Dynamics Induced by Delay in a Nutrient-Phytoplankton Model with Multiple Delays

Abstract

A nutrient-phytoplankton model with multiple delays is studied analytically and numerically. The aim of this paper is to study how the delay factors influence dynamics of interaction between nutrient and phytoplankton. The analytical analysis indicates that the positive equilibrium is always globally asymptotically stable when the delay does not exist. On the contrary, the positive equilibrium loses its stability via Hopf instability induced by delay and then the corresponding periodic solutions emerge. Especially, the stability switches for positive equilibrium occur as the delay is increased. Furthermore, the numerical simulations show that periodic-2 and periodic-3 solutions can appear due to the existence of delays. Numerical results are consistent with the analytical results. Our results demonstrate that the delay has a great impact on the nutrient-phytoplankton dynamics.

Impact of Cell Size Effect on Nutrient-Phytoplankton Dynamics

“…However, in the real aquatic environments, the growth of phytoplankton is generally in uenced by many biotic and abiotic factors, such as light [5], cell size [6], climate [7], grazer [8], carbon dioxide [9], nutrient [10], and temperature [11], which make it difficult to determine a clear mechanism of phytoplankton blooms only through experimental studies. Actually, 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].…”

“…In 1949, Riley et al [38] first used the mathematical model to study the nutrient-plankton dynamics, which leads to the formulation of a growing number of mathematical models to describe the nutrient-phytoplankton dynamics or nutrient-plankton dynamics, and many dynamic mechanisms of phytoplankton growth response to various factors have been revealed [14,15,22,[39][40][41][42][43][44][45][46][47]. For example, Chen et al [43] showed that the proper control of the ratio for nitrogen and phosphorus can more effectively control and eliminateblue-green algae blooms.…”

“…Most of the existing mathematical modeling studies on the nutrient-phytoplankton dynamics or nutrient-plankton dynamics usually assume that the nutrient uptake rate, phytoplankton sinking rate, phytoplankton growth rate, and so on are independent of cell size [12,14,15,22,[39][40][41][42][43][44][45], which have, in part, been considered to be unrealistic because the factor of cell size is capable to significantly affect the dynamic mechanisms of phytoplankton growth [48]. In fact, cell size is a master functional trait that virtually affects every aspect of phytoplankton biology at the cellular, population, and community levels [49].…”

Stability and bifurcation analysis of a nutrient‐phytoplankton model with time delay

“…Furthermore, since time delay indeed exists in various growth process of phytoplankton, for example, nutrient convert by phytoplankton, and mathematical models can provide quantitative insights into the dynamics of phytoplankton growth systems. Therefore, mathematical models interactions with time delay are more preferred used in study of phytoplankton growth . In addition, Dai et al indicated that patchy spatial patterns can emerge because of a time delay in the growth response of phytoplankton.…”

“…Therefore, mathematical models interactions with time delay are more preferred used in study of phytoplankton growth. [40][41][42][43][44] In addition, Dai et al 45 indicated that patchy spatial patterns can emerge because of a time delay in the growth response of phytoplankton.…”