Hopf Bifurcation Analysis for a Modified Time-Delay Predator-Prey System with Harvesting ()
1. Introduction
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:
(1)
with
(2)
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.
2. Positive Equilibrium and Locally Asymptotically Stabiliy
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
![](//html.scirp.org/file/57638x13.png)
![](//html.scirp.org/file/57638x14.png)
The two distinct interior equilibrium points
exist whenever
![](//html.scirp.org/file/57638x16.png)
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
(3)
First, we give the condition such that
is locally stable. For simplicity, we denote
(4)
The characteristic polynomial of
is
(5)
where
![]()
![]()
Now we consider the locally asymptotically stabiliy of the system without time-delay. Then we have
(6)
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.
3. Hopf Bifurcaion
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
(7)
where
is defined in Equation (5).
If
is a root of Equation (7), then we have
(8)
which leads to
(9)
Let
, then Equation (9) takes the form
(10)
Since
holds, we have
, which leads to
. Thus Equation (10) has at least one positive root, which leads to
(11)
Set
as the root of Equation (8) with
, we have
(12)
where
(13)
Then
are a pair of simple purely imaginary roots of Equation (8) with
, and we have
(14)
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
(15)
From Equation (7), we have
(16)
Substituting Equation (8) into Equation (16), we have
(17)
Substituting Equation (14) into the above equation, we have
![]()
Therefore, the transversality condition is satisfied. Therefore Hopf bifurcation occurs at
.
4. The Direction and Stability of the Hopf Bifurcation
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
(18)
We define
as nonnegative integer, define
as follows
![]()
Rewrite system (18) to
(19)
where
(20)
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
(21)
where
and
are respectively represented by
(22)
and
(23)
where
. By the Riesz representation theorem, there exist a
matrix
, whose elements are of bounded variation functions such that
(24)
In fact, we can choose
(25)
where
is the Dirac delta function. For
, we define
(26)
and
(27)
Thus system (21) is equivalent to
(28)
where
for
.
For
, define
(29)
and a bilinear inner product
(30)
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
(31)
Similarly, let
be the eigenvector of
corresponding to
. Then by
and the definition of
, we obtain
![]()
Therefore
(32)
In order to ensure, we need to determine the value of
, from Equation (29) we have
(33)
Then we can choose
such as
(34)
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
(35)
On the center manifold
, we have
, where
(36)
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
(37)
We rewrite above equation as
(38)
where
(39)
From Equation (35) and Equation (36), we obtain that
(40)
Substituting Equation (23) and Equation (40) into Equation (39), we have
(41)
where
stands for higher order terms, and
![]()
![]()
Comparing Equation (39) and Equation (41), we get
(42)
Since
depends on
and
, we need to find the values of
and
. From Equation (21) and Equation (35), we have
(43)
where
(44)
From Equation (36), we have
(45)
It follows from Equation (39) that
(46)
Comparing the coefficients of
and
from Equation (45) and Equation (46), we get
(47)
Then for
, we have
(48)
Comparing the coefficients of
and
between Equation (44) and Equation (48), we get
(49)
From the definition of
and Equation (49), we have
(50)
Since
, we obtain
(51)
where
is a constant vector. Similarly, we have
(52)
where
is a constant vector. Now, we shall find the values of
and
. From the definition of
and Equation (50), we have
(53)
and
(54)
where
. In view of Equation (43), we induce that when
.
(55)
Then we have
(56)
Comparing both sides of Equation (56), we obtain
(57)
where
and
are respectively the coefficients of
and
of
. Thus we have
(58)
where
.
Since
is the eigenvalue of
and
is the corresponding eigenvector, we get
(59)
(60)
Therefore, substituting Equation (53) and Equation (59) into Equation (60), we have
(61)
that is
(62)
where
(63)
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
(64)
where
(65)
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
(66)
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
.
5. Conclusion
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.
Acknowledgements
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.