Impact of Cell Size Effect on Nutrient-Phytoplankton Dynamics

Tiancai Liao,Hengguo Yu,Chuanjun Dai,Min Zhao

doi:10.1155/2019/8205696

Abstract

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.

Highlights

Phytoplankton blooms, which can negatively a ect the aquatic ecosystems, human health, marine sheries, and local economy, are growing in frequency, magnitude, and duration globally in recent years [1, 2]
Many ecologists, biologists, and biomathematicians increasingly realize that a mathematical model is a powerful tool for exploring biological and physical processes on the dynamic mechanisms of phytoplankton growth in relation to different factors qualitatively and quantitatively [12, 13], as the research results can help us to find out the key factors that may induce the blooms of phytoplankton but are difficult to predict in the experimental analysis, to answer that what the growth mechanism of phytoplankton is, to predict possibly when the phytoplankton blooms will occur, and to determine the optimal strategy for possible control of phytoplankton blooms [14,15,16,17,18,19,20,21,22,23,24,25]. e application of mathematical models in other research fields, such as investigating other predator-prey dynamics or infectious disease dynamics, can be found in [26,27,28,29,30,31,32,33,34,35,36,37]
Many field and laboratory evidences indicated that the plankton body size, especially the cell size of phytoplankton, plays a key role in the metabolism, growth, and interaction of phytoplankton [6, 51, 54]

Summary

Introduction

Phytoplankton blooms, which can negatively a ect the aquatic ecosystems, human health, marine sheries, and local economy, are growing in frequency, magnitude, and duration globally in recent years [1, 2]. By following the method in [65], we assume that stochastic environmental fluctuations mainly affect the growth of phytoplankton μ(x) In this way, μ(x) changes to a random variable μ(x), and μ(x) μ(x) + δB_(t), where B(t) is a standard Brownian motion defined on a complete probability space (Ω, F, (Ft)t≥0, P), B_(t) indicates the white noise, and δ represents the intensity of the white noise. We study the existence and uniqueness of the solution, stochastic extinction, persistence in the mean, and a unique ergodic stationary distribution of model (3). Following the research [65], we prove that model (3) has a unique global positive solution.

Dε ε

Mλ MIε

Numerical Results

Stochastic Deterministic

Conclusion