The effects of additional food and environmental stochasticity on the asymptotic properties of a nutrient–phytoplankton model

In this paper, we investigate the deterministic and stochastic features of an additional food provided nutrient–phytoplankton model, and discuss the effects of additional food and environmental stochasticity (noise) on its asymptotic properties. A detailed global dynamics analysis of the deterministic model is firstly provided, and then the critical conditions for persistence and extinction of phytoplankton is found. For the stochastic model, we establish some sufficient conditions for the exponential extinction of phytoplankton and the existence of stationary distribution. By solving the Fokker–Planck equation, the explicit expression of local probability density of the stochastic model is further derived. With the help of probability density, we obtain the mean, variance, skewness and kurtosis of the biomass of phytoplankton, and numerically find the additional food and stochasticity have complex effects on these statistical properties. Both theoretical and numerical findings indicate that the additional food source is favorable for the growth of phytoplankton, whereas the stochasticity is adverse to its growth. Moreover, the additional food source and stochasticity play important roles in the formation of phytoplankton bloom.