The impact of plankton body size on phytoplankton-zooplankton dynamics in the absence and presence of stochastic environmental fluctuation

Abstract

In this paper, we study the dynamics of a stochastic phytoplankton-zooplankton (PZ) model with phytoplankton cell size and zooplankton body size. In the case of PZ model without stochastic environmental fluctuations, we provide the positivity and boundedness of the solutions, investigate the dissipativity and permanence of the model, prove the existence of Hopf bifurcation, and give the local and global stability of the boundary and positive equilibria. In the case of PZ model with stochastic environmental fluctuations, the stochastic dynamics including a unique ergodic stationary distribution, stochastic permanence, stochastic extinction and persistence in the mean are explored in detail. Based on the theoretical analysis above, via numerical simulations, we find that the increase of environmental capacity can not only destabilize the deterministic model via Hopf bifurcation and induce periodic solutions, but can also stabilize the deterministic model by rejecting the periodic solutions. Interestingly, the increase of phytoplankton cell size or zooplankton body size can stabilize the deterministic model by excluding the periodic solutions induced by environmental capacity. Additionally, it is worth emphasizing that the small phytoplankton cell size can lead to the inability of plankton to survive in both deterministic and random environments, while the small zooplankton body size can destabilize the deterministic model and induce periodic solutions. Furthermore, it should be noted that the large phytoplankton cell size can weaken the effect of random environmental disturbance, but large zooplankton body size can not. These results may provide new insights in understanding the complex dynamics of phytoplankton-zooplankton models.