Dynamical behavior of a nutrient–plankton model with Ornstein–Uhlenbeck process and nutrient recycling

In this paper, we investigate the dynamics of a nutrient–phytoplankton–zooplankton model with nutrient recycling, in which the maximal nutrient uptake rate and maximal zooplankton ingestion rate are given by a continuous, mean-reverting, stochastic process. We first prove the existence and uniqueness of the global solution. Then conditions for the extinction of plankton are derived in two cases. Moreover, we establish sufficient condition for the existence of stationary distribution by constructing appropriate Lyapunov functions. It is worth noting that we further give the exact expression of density function around the positive equilibrium of deterministic system. Finally, some simulations are carried out to demonstrate our theoretical results.