In this paper, we formulate a phytoplankton-zooplankton-fish model with distributed delays and hybrid stochastic noises involving Brownian motion and Markov chain, and propose an optimal harvesting problem pursuing the maximum of total economic income. By global analysis in terms of some system parameters, we investigate the dynamical behaviors on the well-posedness, bounded- ness, persistence, extinction, stability and attractiveness of the solutions for the stochastic delayed system. Moreover, we provide sufficient and necessary condition ensuring the existence of the optimization solution for the optimization problem and obtain the optimal harvesting effect and the maximum of sustainable yield. Lastly, two numerical examples and their simulations are given to illustrate the effectiveness of our results.

中文翻译：

---

具有分布时滞的混合随机浮游植物-浮游动物-鱼类模型的全局分析和最佳收获

本文以浮游植物-浮游动物-浮游动物-鱼为模型，建立了包含布朗运动和马尔可夫链的时滞和混杂随机噪声的浮游植物-浮游动物-浮游生物-鱼，并提出了追求最大经济收入的最优捕捞问题。通过对某些系统参数进行全局分析，我们研究了关于随机时滞系统解的适定性，有界性，持久性，消光性，稳定性和吸引力的动力学行为。此外，我们提供了充分和必要的条件，以确保存在针对优化问题的优化解决方案，并获得最佳的收获效果和最大的可持续产量。最后，给出了两个数值示例及其仿真，以说明我们的结果的有效性。