeneity, facilitating intraspecific comparisons of life-history traits and evolutionary analyses of maturation schedules.

In oceans across the globe, a silent, pervasive change is underway: fish are getting smaller. This isn’t a random fluctuation; it’s a consistent, troubling pattern driven by a warming climate and the relentless pressure of our fishing nets. This raises a critical question: how do we accurately track these fundamental changes to animal populations?

For decades, the standard approach has often involved focusing on the “average” individual to understand population trends. However, this method can overlook crucial details and variations within a population, potentially hiding a more complex reality. Now, a new approach combining the power of deep learning with a statistical method called quantile regression offers a more nuanced and powerful way to understand what is truly happening to life in our oceans.

“Average” Is a Myth—And It’s Hiding a Dangerous Truth

Traditional scientific models often rely on “mean regression,” a method that calculates the average response to understand trends like growth rates. But imagine trying to understand the academic performance of a school by only looking at the average grade. You’d miss the story of the struggling students who need help and the high-achievers who are pushing the boundaries.

Quantile regression is like looking at the full report card. Instead of focusing only on the average, this method models relationships across the entire distribution of a population. It allows scientists to analyze the bottom 10%, the top 10%, and every slice in between, revealing a complete and often surprising picture. They can compare the slowest-growing 10% of individuals to the median individual (the 50th percentile) or the fastest-growing 90%, providing a much deeper understanding of how different parts of a population are responding to environmental pressures.

The real-world implication of this difference is critical. As new research highlights, focusing only on the average can lead to flawed conclusions with serious consequences for conservation.

Relying solely on the median growth curve fails to capture the disproportionate contribution of larger, highly fecund individuals, potentially leading to a significant underestimation of the population’s total reproductive potential.

In short, if we only look at the “average” fish, we completely miss the vital role that the largest, most fertile individuals play in sustaining the population. This underestimation could lead to the mismanagement of fisheries, threatening their long-term survival.

Deep Learning Becomes a New Microscope for Ecologists

The new method presented in the research uses deep neural networks—a form of AI—to power this advanced quantile regression analysis. A key advantage of this AI-driven approach is that it removes the reliance on scientists having to pre-select a specific biological growth function or make prior assumptions about how the data should behave.

This is a profound shift. Previously, ecologists had to make an educated guess about the mathematical shape of a fish’s growth curve—assuming it was exponential, for instance. If that assumption was wrong, the entire model could be flawed. The deep learning framework, however, makes no such assumptions. It learns the true, often complex, growth patterns directly from the data, acting as a more honest and flexible microscope.

In the study, this method achieved higher predictive accuracy than traditional approaches. The researchers developed two specific models to achieve this:

The deep quantile growth model (deep QGM): To map out the full range of growth trajectories, from the slowest to the fastest-growing individuals.
The deep binary quantile maturation model (deep BQMM): To understand the different ages and sizes at which fish across the population become ready to reproduce.

These sophisticated new tools give ecologists a clearer, more detailed picture of population dynamics than was previously possible.

A Tale of Two Coasts: Fish Life Stories Are Being Rewritten

To test their framework, researchers applied it to a case study of the largehead hairtail (Trichiurus japonicus), a commercially important fish, in two distinct locations off the coasts of Taiwan Island: the cooler northern site (K) and the warmer southern site (T). The results revealed that the life stories of these fish are being dramatically rewritten by their environments.

In the cooler northern waters, the largehead hairtail can afford to play the long game. They mature later, at an average of 1.55 years, and ultimately reach a larger, more fecund size.

But in the warmer, heavily fished southern waters, it’s a race against the net. These fish mature much earlier, at just 1.08 years, and at a fraction of the size. These differences align with established ecological principles like the “temperature-size rule,” which predicts that organisms in cooler climates often grow slower but reach larger adult sizes. However, the study points to another major factor at play: intense, size-selective fishing. The research suggests that heavy fishing pressure at the southern site may have “induced evolutionary changes, resulting in earlier maturation at smaller sizes”—a classic sign of fisheries-induced evolution, where the pressure of the nets favors fish that reproduce before they can be caught.

Conclusion: A More Complex, Complete Picture

By moving beyond simple averages and embracing advanced tools like deep quantile regression, we can gain a richer, more accurate understanding of how populations are responding to mounting environmental pressures. This new framework allows scientists to move from simple point estimates to a “probabilistic understanding of population structure and dynamics,” which is a major leap forward in ecological science.

This ability to see the full spectrum of responses within a population is essential for developing effective conservation strategies. This granular view could allow fishery managers to move beyond one-size-fits-all regulations, tailoring quotas to protect the fast-growing, highly fertile subgroups that are critical to a population’s resilience, or identifying specific areas where younger fish are maturing dangerously early. It leaves us with a final, crucial question to consider: if the “average” fish is a fiction, what other critical ecological truths are we missing by looking only at the mean?