# Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

Authors: Fengying Wei; Lanqi Wu; Yuzhi Fang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτ passes through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.


---

## Body

## 1. Introduction

The classical predator-prey systems have been extensively investigated in recent years, and they will continue to be one of the dominant themes in the future due to their universal existence and importance. Many biological phenomena are always described by differential equations, difference equations, and other type equations. In general, delay differential equations exhibit more complicated dynamical behaviors than ordinary ones; for example, the delay can induce the loss of stability, various oscillations, and periodic solutions. The dynamical behaviors of delay differential equations, stability, bifurcation and chaos, and so forth have been paid much attention by many researchers. Especially, the direction and stability of Hopf bifurcation to delay differential equations have been investigated extensively in recent work (see [1–7] and references therein).After the classical predator-prey model was first proposed and discussed by May in [8], there were some similar topics, regarding persistence, local and global stabilities of equilibria, and other dynamical behaviors (see [5, 9, 10] and references therein). Recently, Song and Wei in [7] had considered a delayed predator-prey system as follows:
(1)
x
˙
(
t
)
=
x
(
t
)
[
r
1
-
a
11
x
(
t
-
τ
)
-
a
12
y
(
t
)
]
,
y
˙
(
t
)
=
y
(
t
)
[
-
r
2
+
a
21
x
(
t
)
-
a
22
y
(
t
)
]
,
where x
(
t
) and y
(
t
) were the densities of prey species and predator species at time t, respectively. The local Hopf bifurcation and the existence of the periodic solution bifurcating of system (1) was investigated in [7]. When selective harvesting was put into the predator-prey model similar to (1), Kar [11] studied two predator-prey models with selective harvesting; that is, in the first model, selective harvesting of predator species:
(2)
x
˙
(
t
)
=
x
(
t
)
[
g
(
x
)
-
y
p
(
x
)
]
,
y
˙
(
t
)
=
y
(
t
)
[
-
d
+
α
x
p
(
x
)
]
-
q
E
y
(
t
-
τ
)
,
and, in the second model, selective harvesting of prey species:
(3)
x
˙
(
t
)
=
x
(
t
)
[
g
(
x
)
-
y
p
(
x
)
]
-
q
E
x
(
t
-
τ
)
,
y
˙
(
t
)
=
y
(
t
)
[
-
d
+
α
x
p
(
x
)
]
had been considered by incorporating time delay on the harvesting term. They found that the delay for selective harvesting could induce the switching of stability and Hopf bifurcation occurred at τ
=
τ
0.Recently, Kar and Ghorai [9] had investigated a predator-prey model with harvesting:
(4)
x
˙
(
t
)
=
r
1
x
(
t
)
-
b
1
x
2
(
t
)
-
a
1
x
(
t
)
y
(
t
)
x
(
t
)
+
k
1
-
c
1
x
(
t
)
,
y
˙
(
t
)
=
y
(
t
)
[
r
2
-
a
2
y
(
t
-
τ
)
x
(
t
-
τ
)
+
k
2
]
-
c
2
y
(
t
)
.
They obtained the local stability, global stability, influence of the harvesting, direction of Hopf bifurcation and the stability to system (4). Motivated by models (1)–(4), we will consider a predator-prey system with delay incorporating harvests to predator and prey:
(5)
x
˙
(
t
)
=
x
(
t
)
[
1
-
x
(
t
)
k
1
-
a
y
(
t
)
]
-
h
1
x
(
t
)
,
y
˙
(
t
)
=
y
(
t
)
[
1
-
y
(
t
-
τ
)
k
2
+
b
x
(
t
-
τ
)
]
-
h
2
y
(
t
)
,
where x
(
t
) and y
(
t
) represent the population densities of prey species and predator species, respectively, at time t; a, b, h
1, h
2, k
1, and k
2 are model parameters assuming only positive values; k
1 measures the scale whose environment provides protection to prey x; k
2 denotes the scale whose environment provides protection to predator y; τ means the period of pregnancy; x
(
t
-
τ
) represents the number of prey species which was born at time t
-
τ and still survived at time t; h
1 and h
2 represent the coefficients of prey species and predator species, respectively. We always assume that 0
≤
h
1
≤
h
2
<
1 in this paper.The organization of the paper is as follows. The stability of the positive equilibrium and the existence of the Hopf bifurcation are discussed in Section2. The effect of harvesting to prey species and predator species is investigated in Section 3. The direction of Hopf bifurcation and stability of the corresponding periodic solution are obtained in Section 4. Numerical simulations are carried out to illustrate our results in Section 5.


