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We consider a nutrient-phytoplankton model with a Holling type II functional response and a time delay. By selecting the time delay used as a bifurcation parameter, we prove that the system is stable if the delay value is lower than the critical value but unstable when it is above this value. First, we investigate the existence and stability of the equilibria, as well as the existence of Hopf bifurcations. Second, we consider the direction, stability, and period of the periodic solutions from the steady state based on the normal form and the center manifold theory, thereby deriving explicit formulas. Finally, some numerical simulations are given to illustrate the main theoretical results.

Ecological systems are characterized by the relationships between species and their natural environment. One of the key factors that affect population dynamics is predator. Due to its universal existence and importance in nature, the dynamic interactions between predators and their prey have remained one of the dominant themes in ecological population dynamics since the origin of this discipline [

Phytoplankton has vital effects in aquatic ecosystems where it plays a significant role as the base of the food chain. Phytoplankton controls the global carbon cycle which has an important effect on climate regulation [

Nutrient-phytoplankton systems have a very important effect on aquatic ecosystems as part of predator-prey systems. Recently, many researchers have investigated a predator-prey model that involves nutrients, phytoplankton, and zooplankton [

Huppert et al. [

Mäler [

In this study, based on the idea of Huppert et al. [

Huppert et al. [

In this study, in order to investigate the effects of a time delay on the system, we selected the delay

In this section, we consider the existence of the positive equilibria for system (

According to

According to

First, when the system comprises only nutrients without phytoplankton in the system, according to the vertical isocline,

In addition, according to the vertical isocline,

In the analysis above, we only prove that positive equilibria exist in the system when certain conditions are satisfied but we do not indicate that positive equilibria do not exist when the conditions are not satisfied.

The Jacobian matrix of the system without time delay at the equilibrium

The index of equilibrium

From the above discussion, we find a positive equilibrium in the system without time delay under some preconditions, defined by

When

The linearization of (

If

There are two eigenvalues where

In the following, we focus on the existence of a local Hopf bifurcation at positive equilibrium

Its characteristic equation is

For system (

If

When

Let

When

However, we can solve the first formula in (

For arbitrary

Differentiating characteristic equation (

From Lemmas

In the previous section, we obtained some conditions for the occurrence of Hopf bifurcations. In this section, we consider the direction, stability, and period of the periodic solutions from the steady state, as well as deriving the explicit formulae that determine these factors at the critical value

If we let

In the following discussion, we omit the

In the following, we first construct the coordinates to describe the center manifold

if

if

if

According to the analysis above, the positive equilibrium does not always exist. Thus, based on the numerical technology, we can obtain the positive equilibria that exist under certain conditions. When some parameters are set, we select

(a) The intersections between the vertical isocline and horizontal isocline with different nutrient concentrations

Based on the two previous sections, we know that the stability of the boundary equilibria

The solution for nutrients where

Furthermore, in order to study the relationship between nutrients and phytoplankton, we set the parameter

(a) Solutions for nutrients and phytoplankton where

In this study, we considered a biological system that comprises nutrients and phytoplankton, where we focused on the effects of a delay on the system dynamics.

In Section

Based on the computer simulation, we showed that the relationship between nutrients and phytoplankton is mutually constrained. Thus, an abundance of nutrients leads to a major increase of phytoplankton, which depleted the nutrients via consumption. Due to the effect of time delay, the system shows oscillation. It reveals that time delay has a vital effect on the system.

The main limitations of this study are that the main theoretical results were confirmed by numerical simulations with hypothetical parameter values. Thus, we aim to obtain some real data to confirm the validity of our system. Furthermore, we did not consider the diffusion and advection of nutrients and phytoplankton; therefore, these aspects should be considered comprehensively in our future research.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 31170338), by the Key Program of Zhejiang Provincial Natural Science Foundation of China (Grant no. LZ12C03001), and by the National Key Basic Research Program of China (973 Program, Grant no. 2012CB426510).