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

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.