## 2. Stability of Positive Equilibrium and Hopf Bifurcation

By simple computation, ifk
1
+
k
2
a
(
h
2
-
1
)
>
0 holds, system (5) admits a unique positive equilibrium E
*
(
x
*
,
y
*
):
(6)
x
*
=
(
1
-
h
1
)
[
k
1
+
k
2
a
(
h
2
-
1
)
]
1
+
a
b
(
1
-
h
1
)
(
1
-
h
2
)
,
y
*
=
(
1
-
h
2
)
[
k
2
+
k
1
b
(
1
-
h
1
)
]
1
+
a
b
(
1
-
h
1
)
(
1
-
h
2
)
.
Let x
1
=
x
-
x
*, x
2
=
y
-
y
*, and then we get the linear system of (5):
(7)
x
˙
1
(
t
)
=
-
a
11
x
1
(
t
)
-
a
12
x
2
(
t
)
,
x
˙
2
(
t
)
=
a
21
x
1
(
t
-
τ
)
-
a
22
x
2
(
t
-
τ
)
,
where a
11
=
x
*
/
(
k
1
-
a
y
*
), a
12
=
a
x
*
2
/
(
k
1
-
a
y
*
)
2, a
21
=
b
y
*
2
/
(
k
2
+
b
x
*
)
2, a
22
=
y
*
/
(
k
2
+
b
x
*
). From linear system (5) the characteristic equation is as follows:
(8)
λ
2
+
λ
(
a
11
+
a
22
e
-
λ
τ
)
+
(
a
11
a
22
+
a
12
a
21
)
e
-
λ
τ
=
0
.
Roots of system (8) imply the stability of the equilibrium E
* and Hopf bifurcation of system (5). Obviously, λ
=
0 is not a root of system (8). For τ
=
0, system (8) becomes
(9)
λ
2
+
(
a
11
+
a
22
)
λ
+
(
a
11
a
22
+
a
12
a
21
)
=
0
.
It is obvious that the root of system (9) has negative real part. Now, for τ
>
0, if λ
=
i
ω
(
ω
>
0
) is a root of (8), then we have
(10)
-
ω
2
+
i
ω
(
a
11
+
a
22
e
-
i
ω
τ
)
+
(
a
11
a
22
+
a
12
a
21
)
e
-
i
ω
τ
=
0
.
Furthermore,
(11)
-
ω
2
+
a
22
ω
sin
⁡
ω
τ
+
(
a
11
a
22
+
a
12
a
21
)
cos
⁡
⁡
ω
τ
=
0
,
a
11
ω
+
a
22
ω
cos
⁡
⁡
ω
τ
-
(
a
11
a
22
+
a
12
a
21
)
sin
⁡
ω
τ
=
0
,
which lead to polynomial equation
(12)
ω
4
+
(
a
11
2
-
a
22
2
)
ω
2
-
(
a
11
a
22
+
a
12
a
21
)
2
=
0
.
It is easy to see that (12) has one positive root
(13)
ω
=
2
2
(
a
22
2
-
a
11
2
+
Δ
)
1
/
2
,
where Δ
=
(
a
11
2
-
a
22
2
)
2
+
4
(
a
11
a
22
+
a
12
a
21
). By (11), one gets that
(14)
τ
j
=
1
ω
arc
cos
⁡
a
12
a
21
ω
2
ω
2
a
22
2
+
(
a
11
a
22
+
a
12
a
21
)
2
+
2
π
j
ω
,
a
12
a
21
ω
2
ω
2
a
22
2
+
(
a
11
a
22
+
a
12
a
21
)
2
1
ω
acco
j
=
0,1
,
…
.
Let
(15)
λ
(
τ
)
=
α
(
τ
)
+
i
ω
(
τ
)
be a pair of purely imaginary roots of (8), such that
(16)
α
(
τ
j
)
=
0
,
ω
(
τ
j
)
=
ω
.
Next, we will prove λ
(
τ
j
) meets the transversality conditions; taking the derivative of system (8) with respect to τ, one derives that
(17)
[
d
λ
d
τ
]
-
1
=
2
λ
+
a
11
+
a
22
e
-
λ
τ
-
τ
e
-
λ
τ
[
λ
a
22
+
(
a
11
a
22
+
a
12
a
21
)
]
λ
e
-
λ
τ
[
λ
a
22
+
(
a
11
a
22
+
a
12
a
21
)
]
=
2
λ
+
a
11
+
a
22
e
-
λ
τ
λ
e
-
λ
τ
[
λ
a
22
+
(
a
11
a
22
+
a
12
a
21
)
]
-
τ
λ
,
which, together with (11), leads to
(18)
Re
[
d
λ
d
τ
]
τ
=
τ
j
-
1
=
Re
{
2
λ
+
a
11
+
a
22
e
-
λ
τ
λ
e
-
λ
τ
[
λ
a
22
+
(
a
11
a
22
+
a
12
a
21
)
]
-
τ
λ
}
τ
=
τ
j
=
Re
{
2
λ
+
a
11
+
a
22
e
-
λ
τ
λ
e
-
λ
τ
[
λ
a
22
+
(
a
11
a
22
+
a
12
a
21
)
]
}
τ
=
τ
j
=
(
2
a
22
2
ω
4
+
[
2
(
a
11
a
22
+
a
12
a
21
)
2
m
m
m
m
m
m
m
l
l
l
+
(
a
11
2
-
a
22
2
)
a
22
2
(
a
11
a
22
+
a
12
a
21
)
2
]
ω
2
m
m
l
l
l
+
(
a
11
2
-
a
22
2
)
(
a
11
a
22
+
a
12
a
21
)
2
a
22
2
ω
4
+
[
2
(
a
11
a
22
+
a
12
a
21
)
2
+
(
a
11
2
-
a
22
2
)
a
22
2
]
)
m
m
l
l
×
(
ω
2
a
22
2
+
(
a
11
a
22
+
a
12
a
21
)
2
)
-
1
>
0
.
So, we have
(19)
sign
⁡
{
Re
[
d
λ
d
τ
]
}
τ
=
τ
j
>
0
.
Thus, we can obtain the following lemma.Lemma 1.
Ifk
1
+
k
2
a
(
h
2
-
1
)
>
0 holds, then the following results are true:(i)
whenτ
=
0, the positive equilibrium of E
* of system (5) is locally asymptotically stable;
(ii)
when0
<
τ
<
τ
0, the positive equilibrium of E
* of system (5) is locally asymptotically stable, and E
* is unstable when τ
>
τ
0, where ω, τ
j  (j
=
0,1
,
…) can be defined in (13), (14).


