Gamma-Based Expansion for the First-Passage Time Distribution of Stochastic Logistic Models with Harvesting

2026-04-20

AI Summary

This study addresses the first-passage time (FPT) distribution for a stochastic logistic growth model incorporating constant harvesting and multiplicative noise. By employing a power series expansion of the Laplace transform, the authors derive moments and cumulants for both upward and downward FPTs. They further introduce, for the first time, a high-accuracy analytical approximation of the FPT density based on a Laguerre orthogonal expansion weighted by a Gamma function. The proposed method exhibits numerical stability, controllable precision, and superior performance under moderate noise intensities, making it well-suited for modeling large-scale population dynamics. The approach is successfully applied to parameter estimation in fisheries management, demonstrating its practical utility.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

The first passage time problem is considered for stochastic logistic growth model with constant harvesting and multiplicative environmental noise. Explicit expressions for the moments and cumulants of both upcrossing and downcrossing FPTs in the presence of constant thresholds are obtained through a power-series expansion of the Laplace transform. Then a closed-form representation of the FPT density is recovered via an orthogonal Laguerre--Gamma expansion . This representation is used to numerically evaluate FPT densities, with the truncation order controlling the trade-off between accuracy and stability. Numerical experiments based on Monte Carlo simulations confirm the high accuracy of the method in regimes of moderate dispersion and highlight its limitations when higher-order moments grow rapidly. Application to fisheries management models shows that the method remains effective even for large-scale population. Finally, the approximated density is satisfactory used to estimate some parameters of the model.

Problem

• first-passage time
• stochastic logistic model
• harvesting
• environmental noise
• population dynamics

Innovation

• first-passage time
• Laguerre–Gamma expansion
• stochastic logistic model
• harvesting
• orthogonal expansion