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A delayed Lotka-Volterra predator-prey system with time delayed feedback is studied by using the theory of functional differential equation and Hassard’s method. By choosing appropriate control parameter, we investigate the existence of Hopf bifurcation. An explicit algorithm is given to determine the directions and stabilities of the bifurcating periodic solutions. We find that these control laws can be applied to control Hopf bifurcation and chaotic attractor. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.

Lotka-Volterra system is one of the most classical and important systems in the field of mathematical biology. Since the word of Volterra, there have been extensively detailed investigations on Lotka-Volterra system including stability, attractivity, persistence, periodic oscillation, bifurcation and chaos (see [

Reference [

Waveform plot and phase plot of system (

In the sense of biology, chaotic behavior sometimes is to the disadvantage of virtuous cycle and develop of the ecosystem, so we want to control this chaos phenomenon and create periodic orbits. So far, many researchers have proposed chaos control schemes in recent years [

To the end of controlling chaos in system (

This paper is organized as follows. In Section

In this section, by analyzing the characteristic equation of the linearized system of system (

Clearly, under the hypothesis (H1), system (

Denote

Equation (

From (

Define

Through tedious calculating, we can get

In order to get the main results, it is necessary to make the following assumption.

Note that when

Consider the exponential polynomial

From Lemma

Suppose that

The positive equilibrium

system (

In this section, we obtain the conditions under which a family periodic solutions bifurcate form the steady state at the critical value of

For the sake of simplicity of notation, we denote the critical values

Obviously,

Suppose that

Similarly, let

In order to assure

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

In order to assure the value of

Notice that

In what follows, we will compute

From (

Substituting (

Similarly, substituting (

Thus, we can determine

In this section, we present numerical results to verify the analytical predictions obtained in the previous sections and use the delayed-feedback controller to control the Hopf bifurcation and chaos of system (

For the convenience of the calculation, we take the parameters of system (

From [

The bifurcation diagram of

Under the delayed-feedback control, if we choose

By comparing the two bifurcation plots of Figure

Waveform plot and phase plot of system (

Waveform plot and phase plot of system (

Waveform plot and phase plot of system (

In this paper, we have studied a delayed Lotka-Volterra predator-prey system with time delayed feedback by using the theory of functional differential equation and Hassard’s method. By analyzing the corresponding characteristic equations, the local stability of the positive equilibrium of system (

We have obtained the estimated length of gestation delay which would not affect the stable coexistence of both prey and predator species at their equilibrium values. The existence of Hopf bifurcation for system (

As the estimated length of delay to preserve stability and the critical length of time delay for Hopf-bifurcation are dependent upon the system parameters, it is possible to impose some control, which will prevent the possible abnormal oscillation in population density. Our results show that if we choose some appropriate parameters, the oscillation can be controlled to a stable equilibrium or a stable periodic orbit; that is to say, we can achieve the ecological equilibrium by adjusting the capture (or release) level.

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

This research is supported by Science and Technology Department of Henan Province (no.122300410417), Education Department of Henan Province (no.13A110108), 2013 Narure Science Foundation of Ningxia (no. NZ13096), 2013 Higher educational scientific research project of Ningxia (no. NGY2013086).