## 3. The Influence of Harvesting

Next, we will discuss the influence of the harvesting on system (5).Case 1 (only predator species is harvested).
Forh
1
=
0, and the positive equilibrium of system (5) changes to E
1
*
(
x
1
*
,
y
1
*
), where
(20)
x
1
*
=
k
1
+
k
2
a
(
h
2
-
1
)
1
+
a
b
(
1
-
h
2
)
,
y
1
*
=
(
1
-
h
2
)
(
k
2
+
k
1
b
)
1
+
a
b
(
1
-
h
2
)
,
it is obvious that y
1
*
>
0 and x
1
*
>
0 if and only if k
1
+
k
2
a
(
h
2
-
1
)
>
0. Obviously, x
1
* and y
1
* are the continuous differentiable functions with respect to h
2; then, we have
(21)
d
x
1
*
d
h
2
=
k
2
a
+
k
1
a
b
[
1
+
a
b
(
1
-
h
2
)
]
2
>
0
,
d
y
1
*
d
h
2
=
-
k
2
-
k
1
b
[
1
+
a
b
(
1
-
h
2
)
]
2
<
0
.Theorem 2.
Ifk
1
+
k
2
a
(
h
2
-
1
)
>
0 holds, then x
1
* is the monotonic increasing function of h
2, y
1
* is the monotonic decreasing function of h
2; that is, when h
2 increases, the density of prey species will increase, the density of predator species will decrease.Case 2 (only prey species is harvested).
Forh
2
=
0, and the positive equilibrium of system (5) changes to E
2
*
(
x
2
*
,
y
2
*
), where
(22)
x
2
*
=
(
1
-
h
1
)
(
k
1
-
k
2
a
)
1
+
a
b
(
1
-
h
1
)
,
y
2
*
=
k
2
+
k
1
b
(
1
-
h
1
)
1
+
a
b
(
1
-
h
1
)
,
it is obvious that y
2
*
>
0 and x
2
*
>
0 if and only if k
1
-
k
2
a
>
0. Obviously, x
2
* and y
2
* are the continuous differentiable functions with respect to h
1; then, one get that
(23)
d
x
2
*
d
h
1
=
k
2
a
-
k
1
[
1
+
a
b
(
1
-
h
1
)
]
2
<
0
,
d
y
2
*
d
h
1
=
k
2
a
b
-
k
1
b
[
1
+
a
b
(
1
-
h
1
)
]
2
<
0
.Theorem 3.
Ifk
1
-
k
2
a
>
0 holds, then x
2
* and y
2
* are the monotonic decreasing functions of h
1; that is, if h
1 increases, then the density of prey species and predator species will decrease; on the contrary, if h
1 decreases, the density of prey species and predator species will increase.Case 3 (predator species and prey species are harvested simultaneously).
Forh
1
h
2
≠
0, the mixed derivative of x
* and y
* are given by
(24)
∂
x
*
∂
h
1
=
-
[
k
1
+
a
k
2
(
h
2
-
1
)
]
[
1
+
a
b
(
1
-
h
1
)
(
1
-
h
2
)
]
2
<
0
,
∂
x
*
∂
h
2
=
(
1
-
h
1
)
[
k
2
+
k
1
b
(
1
-
h
1
)
]
a
[
1
+
a
b
(
1
-
h
1
)
(
1
-
h
2
)
]
2
>
0
,
∂
y
*
∂
h
1
=
b
(
h
2
-
1
)
[
k
1
+
a
k
2
(
h
2
-
1
)
]
[
1
+
a
b
(
1
-
h
1
)
(
1
-
h
2
)
]
2
<
0
,
∂
y
*
∂
h
2
=
-
[
k
2
+
k
1
b
(
1
-
h
1
)
]
[
1
+
a
b
(
1
-
h
1
)
(
1
-
h
2
)
]
2
<
0
.Theorem 4.
Ifk
1
+
k
2
a
(
h
2
-
1
)
>
0 is valid, then the densities of prey species and predator species will both decrease when harvesting rate h
1 increases; on the contrary, the density of prey species will increase and predator species will decrease when harvesting rate h
2 increases.


