Dynamical analysis of a stochastic toxin-producing phytoplankton–fish system with harvesting

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.