Abstract

In this paper, we analytically and numerically study the dynamics of a stochastic toxin-producing phytoplankton–fish system with harvesting. Mathematically, we give the existence and stability of the positive equilibrium in the deterministic system (i.e., the system without environmental noise fluctuations). In the case of the stochastic system (i.e., the system with environmental noise fluctuations), in addition to the existence and uniqueness of the positive solution, we provide the properties of the stochastic dynamics including the stochastic extinction and persistence in the mean, almost sure permanence and uniform boundedness, and the existence of ergodic stationary distribution for the phytoplankton and fish. Ecologically, via numerical analysis, we find that (1) the small random environmental fluctuations can ensure the persistence of phytoplankton and fish, but the larger one can result in the extinction of these populations; (2) an appropriate increase in harvest rate can reduce the irregular random variation of phytoplankton and fish; (3) the increase of toxin liberate rate is capable to decrease the height of probability density function of phytoplankton. These results may help us to better understand the phytoplankton–fish dynamics.

中文翻译：

---

具有收获的随机产生毒素的浮游植物－鱼类系统的动力学分析

摘要

在本文中，我们通过分析和数值研究了收获时随机产生毒素的浮游植物－鱼类系统的动力学。在数学上，我们给出确定性系统（即，没有环境噪声波动的系统）中正平衡的存在性和稳定性。在随机系统（即具有环境噪声波动的系统）的情况下，除了正解的存在和唯一性之外，我们还提供了随机动力学的性质，包括随机灭绝和持久性，几乎可以肯定永久性和均匀有界性，以及浮游植物和鱼类的遍历平稳分布的存在。通过数值分析，我们发现（1）较小的随机环境波动可以确保浮游植物和鱼类的持久性，但是较大的波动可能导致这些种群的灭绝；（2）适当提高收获率可以减少浮游植物和鱼类的不规则随机变化；（3）毒素释放速率的增加能够降低浮游植物概率密度函数的高度。这些结果可能有助于我们更好地了解浮游植物－鱼类的动态。