## 4. Direction and Stability of Hopf Bifurcation

Motivated by the ideas of Hassard et al. [12], by applying the normal form theory and the center manifold theorem, the properties of the Hopf bifurcation at the critical value τ
=
τ
j are derived in this section.Lett
=
s
τ, x
i
(
s
τ
)
=
x
^
i
(
s
), i
=
1,2, τ
=
τ
0
+
μ, μ
∈
R
;
τ
0 is defined by (14), we still denote x
^
i
(
s
)
=
u
i
(
s
) and s
=
t, then system (5) is transformed into functional differential equations in C
(
[
-
1,0
]
,
R
2
) as
(25)
u
˙
(
t
)
=
L
μ
(
u
t
)
+
f
(
μ
,
u
t
)
,
where u
(
t
)
=
(
u
1
(
t
)
,
u
2
(
t
)
)
T
∈
R
2, u
t
(
θ
)
=
u
(
t
+
θ
), θ
∈
[
-
1,0
], and L
μ
:
C
(
[
-
1,0
]
;
R
2
)
→
R
,
f
:
R
×
C
(
[
-
1,0
]
;
R
2
)
→
R are given by(26)
L
μ
(
ϕ
)
=
(
τ
0
+
μ
)
(
-
a
11
-
a
12
0
0
)
(
ϕ
1
(
0
)
ϕ
2
(
0
)
)
+
(
τ
0
+
μ
)
(
0
0
a
21
-
a
22
)
(
ϕ
1
(
-
1
)
ϕ
2
(
-
1
)
)
,
(27)
f
(
μ
,
ϕ
)
=
(
τ
0
+
μ
)
(
c
1
ϕ
2
2
(
0
)
+
c
2
ϕ
1
(
0
)
ϕ
2
(
0
)
+
c
3
ϕ
1
2
(
0
)
e
1
-
e
2
ϕ
2
(
0
)
c
4
ϕ
1
(
-
1
)
ϕ
2
(
0
)
-
c
5
ϕ
1
2
(
-
1
)
+
c
6
ϕ
1
(
-
1
)
ϕ
2
(
-
1
)
-
c
7
ϕ
2
(
0
)
ϕ
2
(
-
1
)
e
3
+
e
4
ϕ
1
(
-
1
)
)
,where
(28)
c
1
=
a
2
(
x
*
)
2
,
c
2
=
2
a
2
x
*
y
*
-
2
a
k
1
x
*
,
c
3
=
2
a
k
1
y
*
-
a
2
c
k
1
2
y
*
,
c
4
=
b
k
2
y
*
+
b
2
x
*
y
*
,
c
5
=
b
2
(
y
*
)
2
,
c
6
=
k
2
b
y
*
+
b
2
x
*
y
*
,
c
7
=
2
k
2
b
x
*
+
k
2
2
+
b
2
(
x
*
)
2
,
e
1
=
(
k
1
-
a
y
*
)
3
,
e
2
=
a
(
k
1
-
a
y
*
)
2
,
e
3
=
(
k
2
+
b
x
*
)
3
,
e
4
=
b
(
k
2
+
b
x
*
)
2
.By Riesz representation theorem, there exists a functionη
(
θ
,
μ
) of bounded variation for θ
∈
[
-
1,0
], such that
(29)
L
μ
ϕ
=
∫
-
1
0
‍
d
η
(
θ
,
μ
)
ϕ
(
θ
)
.
We choose
(30)
η
(
θ
,
μ
)
=
(
τ
0
+
μ
)
(
-
a
11
-
a
12
0
0
)
δ
(
θ
)
+
(
τ
0
+
μ
)
(
0
0
a
21
-
a
22
)
δ
(
θ
+
1
)
,
where δ is the Dirac delta function. For ϕ
∈
C
1
(
[
-
1,0
]
,
R
2
), we define
(31)
A
(
μ
)
ϕ
(
θ
)
=
{
d
ϕ
(
θ
)
d
θ
,
-
1
≤
θ
<
0
∫
-
1
0
‍
d
η
(
s
,
μ
)
ϕ
(
s
)
,
θ
=
0
,
R
(
μ
)
ϕ
(
θ
)
=
{
0
,
-
1
≤
θ
<
0
f
(
μ
,
ϕ
)
,
θ
=
0
.
Then, system (25) can be transformed into an operator differential equation of the form
(32)
u
˙
t
=
A
(
μ
)
u
t
+
R
(
μ
)
u
t
,
where u
t
(
θ
)
=
u
(
t
+
θ
), for θ
∈
[
-
1,0
]. For ψ
∈
C
1
(
[
0,1
]
,
(
R
2
)
*
), we define
(33)
A
*
(
μ
)
ψ
(
s
)
=
{
-
d
ψ
(
s
)
d
s
,
0
<
s
≤
1
∫
-
1
0
‍
d
η
T
(
t
,
0
)
ψ
(
-
t
)
,
s
=
0
and a bilinear inner product
(34)
〈
ψ
(
θ
)
,
ϕ
(
θ
)
〉
=
ψ
¯
T
(
0
)
ϕ
(
0
)
-
∫
-
1
0
‍
∫
ξ
=
0
θ
‍
ψ
¯
T
(
ξ
-
θ
)
d
η
(
θ
)
ϕ
(
ξ
)
d
ξ
,
where η
(
θ
)
=
η
(
θ
,
0
); then, A
(
0
) and A
* are adjoint operators. Noting that ±
i
ω
τ
0 are eigenvalues of A
(
0
), thus, they are also eigenvalues of A
*. In order to calculate the eigenvector q
(
θ
) of A
(
0
) corresponding to the eigenvalue i
ω
τ
0 and p
(
s
) of A
* corresponding to the eigenvalue -
i
ω
τ
0, let q
(
θ
)
=
(
1
,
α
)
T
e
i
ω
τ
0
θ be the eigenvector of A
(
0
) corresponding to i
ω
τ
0; then, A
(
0
)
q
(
θ
)
=
i
ω
τ
0
q
(
θ
).By the definition ofA
(
0
) and (26), (30), then,
