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This paper brings up the idea of a biological economic system with time delay in a polluted environment. Firstly, by proper linear transformation and parametric method, the singular time-delay systems are transformed to differential time-delay systems. Then, using center manifold theory and Poincare normal form method, the direction of Hopf bifurcation and the stability and period of its periodic orbits are analysed. At last, we have performed numerical simulation to support the analytical results.

Environmental pollution has been increasingly influencing the biological systems. In order to investigate the development and dynamics of population of the biological systems, it is necessary to consider the factor of pollution when establishing a mathematical model. In addition, delay is also a kind of common phenomenon in reality and it has great influence on the dynamic behavior of system. Therefore, the delay differential equations are needed to describe the system when the influence of time delay is considered. Time delay can lead to the imbalance of the system and the emergence of a variety of bifurcations, among which Hopf bifurcation is the most common. The properties of Hopf bifurcation consist of the stability of the periodic solutions, the direction of bifurcations, the period, and so forth. In recent years, the theory of delay system has gradually been generalised to many important fields by domestic and foreign scholars, including the applications in circuit communiment-system [

There are many kinds of research methods for delay differential systems. Among them, the most commonly used ones [

A single-creature model with stage structure is investigated in [

A single-creature model in the polluted environment is investigated in [

Considering the need of a period of time when the immature creatures change into mature creatures, based on system (

The positive equilibrium of system (

System (

Considering system (

When

When

The stability of the unique positive equilibrium point of system

In order to make the research more convenient, system (

In order to investigate the local stability of the positive equilibrium point, make the following transformation on system (

In order to derive the formula determining the properties of the positive equilibrium of system (

The characteristic equation of system (

Separate the imaginary part and real part from (

According to (

The properties of Hopf bifurcation are determined by (

The direction of Hopf bifurcation: the Hopf bifurcation is supercritical (resp., subcritical) when

The stability of the bifurcating periodic solutions: the bifurcation periodic solutions are stable (resp., unstable) if

The period of the bifurcating periodic solutions: the period increases (resp., decreases) if

Considering system (

By Riesz representation theorem, there exist

Then, (

For

It is easy to calculate that

Next, the coordinates to describe the center manifold

We have the following results:

By selecting some related data from China environment protection database and doing the appropriate treatment, the following parameters can be obtained [

Then, system (

Dynamical responses of system (

Dynamical responses of system (

The bifurcation diagram of system (

Based on the mathematical biology theory, the Hopf bifurcation theory of differential system, and the singular system theory, this paper considers a singular biological economic system with time delay in a polluted environment. The Hopf bifurcation occurs at the positive equilibrium with the change of time delay. We can proof that time delay has a great influence on the development of the population and economic development. In order to make the population development sustainable and ensure the maximization of economic benefits, the properties of Hopf bifurcation is necessary to be studied.

The authors declare that there are no competing interests regarding the publication of this paper.

This work was supported by National Natural Science Foundation of China under Grant no. 61273008, National Natural Science Foundation of Liaoning Province under Grant no. 2015020007, Science and Technology Research Fund of Liaoning Education Department under Grant no. L2013051, and Jiangsu Planned Projects for Postdoctoral Research Funds under Grant no. 1401044, and Doctor Startup Fund of Liaoning Province under Grant no. 20141069.