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In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a delayed predator-prey system with harvesting. We show that the positive equilibrium point is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Furthermore, using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation.

Due to its universal existence and importance, the study on the dynamics of predator-prey systems is one of the dominant subjects in ecology and mathematical ecology since Lotka [

with

where dot means differentiation with respect to time

The purpose of this paper is to investigate the effect of time-delay on a modified predator-prey model with harvesting. We discussed the existence of Hopf bifurcation of system (1) and the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are given.

After some calculations, we note system (1) has no boundary equilibria. However, it is more interesting to study the dynamical behaviors of the interior equilibrium points

The two distinct interior equilibrium points

holds.

We transform the interior equilibrium

First, we give the condition such that

The characteristic polynomial of

where

Now we consider the locally asymptotically stabiliy of the system without time-delay. Then we have

If

holds, then it follows from the Routh-Hurwitz criterion that two roots of (6) have negative real parts.

Theorem 1. If

In the section, we study whether there exists periodic solutions of system (1) about the interior equilibrium point

Theorem 2. If the system (1) satisfies the hypothesis

By the use of the instability result for the delayed differential Equations, in order to prove the instability of the equilibrium point, it is sufficient to show that there exists a purely imaginary

where

If

which leads to

Let

Since

Set

where

Then

Then by the Butlerâ€™s Lemma,

Theorem 3. If the system (1) satisfies the hypothesis

Proof. The Hopf bifurcation will be proved if we can show that

From Equation (7), we have

Substituting Equation (8) into Equation (16), we have

Substituting Equation (14) into the above equation, we have

Therefore, the transversality condition is satisfied. Therefore Hopf bifurcation occurs at

In this section, we analyze the direction and stability of the Hopf bifurcation of (3) obtained in Theorem 3 by taking

Let

We define

Rewrite system (18) to

where

We use the method which is based on the center manifold and normal form theory, and define

where

and

where

In fact, we can choose

where

and

Thus system (21) is equivalent to

where

For

and a bilinear inner product

where

Suppose

Then we have

Similarly, let

Therefore

In order to ensure, we need to determine the value of

Then we can choose

where

Next we will compute the coordinate to describe the center manifold

On the center manifold

We rewrite above equation as

where

From Equation (35) and Equation (36), we obtain that

Substituting Equation (23) and Equation (40) into Equation (39), we have

where

Comparing Equation (39) and Equation (41), we get

Since

where

From Equation (36), we have

It follows from Equation (39) that

Comparing the coefficients of

Then for

Comparing the coefficients of

From the definition of

Since

where

where

and

where

Then we have

Comparing both sides of Equation (56), we obtain

where

where

Since

Therefore, substituting Equation (53) and Equation (59) into Equation (60), we have

that is

where

Thus

where

Thus

Then the Hopf bifurcating periodic solutions of system (1) at

Here

Theorem 4. The Hopf bifurcation of the system (1) occurring at

This paper introduces modified time-delay predator- prey model. Then we study the Hopf bifurcation and the stability of the system. Our results reveal the conditions on the parameters so that the periodic solutions exist surrounding the interior equilibrium point. It shows that

This project is jointly supported by the National Natural Science Foundations of China (Grant No. 61074192). We also would like to thank the anonymous referees which have improved the quality of our study.

Yang Ni,Yan Meng,Yiming Ding, (2015) Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting. Journal of Applied Mathematics and Physics,03,771-780. doi: 10.4236/jamp.2015.37094