(35)
τ
0
(
-
i
ω
-
a
11
-
a
12
a
21
e
-
i
ω
τ
0
-
i
ω
-
a
22
e
-
i
ω
τ
0
)
q
(
0
)
=
(
0
0
)
.
Thus, we can get
(36)
q
(
0
)
=
(
1
,
α
)
T
=
(
1
,
a
11
+
i
ω
-
a
12
)
T
.
Similarly, let p
(
s
)
=
D
(
1
,
β
)
T
e
i
ω
τ
0
s be the eigenvector of A
* corresponding to -
i
ω
τ
0; by similar discussion, we get β
=
(
a
11
-
i
ω
)
/
a
21
e
i
ω
τ
0.In view of standardization ofp
(
s
) and q
(
θ
); that is, 〈
p
(
s
)
,
q
(
θ
)
〉
=
1, we have
(37)
〈
p
(
s
)
,
q
(
θ
)
〉
=
D
¯
(
1
,
β
¯
)
(
1
,
α
)
T
l
m
m
-
∫
-
1
0
‍
∫
ξ
=
0
θ
‍
D
¯
(
1
,
β
¯
)
e
-
i
ω
τ
0
(
ξ
-
θ
)
d
η
(
θ
)
(
1
,
α
)
T
e
i
ω
τ
0
ξ
d
ξ
=
D
¯
{
1
+
α
β
¯
-
∫
-
1
0
‍
(
1
,
β
¯
)
θ
e
i
ω
τ
0
θ
d
η
(
θ
)
(
1
,
α
)
T
}
=
D
¯
{
1
+
α
β
¯
+
τ
0
β
¯
e
-
i
ω
τ
0
(
a
21
-
α
a
22
)
}
.
Thus, choose D
=
[
1
+
β
α
¯
+
τ
0
β
e
i
ω
τ
0
(
a
21
-
α
¯
a
22
)
]
-
1. Next, we will quote the same notation (see [13]), we first compute the coordinates to describe the center manifold C
0 at μ
=
0. Define
(38)
z
(
t
)
=
〈
p
,
u
t
〉
,
W
(
t
,
θ
)
=
u
t
(
θ
)
-
2
Re
{
z
(
t
)
q
(
θ
)
}
.
On the center manifold C
0, we have
(39)
W
(
t
,
θ
)
=
W
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
=
W
20
(
θ
)
z
2
2
+
W
11
(
θ
)
z
z
¯
+
W
02
(
θ
)
z
¯
2
2
+
⋯
z and z
¯ are local coordinates for center manifold C
0 in the direction p and p
¯; noting that W is real if u
t is real, we only consider real solution u
t
∈
C
0 of (25). Since μ
=
0, then we have
(40)
z
˙
(
t
)
=
i
ω
τ
0
z
+
p
¯
(
0
)
f
(
0
,
W
(
z
,
z
¯
,
θ
)
+
2
Re
{
z
(
t
)
q
(
0
)
}
)
=
def
i
ω
τ
0
z
+
p
¯
(
0
)
f
0
(
z
,
z
¯
)
.
We rewrite this equation as
(41)
z
˙
(
t
)
=
i
ω
τ
0
z
+
g
(
z
,
z
¯
)
,
where
(42)
g
(
z
,
z
¯
)
=
p
¯
(
0
)
f
0
(
z
,
z
¯
)
=
g
20
(
θ
)
z
2
2
+
g
11
(
θ
)
z
z
¯
+
g
02
(
θ
)
z
¯
2
2
+
⋯
.
Noting u
t
(
θ
)
=
(
ϕ
1
(
θ
)
,
ϕ
2
(
θ
)
)
T
=
W
(
t
,
θ
)
+
z
q
(
θ
)
+
z
¯
q
¯
(
θ
) and q
(
θ
)
=
(
1
,
α
)
T
e
i
ω
τ
0
θ, we have
(43)
ϕ
1
(
0
)
=
z
+
z
¯
+
W
20
(
1
)
(
0
)
z
2
2
+
W
11
(
1
)
(
0
)
z
z
¯
+
W
02
(
1
)
(
0
)
z
¯
2
2
+
⋯
,
ϕ
2
(
0
)
=
z
α
+
z
¯
α
¯
+
W
20
(
2
)
(
0
)
z
2
2
+
W
11
(
2
)
(
0
)
z
z
¯
+
W
02
(
2
)
(
0
)
z
¯
2
2
+
⋯
,
ϕ
1
(
-
1
)
=
z
e
-
i
ω
τ
0
+
z
¯
e
i
ω
τ
0
+
W
20
(
1
)
(
-
1
)
z
2
2
+
W
11
(
1
)
(
-
1
)
z
z
¯
+
W
02
(
1
)
(
-
1
)
z
¯
2
2
+
⋯
,
ϕ
2
(
-
1
)
=
z
α
e
-
i
ω
τ
0
+
z
¯
α
¯
e
i
ω
τ
0
+
W
20
(
2
)
(
-
1
)
z
2
2
+
W
11
(
2
)
(
-
1
)
z
z
¯
+
W
02
(
2
)
(
-
1
)
z
¯
2
2
+
⋯
.
From (27), (42), we obtain that
(44)
g
20
=
2
D
-
τ
0
[
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
+
β
-
e
3
m
m
m
m
m
×
(
-
c
5
e
-
2
i
ω
τ
0
+
c
6
α
e
-
2
i
ω
τ
0
m
m
m
m
m
m
l
l
+
c
4
α
e
-
i
ω
τ
0
-
c
7
α
2
e
-
i
ω
τ
0
)
1
e
1
β
-
e
3
]
,
g
11
=
D
-
τ
0
{
1
e
1
[
2
c
3
+
2
c
1
α
α
-
+
c
2
(
α
+
α
-
)
]
+
β
-
e
3
m
m
m
m
l
l
×
[
-
2
c
5
+
c
6
(
α
+
α
-
)
+
c
4
(
α
-
e
-
i
ω
τ
0
+
α
e
i
ω
τ
0
)
m
m
m
m
m
m
-
c
7
(
α
α
-
e
-
i
ω
τ
0
+
α
α
-
e
i
ω
τ
0
)
]
β
-
e
3
}
,
g
02
=
2
D
-
τ
0
[
1
e
1
(
c
3
+
c
1
α
-
2
+
c
2
α
-
)
+
β
-
e
3
m
m
m
m
m
×
(
-
c
5
e
2
i
