Dynamics of a stochastic nutrient-plankton system with seasonal fluctuation and toxin-producing phytoplankton

This paper focuses on a stochastic autonomous nutrient-plankton system, incorporating toxin-producing phytoplankton and white noise, along with its non-autonomous counterpart. For the autonomous system, we derive sufficient conditions for the stochastic extinction and persistence in the mean of the system, and further prove the existence of a unique ergodic stationary distribution. For the non-autonomous system, we investigate the existence of a unique positive stochastic periodic solution. Ecologically, through numerical simulations, we demonstrate that the constant nutrient input concentration, toxic phytoplankton and environmental noise exert significant influences on phytoplankton growth. Notably, increasing the nutrient input concentration can enhance the likelihood of phytoplankton blooms, whereas elevated levels of toxic substances released by phytoplankton or intensified random environmental disturbances can mitigate or even terminate such blooms. Furthermore, the dilution rate, the constant input of nutrient concentration and toxin release rate can regulate the skewness of stationary distribution for plankton and nutrient.