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Quantifying Environmental and Demographic Stochasticity in Population Viability Analysis: The Case of Rissa tridactyla

This study utilizes stochastic differential equations derived from the best-fitting logistic growth model to analyze Black-legged Kittiwake population dynamics, revealing that demographic stochasticity is the primary driver of variability and predicting a 53.1% probability of quasi-extinction within the next 30 years.

Original authors: Riddhimann Chattopadhyay, Richik Bandyopadhyay, Tanoy Mukherjee

Published 2026-09-21

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CC BY 4.0

Original authors: Riddhimann Chattopadhyay, Richik Bandyopadhyay, Tanoy Mukherjee

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Understanding whether a species will survive or vanish into history is one of the most urgent questions in conservation biology. In the natural world, populations rarely grow in a straight, predictable line. Instead, they are subject to two distinct types of randomness. One type comes from the environment: a harsh winter, a shortage of food, or a sudden shift in climate that affects every single individual in a group at the same time. The other type is far more intimate and internal. It arises from the simple, random luck of birth and death. In any given year, a few more birds might hatch than expected, or a few more adults might die by chance. In small populations, this internal randomness can be powerful enough to push a species toward extinction, even if the overall environment seems stable and the population appears to be growing. Scientists use mathematical models to try to see through this fog of chance, hoping to predict which populations are safe and which are on a slippery slope toward disappearance.

A recent study focused on the Black-legged Kittiwake, a seabird that nests on steep coastal cliffs across the North Atlantic and North Pacific. These birds are long-lived and have a slow pace of life; they take several years to reach adulthood and produce only a few chicks each year. Because they rely so heavily on specific marine conditions and small fish for food, their numbers can swing dramatically. Researchers gathered decades of population records for this bird, spanning from 1949 to 1984, and set out to understand what drives these changes. They began by testing three different ways to describe how the population grew over time. One model assumed the birds would grow forever without limits, another assumed they would grow quickly at first and then slow down in a specific curved pattern, and the third assumed they would grow quickly, slow down, and eventually level off at a maximum number the environment could support. By comparing how well each model matched the actual historical data, the researchers found that the third option, the one that levels off, provided the most accurate picture of the bird's history.

With the best growth pattern identified, the team then asked a deeper question: what kind of randomness is most responsible for the ups and downs in the bird's numbers? They built a new version of their model that included two types of noise. The first type represented the changing environment, where a bad year for everyone would shake the entire population. The second type represented the internal luck of individual births and deaths, which becomes more significant when there are fewer birds around. When they fitted these complex models to the data, the results pointed clearly to one source of trouble. The study found that the random variation in individual births and deaths was the dominant force driving the population's fluctuations, far more so than the shifting external environment. This means that even if the ocean conditions were perfect, the sheer luck of who survived and who reproduced in any given year was the primary engine of uncertainty for this species.

Using this understanding, the researchers ran thousands of computer simulations to look into the future. They projected the population forward for thirty years, allowing the random luck of births and deaths to play out in each scenario. The results were sobering. In more than half of the simulated futures, the population dropped below a critical threshold of ten individuals, a point where recovery becomes nearly impossible due to the lack of breeding partners and the increased risk of inbreeding. The study calculated a 53.1 percent probability of this quasi-extinction occurring within the next three decades. This is a stark reminder that a population can appear to be stable or even growing on average, yet still face a high risk of collapse simply because of the unpredictable nature of individual life and death.

The findings suggest that for species like the Black-legged Kittiwake, conservation efforts must look beyond just protecting habitats or ensuring food supplies. While those factors are vital, the study highlights that the inherent instability of small populations is a major threat that cannot be ignored. The researchers noted that their model assumed a steady environment, but in reality, climate change and shifting fish stocks could make the situation even more precarious. By combining historical data with rigorous statistical testing, the study provides a clear, quantitative warning: without intervention to boost survival and recruitment, the random fluctuations of nature alone could drive this seabird to the brink. The work offers a practical framework for other scientists to assess similar risks in vulnerable species, turning vague fears about extinction into concrete probabilities that can guide real-world protection strategies.

Technical Summary: Quantifying Environmental and Demographic Stochasticity in Population Viability Analysis: The Case of Rissa tridactyla

Problem Statement
Understanding population dynamics and assessing extinction risk are central objectives in conservation biology, particularly under increasing environmental variability. While deterministic models (e.g., logistic, Gompertz) provide a foundational framework for describing density-dependent growth, they fail to capture the inherent randomness of natural populations. Real-world populations are subject to two primary sources of stochasticity: demographic stochasticity (randomness in individual birth and death events, critical in small populations) and environmental stochasticity (temporal fluctuations in external conditions affecting all individuals simultaneously). The challenge lies in quantifying these stochastic processes to accurately estimate extinction probabilities and population viability, as deterministic models often predict long-term survival even when stochastic fluctuations drive populations to extinction. This study addresses the need for a rigorous framework that integrates deterministic growth modeling with statistical inference and stochastic simulation to assess the extinction risk of the Black-legged Kittiwake (Rissa tridactyla).