ω
τ
0
+
c
6
α
-
e
2
i
ω
τ
0
m
m
m
m
m
m
l
l
+
c
4
α
-
e
i
ω
τ
0
-
c
7
α
-
2
e
i
ω
τ
0
)
β
-
e
3
]
,
g
21
=
D
-
τ
0
{
1
e
1
[
e
2
c
2
e
1
c
3
(
2
W
20
(
1
)
(
0
)
+
4
W
11
(
1
)
(
0
)
)
m
m
m
m
m
m
l
l
+
c
1
(
2
α
-
W
20
(
2
)
(
0
)
+
4
α
W
11
(
2
)
(
0
)
)
m
m
m
m
m
m
l
l
+
c
2
(
α
-
W
20
(
1
)
(
0
)
+
W
20
(
2
)
(
0
)
+
2
α
W
11
(
1
)
(
0
)
m
m
m
m
m
m
m
m
m
m
+
2
W
11
(
2
)
(
0
)
)
+
6
c
1
e
2
e
1
+
2
(
α
-
+
2
α
)
m
m
m
m
m
m
l
l
×
e
2
c
3
e
1
+
2
(
α
2
+
2
α
α
-
)
e
2
c
2
e
1
]
+
β
-
e
3
m
m
m
m
l
l
l
×
[
-
c
5
(
2
e
i
ω
τ
0
W
20
(
1
)
(
-
1
)
+
4
e
-
i
ω
τ
0
W
11
(
1
)
(
-
1
)
)
m
m
m
m
m
m
l
+
c
6
(
α
-
e
i
ω
τ
0
W
20
(
1
)
(
-
1
)
+
e
i
ω
τ
0
W
20
(
2
)
(
-
1
)
m
m
m
m
m
m
m
m
m
l
+
2
e
-
i
ω
τ
0
W
11
(
2
)
(
-
1
)
+
2
α
e
-
i
ω
τ
0
W
11
(
1
)
(
-
1
)
)
m
m
m
m
m
m
l
+
c
4
(
α
-
W
20
(
1
)
(
-
1
)
+
e
i
ω
τ
0
W
20
(
2
)
(
0
)
m
m
m
m
m
m
m
m
m
l
+
2
e
-
i
ω
τ
0
W
11
(
2
)
(
0
)
+
2
α
W
11
(
1
)
(
-
1
)
)
m
m
m
m
m
m
l
-
c
7
(
α
-
W
20
(
2
)
(
-
1
)
+
α
-
e
i
ω
τ
0
W
20
(
2
)
(
0
)
m
m
m
m
m
m
m
m
m
l
+
2
α
e
-
i
ω
τ
0
W
11
(
2
)
(
0
)
+
2
α
W
11
(
2
)
(
-
1
)
)
m
m
m
m
m
m
l
+
6
e
4
c
5
e
3
e
-
i
ω
τ
0
-
2
e
4
c
6
e
3
m
m
m
m
m
m
l
×
(
α
-
e
-
i
ω
τ
0
+
2
α
e
-
i
ω
τ
0
)
-
2
e
4
c
4
e
3
m
m
m
m
m
m
l
×
(
α
-
e
-
2
i
ω
τ
0
+
2
α
)
+
2
e
4
c
7
e
3
m
m
m
m
m
m
l
×
(
α
-
α
e
-
2
i
ω
τ
0
+
α
α
-
+
α
2
)
2
e
i
ω
τ
0
W
20
(
1
)
(
-
1
)
]
1
e
1
}
.
Because g
21 contains W
20 and W
11, from (32) and (38), we have
(45)
W
˙
=
u
˙
t
-
z
˙
q
-
z
¯
˙
q
-
=
{
A
W
-
2
Re
{
p
¯
(
0
)
f
0
q
(
θ
)
}
,
-
1
≤
θ
<
0
A
W
-
2
Re
{
p
¯
(
0
)
f
0
q
(
0
)
}
+
f
0
,
θ
=
0
=
def
A
W
+
H
(
z
,
z
¯
,
θ
)
,
where
(46)
H
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
=
H
20
(
θ
)
z
2
2
+
H
11
(
θ
)
z
z
¯
+
H
02
(
θ
)
z
¯
2
2
+
⋯
.
Substituting the corresponding series into (45) and comparing the coefficients, we have
(47)
(
A
-
2
i
ω
τ
0
)
W
20
(
θ
)
=
-
H
20
(
θ
)
,
A
W
11
(
θ
)
=
-
H
11
(
θ
)
,
…
.
From (45), we know that for θ
∈
[
-
1,0
), we have
(48)
H
(
z
(
t
)
,
z
¯
(
t
)
,
θ
)
=
-
p
¯
(
0
)
f
0
q
(
θ
)
-
p
(
0
)
f
¯
0
q
¯
(
θ
)
=
-
g
(
z
,
z
¯
)
q
(
θ
)
-
g
¯
(
z
,
z
¯
)
q
¯
(
θ
)
.
Comparing the coefficient with (46) yields that for θ
∈
[
-
1,0
)
(49)
H
20
(
θ
)
=
-
g
20
(
θ
)
-
g
¯
02
q
¯
(
θ
)
,
(50)
H
11
(
θ
)
=
-
g
11
q
(
θ
)
-
g
¯
11
q
¯
(
θ
)
.
From (47), (49) and the definition of A, it follows that
(51)
W
˙
20
(
θ
)
=
2
i
ω
τ
0
W
20
(
θ
)
+
g
20
q
(
θ
)
+
g
¯
02
q
¯
(
θ
)
,
taking notice of q
(
θ
)
=
(
1
,
α
)
T
e
i
ω
τ
0
θ; hence,
(52)
W
20
(
θ
)
=
i
g
20
ω
τ
0
q
(
0
)
e
i
ω
τ
0
θ
-
g
02
¯
3
i
ω
τ
0
q
¯
(
0
)
e
-
i
ω
τ
0
θ
+
E
1
e
2
i
ω
τ
0
θ
,
where E
1
=
(
E
1
(
1
)
,
E
1
(
2
)
)
∈
R
2 is a constant vector. By the similar way, we have
(53)
W
11
(
θ
)
=
-
i
g
11
ω
τ
0
q
(
0
)
e
i
ω
τ
0
θ
-
g
¯
11
i
ω
τ
0
q
¯
(
0
)
e
-
i
ω
τ
0
θ
+
E
2
,
where E
2
=
(
E
2
(
1
)
,
E
2
(
2
)
)
∈
R
2 is a constant vector.Next, computingE
1 and E
2, from the definition of A and (47), one then obtains
(54)
∫
