Impact of Cell Size Effect on Nutrient-Phytoplankton Dynamics

In this paper, a nutrient-phytoplankton model, which is described by a system of ordinary differential equations incorporating the effect of cell size, and its corresponding stochastic differential equation version are studied analytically and numerically. A key advantage of considering cell size effect is that it can more accurately reveal the intrinsic law of interaction between nutrient and phytoplankton. The main purpose of this paper is to research how cell size affects the nutrient-phytoplankton dynamics within the deterministic and stochastic environments. Mathematically, we show that the existence and stability of the equilibria in the deterministic model can be determined by cell size: the smaller or larger cell size can lead to the disappearance of the positive equilibrium, but the boundary equilibrium always exists and is globally asymptotically stable; the intermediate cell size is capable to drive the positive equilibrium to appear and be globally asymptotically stable, whereas the boundary equilibrium becomes unstable. In the case of the stochastic model, the stochastic dynamics including the stochastic extinction, persistence in the mean, and the existence of ergodic stationary distribution is found to be largely dependent on cell size and noise intensity. Ecologically, via numerical simulations, it is found that the smaller cell size or larger cell size can result in the extinction of phytoplankton, which is similar to the effect of larger random environmental fluctuations on the phytoplankton. More interestingly, it is discovered that the intermediate cell size is the optimal size for promoting the growth of phytoplankton, but increasing appropriately the cell size can rapidly reduce phytoplankton density and nutrient concentrations at the same time, which provides a possible strategy for biological control of algal blooms.