^{1,2}

^{2,3}

^{2,3}

^{1,2}

^{1,2}

^{1,2}

^{1}

^{2}

^{3}

We analyze a nutrient-plankton system with a time delay. We choose the time delay as a bifurcation parameter and investigate the stability of a positive equilibrium and the existence of Hopf bifurcations. By using the center manifold theorem and the normal form theory, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are researched. The theoretical results indicate that the time delay can induce a positive equilibrium to switch from a stable to an unstable to a stable state and so on. Numerical simulations show that the theoretical results are correct and feasible, and the system exhibits rich complex dynamics.

Plankton is the basis of all aquatic food chains and its importance for marine ecosystems is widely recognized [

Algal blooms are a common feature of marine ecosystems, and a high nutrient concentration has an important influence on algal blooms [

Harmful algal blooms have become a serious environmental problem worldwide and have been widely studied [

To reflect maturation time, capture time, and other factors, a time delay is often included in mathematical models of population dynamics. Incorporation of a time delay can provide a better understanding of the dynamics of biological models. In recent years, many researchers have studied the fact that time delay plays important roles in biological dynamical systems [

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

In this section, we study the local stability of the positive equilibrium of system (

Using the translations

By Corollary 2.4 in the paper of Ruan and Wei [

For (

If

If

Hence, if the condition,

If (

Differentiating both sides of (

If (

On the basis of the above analysis, we have the following theorem.

Suppose that (

If

If (

From Theorem

Let

For

Let

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

On the center manifold

For

For (

According to the above analysis, we can obtain the following theorem about the properties of Hopf bifurcation.

For

If

If

If

In this section, we verify the theoretical results proved in previous sections using numerical simulations for the following parameter values:

(a) Relationship between the conversion frequency

On the basis of the above analysis, we present examples in Figures

Numerical solutions of system (

Numerical solutions of system (

Numerical solutions of system (

Numerical solutions of system (

The numerical results in Figures

(a) Bifurcation diagram of system (

In this paper, we have studied the dynamics of a nutrient-plankton system with a time delay. We have investigated the stability of the positive equilibrium and the existence of Hopf bifurcation. By using center manifold theory and the normal form method, we determined the direction of Hopf bifurcation and the stability of bifurcating periodic solutions.

In detail, we have found that if some conditions are satisfied, the phenomenon of stability switches arises from system (

Numerical simulations also showed that as the time delay further increases, the periodic solutions disappear and chaos appears. All these results not only will help in further investigating the dynamics of pelagic ecosystem in theory but also are very useful to understand the complex phenomena really happening in marine ecosystem.

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 nos. 31570364 and 31370381).