-
1
0
‍
d
η
(
θ
)
W
20
(
θ
)
=
2
i
ω
τ
0
W
20
(
0
)
-
H
20
(
0
)
,
(55)
∫
-
1
0
‍
d
η
(
θ
)
W
11
(
θ
)
=
-
H
11
(
0
)
,
where η
(
θ
)
=
η
(
0
,
θ
). Furthermore, we have(56)
H
20
(
0
)
=
-
g
20
q
(
0
)
-
g
¯
02
q
¯
(
0
)
+
2
τ
0
(
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
1
e
3
(
-
c
5
e
-
2
i
ω
τ
0
+
c
6
α
e
-
2
i
ω
τ
0
+
c
4
α
e
-
i
ω
τ
0
-
c
7
α
2
e
-
i
ω
τ
0
)
)
,
(57)
H
11
(
0
)
=
-
g
11
q
(
0
)
-
g
¯
11
q
¯
(
0
)
+
2
τ
0
(
1
e
1
(
c
3
+
c
1
α
α
¯
+
c
2
Re
{
α
}
)
1
e
3
(
-
c
5
+
c
6
Re
{
α
}
+
c
4
Re
{
α
e
i
ω
τ
0
}
-
c
7
Re
{
α
α
¯
e
i
ω
τ
0
}
)
)
.Substituting (52) and (56) into (54) and noting that
(58)
(
i
ω
τ
0
I
-
∫
-
1
0
‍
e
i
ω
τ
0
θ
d
η
(
θ
)
)
q
(
0
)
=
0
,
(
-
i
ω
τ
0
I
-
∫
-
1
0
‍
e
-
i
ω
τ
0
θ
d
η
(
θ
)
)
q
¯
(
0
)
=
0
,
it implies that
(59)
(
2
i
ω
τ
0
I
-
∫
-
1
0
‍
e
2
i
ω
τ
0
θ
d
η
(
θ
)
)
E
1
=
2
(
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
1
e
3
(
-
c
5
e
-
2
i
ω
τ
0
+
c
6
α
e
-
2
i
ω
τ
0
+
c
4
α
e
-
i
ω
τ
0
-
c
7
α
2
e
-
i
ω
τ
0
)
)
,
Namely,
(60)
(
2
i
ω
+
a
11
a
12
-
a
21
e
-
2
i
ω
τ
0
2
i
ω
+
a
22
e
-
2
i
ω
τ
0
)
E
1
=
2
(
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
1
e
3
(
-
c
5
e
-
2
i
ω
τ
0
+
c
6
α
e
-
2
i
ω
τ
0
+
c
4
α
e
-
i
ω
τ
0
-
c
7
α
2
e
-
i
ω
τ
0
)
)
.
Then it yields that(61)
E
1
(
1
)
=
2
A
1
|
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
a
12
1
e
3
[
(
-
c
5
+
c
6
α
)
e
-
2
i
ω
τ
0
+
(
c
4
α
-
c
7
α
2
)
e
-
i
ω
τ
0
]
2
i
ω
+
a
22
e
-
2
i
ω
τ
0
|
,
E
1
(
2
)
=
2
A
1
|
2
i
ω
+
a
11
1
e
1
(
c
3
+
c
1
α
2
+
c
2
α
)
-
a
21
e
-
2
i
ω
τ
0
1
e
3
[
(
-
c
5
+
c
6
α
)
e
-
2
i
ω
τ
0
+
(
c
4
α
-
c
7
α
2
)
e
-
i
ω
τ
0
]
|
,where
(62)
A
1
=
|
2
i
ω
+
a
11
a
12
-
a
21
e
-
2
i
ω
τ
0
2
i
ω
+
a
22
e
-
2
i
ω
τ
0
|
.
Similarly, we get(63)
E
2
(
1
)
=
2
A
2
|
1
e
1
(
c
3
+
c
1
α
α
¯
+
c
2
Re
{
α
}
)
a
12
1
e
3
(
-
c
5
+
c
6
Re
{
α
}
+
c
4
Re
{
α
e
i
ω
τ
0
}
-
c
7
Re
{
α
α
¯
e
i
ω
τ
0
}
)
a
22
|
,
E
2
(
2
)
=
2
A
2
|
a
11
1
e
1
(
c
3
+
c
1
α
α
¯
+
c
2
Re
{
α
}
)
-
a
21
1
e
3
(
-
c
5
+
c
6
Re
{
α
}
+
c
4
Re
{
α
e
i
ω
τ
0
}
-
c
7
Re
{
α
α
¯
e
i
ω
τ
0
}
)
|
,where
(64)
A
2
=
|
a
11
a
12
-
a
21
a
22
|
.
Through simple computation, we determine W
20, W
11 from (52) and (53); further, we can determine g
21. Therefore, g
i
j in (44) can be expressed by the parameter and delay; hence,
(65)
C
1
(
0
)
=
i
2
ω
τ
0
(
g
20
g
11
-
2
|
g
11
|
2
-
|
g
02
|
2
3
)
+
g
21
2
,
μ
2
=
-
Re
{
C
1
(
0
)
}
Re
{
λ
′
(
τ
0
)
}
,
ζ
=
2
Re
{
C
1
(
0
)
}
,
T
=
-
Im
⁡
{
C
1
(
0
)
}
+
μ
Im
⁡
{
λ
′
(
τ
0
)
}
ω
τ
0
,
which determine the qualities of bifurcation periodic solution of the critical value τ
0.Theorem 5.
(i)μ
2 determines the direction of Hopf bifurcation: if μ
2
>
0  (<
0), then Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist for τ
>
τ
0 (τ
<
τ
0).
(ii)ζ determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if ζ
<
0  (ζ
>
0). T determines the period of the bifurcating periodic solution: the period increases (decrease) if T
>
0 (T
<
0).


