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.

翻译：本文研究了带恒定捕捞和乘性环境噪声的随机逻辑斯谛增长模型的首达时间问题。通过拉普拉斯变换的幂级数展开，获得了恒定阈值下上跨和下跨首达时间矩与累积量的显式表达式。随后利用正交拉盖尔-伽马展开恢复了首达时间密度的闭式表示。该表示可用于数值评估首达时间密度，其中截断阶数控制着精度与稳定性之间的权衡。基于蒙特卡洛模拟的数值实验证实了该方法在中等离散程度下的高精度，并揭示了高阶矩快速增长时的局限性。应用于渔业管理模型表明，该方法对大规模种群仍然有效。最后，近似密度被满意地用于估计模型的部分参数。