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A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations.

Phytoplankton plays a very important role as the first trophic level in aquatic ecosystems. To describe the complex dynamics of phytoplankton populations, the dynamic relationship between phytoplankton and nutrients has been investigated theoretically for a long time, as well as experimentally. Since the pioneering work of Riley et al. [

The model proposed by Taylor et al. [

In this study, we consider an approximated model of system (

Taylor et al. [

Time delay is known to play important roles in biological dynamical systems, which have been studied by many researchers in recent years [

The remainder of this paper is organized as follows. In Section

In this section, we mainly consider the existence and stability of the nonnegative equilibria of system (

There are two solutions:

We linearize (

There are two eigenvalues where

The equilibrium

We recall that

Similarly, we linearize (

The corresponding characteristic equation is

When

When

Let

Hence,

Let

However, we can solve the first formula in (

Substituting

For

This implies that all of the roots cross the imaginary axis at

Based on the above, we have the following theorem.

The interior equilibrium

In this section, we consider the direction, stability, and period of the periodic solutions from the steady state using the method introduced by Hassard et al. [

Then (

According to the Riesz representation theorem, there exists a matrix

In particular, we can select

For

Then, (

For

For

Suppose that

With the condition

Similarly, let

Given the condition

In the following, we compute the coordinates to describe the center manifold

On the center manifold

Then,

Comparing the coefficients, we have

Note that

Thus, according to the definition of

In the final formula,

Consider

Comparing the coefficients, we have

From (

Solving the formula, we have

Using the same method, we can obtain

For

According to (

Substituting (

From the above, we already know

if

if

if

According to Pardo [

Consider

In this study, we are interested in the interior equilibrium, though the interior does not always exist. Thus, the field where there exists an interior equilibrium is given when

(a) The field where two equilibria exist; (b) the relationship of

Based on Figure

Solution for phytoplankton for

Furthermore, to investigate the relationship between nutrients and phytoplankton, the numerical solutions for nutrients and phytoplankton are shown in Figure

(a) The solutions for nutrients and phytoplankton for

To determine how

The stable state bifurcation: (a)

In this study, we investigate a biological system and consider the effect of time delay. Our results show that the time delay plays a vital role in system (

Pardo [

In our study, we consider the model with delay as (

The limitations of our study are that we only consider a discrete delay in the phytoplankton increase and we assumed that the recycling of nutrients is instantaneous. A model with delayed recycling would be more complicated but more similar to reality. Moreover, the model we considered is only an approximate model based on (

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. 31370381), 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).