Stochastic periodic solution of a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation

Phytoplankton allelopathy is an ecological phenomenon that concerns the interaction among toxic-producing phytoplankton. Recently, researchers pay great attention to whether the cyclic outbreaks of the harmful algal blooms are related with the allelopathy in a random fluctuating environment. In this paper, we are particularly interested in a non-autonomous toxic-producing phytoplankton allelopathy model with environmental fluctuation. For the model, we first consider the existence of the global positive solution and the boundary periodic solution. Then, by using Khasminskii's method and Lyapunov function, we derive the sufficient conditions for the existence of the nontrivial positive stochastically periodic solution. Our results show that the allelopathic effect plays an important role in the existence of the stochastic periodic solution, for example it can lead to the decrease of the peaks of the cyclic outbreaks of the harmful algal blooms. Numerical simulations are carried out to support our theoretical results.