## 5. Numerical Simulations

In this section, we consider a delayed predator-prey system with harvesting as follows:(66)
x
˙
(
t
)
=
x
(
t
)
(
1
-
x
(
t
)
10
-
1.5
y
(
t
)
)
-
0.4
x
(
t
)
,
y
˙
(
t
)
=
y
(
t
)
(
1
-
y
(
t
-
τ
)
6
+
0.3
x
(
t
-
τ
)
)
-
0.5
y
(
t
)
.
Because (
H
1
) holds, from (14), we obtain that
(67)
sign
⁡
{
Re
[
d
λ
d
τ
]
}
τ
=
τ
j
>
0
,
τ
0
≈
2.8015
.
The unique positive equilibrium is E
*
=
(
2.907,3.436
).Ifh
1
=
0.4, when h
2 decreases, then prey species decreases and predator species increases (see Figure 1); when h
2 increases, prey species increases and predator species decreases (see Figure 2); If h
2
=
0.5, when the values of harvesting h
1 decreases, then both predator species and prey species will increase (see Figure 3); on the other hand, when h
1 increases, then both predator species and prey species will decrease (see Figure 4).Figure 1
Whenh
1
=
0.4 and h
2 decreases, prey species x decreases and predator species y increases.Figure 2
Whenh
1
=
0.4 and h
2 increases, prey species x increases and predator species y decreases.Figure 3
Whenh
2
=
0.5 and h
1 decreases, prey species x and predator species y increase.Figure 4
Whenh
2
=
0.5 and h
1 increases, prey species x and predator species y decrease.When parameterτ is little bigger than the critical value τ
0, system (5) will become unstable and predator species and prey species can coexist; when τ increases much more, prey species will go to extinct (see Figure 5). Moreover, from Figure 6, we can see that system (5) is unstable when τ passes through the critical value τ
0. By controlling the harvesting rates h
1 and h
2, respectively, the stability of positive equilibrium to system (5) can been changed. Similarly, when τ
<
τ
0, system (5) is stable; if we decrease the harvesting rate h
2, then the stable system becomes unstable one (see Figure 7).Figure 5
Whenτ
=
2.9
>
τ
0
≐
2.8015, prey species x and predator species y coexist; when τ
=
5
>
τ
0
≐
2.8015, prey species x goes to extinct.Figure 6
Whenτ
=
2.9
>
τ
0
≈
2.8015 and h
1 increases, prey species x and predator species y become stable; when h
2 increases, prey species x and predator species y also become stable.Figure 7
Whenτ
=
2.7
<
τ
0
≈
2.8015 and h
2 decreases, prey species x and predator species y become unstable.Sinceμ
2
<
0, ζ
<
0, Hopf bifurcation is subcritical and the positive equilibrium E
* is asymptotically stable for 0
<
τ
<
τ
0 (see Figure 8); when τ
>
τ
0, E
* loses its stability and Hopf bifurcation occurs; that is, a family of periodic solutions bifurcate from E
* (see Figure 9).Figure 8
Whenτ
=
2.7
<
τ
0
≐
2.8015, the positive equilibrium E
* of system (5) is asymptotically stable.Figure 9
Whenτ
=
2.9
>
τ
0
≐
2.8015, the positive equilibrium E
* of system (5) loses its stability and a Hopf bifurcations occurs.As discussed, our results show that the delayτ affects the stability of system (5) and harvesting rates h
1 and h
2 also affect the stability of system (5).


## 6. Conclusion

In our model, the harvesting term has been introduced into the model (5); by applying the normal form theorem and the center manifold theorem, we investigate the dynamical behaviors of the delayed predator-prey model with harvesting term and obtain the influence of harvesting term on the prey species and predator species. Further, we prove that the influence of the harvesting rates h
1 and h
2 to the stability of system (5), by controlling harvesting rates h
1 and h
2 of prey species and predator species, which makes the unstable (stable) system become stable (unstable).Our results show that Hopf bifurcations occur as the delayτ passes through critical values τ
0
≈
2.8015. When τ
<
τ
0, the positive equilibrium E
* of system (5) is asymptotically stable; when τ
>
τ
0, the positive equilibrium E
* of system (5) loses its stability and Hopf bifurcations occur.



---

*Source: 624162-2014-03-06.xml*