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 [1] and Volterra [2] proposed the well- known predator-prey model [3]-[6]. Recently, a new method of central manifold has been developed to study the stability of delay induced bifurcation. In this paper, we study the following system:

with

where dot means differentiation with respect to time, and are the prey and predator population densities, respectively. Parameter is the specific growth rate of prey in the absence of predation and without environment limitation. is environmental carrying capacity. The functional response of the predator is of Holling’s type with. And all parameters involved with the model are positive.

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 and, where

The two distinct interior equilibrium points exist whenever

holds.

We transform the interior equilibrium to the origin by the transformation,. Respectively, we still denote and by and. Thus, system (1) is transformed into

First, we give the condition such that is locally stable. For simplicity, we denote

The characteristic polynomial of is

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 and hold, the interior equilibrium point of system (1) is locally asymptotically stable.

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

Theorem 2. If the system (1) satisfies the hypothesis and holds, then there exists a critical point such that the positive equilibrium point is locally asymptotically stable for and unstable for, where is defined in Equation (14).

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 and a positive real such that

where is defined in Equation (5).

If is a root of Equation (7), then we have

which leads to

Let, then Equation (9) takes the form

Since holds, we have, which leads to. Thus Equation (10) has at least one positive root, which leads to

Set as the root of Equation (8) with, we have

where

Then are a pair of simple purely imaginary roots of Equation (8) with, and we have

Then by the Butler’s Lemma, is unstable for. On the other hand, if, then Equation (7) have no roots on the imaginary axis. Then Equation (7) for, only has negative real part roots, which implies that is locally asymptotically stable for.

Theorem 3. If the system (1) satisfies the hypothesis and, then the system (1) undergoes Hopf bifurcation at when.

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 as the bifurcation parameter.

Let, then is the Hopf bifurcation value of system (3). Rescale the time by to normalize the delay. The periodic solution of system (3) is equivalent to the solution of the following system

We define as nonnegative integer, define as follows

Rewrite system (18) to

where

We use the method which is based on the center manifold and normal form theory, and define. Then the system (19) is transformed into a functional differential equation as

where and are respectively represented by

and

where. By the Riesz representation theorem, there exist a matrix, whose elements are of bounded variation functions such that

In fact, we can choose

where is the Dirac delta function. For, we define

and

Thus system (21) is equivalent to

where for.

For, define

and a bilinear inner product

where. Then and are adjoint operators. From the discussion in Theorem 2, we know that are eigenvalues of and therefore they are also eigenvalues of.

Suppose is the eigenvector of corresponding to. Thus,. From the definition of we have

Then we have

Similarly, let be the eigenvector of corresponding to. Then by and the definition of, we obtain

Therefore

In order to ensure, we need to determine the value of, from Equation (29) we have

Then we can choose such as

where is the conjugate complex number of.

Next we will compute the coordinate to describe the center manifold at. Let be the solution of Equation (27) when. Define

On the center manifold, we have, where

and are local coordinates for the center manifold in the direction of and. Note that is real if is real. We only concern with the real solutions. For solution of Equation (27), since and Equation (35), we have

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 stands for higher order terms, and

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

Since depends on and, we need to find the values of and. From Equation (21) and Equation (35), we have

where

From Equation (36), we have

It follows from Equation (39) that

Comparing the coefficients of and from Equation (45) and Equation (46), we get

Then for, we have

Comparing the coefficients of and between Equation (44) and Equation (48), we get

From the definition of and Equation (49), we have

Since, we obtain

where is a constant vector. Similarly, we have

where is a constant vector. Now, we shall find the values of and. From the definition of and Equation (50), we have

and

where. In view of Equation (43), we induce that when.

Then we have

Comparing both sides of Equation (56), we obtain

where and are respectively the coefficients of and of. Thus we have

where.

Since is the eigenvalue of and is the corresponding eigenvector, we get

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

that is

where

Thus, , and is the value of the determinant, where is formed by replacing the th column vector of by another column vector for. In a similar way, we have

where

Thus, where and is the value of the determinant that is formed by replacing the th column vector of by another column vector for. Therefore, we can determine and from Equation (51) and Equation (52). Furthermore, we can easily compute.

Then the Hopf bifurcating periodic solutions of system (1) at on the center manifold are determined by the following formulas

Here determines the direction of Hopf bifurcation. If, then the Hopf-bifurcation is forward(backward) and the bifurcating periodic solutions exist for. Again determines the stability of the bifurcating periodic solutions. The bifurcating periodic solutions are stable (unstable) if. determines the period of periodic solutions: the period increases (decreases) if. Therefore, we have the following results.

Theorem 4. The Hopf bifurcation of the system (1) occurring at when is forward (backward) if and the bifurcating periodic solutions on the center manifold are stable (unstable) if .

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 is a critical value for the time delay. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated.

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