Methodology
The study employs a multi-stage analytical workflow using long-term population time-series data (1949–1984) for Rissa tridactyla obtained from the Global Population Dynamics Database (GPDD).

Deterministic Model Selection:
The authors first fitted three classical deterministic growth models—Exponential, Logistic, and Gompertz—to the empirical data using nonlinear least-squares optimization. Model selection was performed using the Akaike Information Criterion (AIC) to identify the best-fitting deterministic framework. The Logistic model was selected as the most appropriate representation of the observed dynamics.
Stochastic Formulation and Parameter Estimation:
Building on the selected Logistic model, the authors developed stochastic differential equation (SDE) formulations to incorporate:
- Demographic Noise: Variance scaling linearly with population size ( $\sigma_d^2 N(t)$ ).
- Environmental Noise: Variance scaling quadratically with population size ( $\sigma_e^2 N(t)^2$ ).
- Combined Noise: A superposition of both sources.
Parameters for these stochastic models (intrinsic growth rate $r_m$ , carrying capacity $K$ , and noise intensities $\sigma_d, \sigma_e$ ) were estimated using Maximum Likelihood Estimation (MLE). The likelihood functions were derived by discretizing the SDEs using the Euler-Maruyama method, assuming Gaussian noise with zero mean.
Uncertainty Quantification:
Using the MLE estimates and the variance-covariance matrix of the parameters, the authors constructed confidence intervals (CIs) for the mean population trajectory and prediction intervals (PIs) for future observations. The PIs were calculated by propagating parameter uncertainty and adding the variance due to demographic process noise.
Extinction Risk Assessment:
Future population trajectories were projected over a 30-year horizon using Monte Carlo simulations (1,000 independent realizations). Extinction was defined as a "quasi-extinction" event where the population size fell below a threshold of 10 individuals. The probability of quasi-extinction was calculated as the proportion of simulations crossing this threshold.

Key Results

Model Selection: The Logistic growth model provided the best fit to the observed data among the deterministic candidates (AIC = 328.69), outperforming the Gompertz (334.24) and Exponential (380.30) models.
Dominant Stochastic Source: Among the stochastic models, the demographic noise model yielded the lowest AIC (292.70) compared to the environmental noise model (336.57) and the combined model (294.70). This indicates that intrinsic stochasticity arising from individual birth-death events is the dominant source of variability in the observed Rissa tridactyla population dynamics.
Extinction Probability: Despite the deterministic Logistic model predicting convergence to a carrying capacity ( $K \approx 168.58$ ), the incorporation of demographic stochasticity revealed a significant extinction risk. Monte Carlo simulations estimated a 53.1% probability of quasi-extinction within the next 30 years.
Model Fit: The MLE-based stochastic trajectory showed improved agreement with observed data, particularly around the inflection point, compared to the deterministic curve fit. The prediction intervals successfully captured the majority of observed data points, validating the model's ability to quantify uncertainty.

Significance and Claims
The authors claim that this study provides a comprehensive framework for assessing population persistence under uncertainty by integrating deterministic growth modeling, statistical inference, and stochastic simulation. Key contributions include:

Demonstration of Stochastic Risk: The study illustrates that even when deterministic dynamics predict long-term persistence, demographic stochasticity alone can generate substantial extinction risk (over 50% in this case), emphasizing the necessity of incorporating stochastic processes into Population Viability Analysis (PVA).
Identification of Variability Drivers: By identifying demographic stochasticity as the dominant source of variability for this species, the findings highlight that random individual-level processes significantly influence long-term ecological outcomes, particularly for species with specific life-history traits like the Black-legged Kittiwake (long-lived, low fecundity).
Methodological Utility: The proposed workflow offers a reproducible, statistically rigorous approach for analyzing long-term ecological time-series data. It provides a practical tool for conservationists to quantify extinction risk and inform management strategies for vulnerable seabird populations in changing environments.

The paper concludes that while the current model assumes a stationary environment and constant carrying capacity, the framework successfully demonstrates the critical role of demographic stochasticity in driving extinction risk, offering a baseline for more complex future models that might include environmental variability, Allee effects, and spatial heterogeneity.

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