Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 738306, 20 pages
doi:10.1155/2010/738306
Research Article
The Permanence and Extinction of a Discrete
Predator-Prey System with Time Delay and
Feedback Controls
Qiuying Li, Hanwu Liu, and Fengqin Zhang
Department of Mathematics, Yuncheng University, Yuncheng 044000, China
Correspondence should be addressed to Qiuying Li,
Received 23 May 2010; Revised 4 August 2010; Accepted 7 September 2010
Academic Editor: Yongwimon Lenbury
Copyright q 2010 Qiuying Li et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
A discrete predator-prey system with time delay and feedback controls is studied. Sufficient
conditions which guarantee the predator and the prey to be permanent are obtained. Moreover,
under some suitable conditions, we show that the predator species y will be driven to extinction.
The results indicate that one can choose suitable controls to make the species coexistence in a long
term.
1. Introduction
The dynamic relationship between predator and its prey has long been and will continue
to be one of the dominant themes in both ecology and mathematical ecology due to
its universal existence and importance. The traditional predator-prey models have been
studied extensively e.g., see 1–10 and references cited therein, but they are questioned by
several biologists. Thus, the Lotka-Volterra type predator-prey model with the Beddington-
DeAngelis functional response has been proposed and has been well studied. The model can
be expressed as follows:
x
,
y
t
y
t
a
21
x
t
1 βx
t
γy
t
− d − a
22
y
11
t
x
t
−
a
12
t
y
t
α
t
β
t
x
t
t
β
t
x
t
γ
t
y
t
− d
t
.
1.2
For the general nonautonomous case, they addressed properties such as permanence,
extinction, and globally asymptotic stability of the system. For the periodic almost periodic
case, they established sufficient criteria for the existence, uniqueness, and stability of a
positive periodic solution and a boundary periodic solution. At the end of their paper,
n
− a
11
n
x
n
−
a
12
n
y
n
1 β
n
x
n
γ
21
n
x
n − τ
1 β
n
x
n − τ
γ
n
y
n − τ
− d
n
− a
22
e
1
n
− 1
u
1
n
− f
1
n
x
n
,
u
2
n 1
1 − e
n denote the death rate and density-dependent coefficient of the predator at time
n, respectively. a
12
n denotes the capturing rate of the predator; a
21
n/a
12
n represents the
rate of conversion of nutrients into the reproduction of the predator. Further, τ is a positive
integer.
For the simplicity and convenience of exposition, we introduce the following
notations. Let R
0, ∞, Z
{1, 2, } and k
1
,k
2
denote the set of integer k satisfying
k
1
≤ k ≤ k
2
. We denote DC
: −τ,0 → R
to be the space of all nonnegative and
bounded discrete time functions. In addition, for any bounded sequence gn, we denote
θ
φ
1
θ
,φ
2
θ
,ψ
1
θ
,ψ
2
θ
,φ
i
,ψ
i
,i 1, 2. 1.5
The main purpose of this paper is to establish a new general criterion for the
permanence and extinction of system 1.3, which is dependent on feedback controls. This
paper is organized as follows. In Section 2, we will give some assumptions and useful
lemmas. In Section 3, some new sufficient conditions which guarantee the permanence of
all positive solutions of system 1.3 are obtained. Moreover, under some suitable conditions,
we show that the predator species y will be driven to extinction.
2. Preliminaries
In this section, we present some useful assumptions and state several lemmas which will be
useful in the proving of the main results.
Throughout this paper, we will have both of the following assumptions:
H
1
rn, bn, dn, βn and γn are nonnegative bounded sequences of real
numbers defined on Z
such that
r
L
> 0,b
L
≥ 0,d
L
> 0, 2.1
H
2
c
i
n, e
x
n
exp
g
n
− a
n
x
n
, 2.3
where functions an, gn are bounded and continuous defined on Z
with a
L
, g
L
> 0. We
have the following result which is given in 23.
Lemma 2.1. Let xn be the positive solution of 2.3 with x0 > 0,then
a there exists a positive constant M>1 such that
n 1
f
n
− e
n
u
n
, 2.5
where functions fn and en are bounded and continuous defined on Z
with f
L
> 0and
0 <e
L
≤ e
M
< 1. The following Lemma 2.2 is a direct corollary of Theorem 6.2ofL.Wangand
M. Q. Wang 24, page 125.
Lemma 2.2. Let un be the nonnegative solution of 2.5 with u0 > 0,then
a f
L
/e
− e
n
u
n
ω
n
, 2.6
where functions fn and en are bounded and continuous defined on Z
with f
L
> 0,
0 <e
L
≤ e
M
< 1andωn ≥ 0. The following Lemma 2.3 is a direct corollary of Lemma 3 of
Xu and Teng 25.
Lemma 2.3. Let un, n
0
,u
0
be the positive solution of 2.6 with u0 > 0, then f or any constants
n, n
0
,u
0
is a positive solution of 2.5 with u
∗
n
0
,n
0
,u
0
u
0
.
Advances in Difference Equations 5
Finally, one considers the following nonautonomous linear equation:
Δu
n 1
−e
n
u
n
ω
Theorem 3.1. Suppose that assumptions H
1
and H
2
hold, then there exists a constant M>0
such that
lim sup
n →∞
x
n
<M, lim sup
n →∞
y
n
<M, lim sup
n →∞
u
1
n
<M, lim sup
n →∞
u
2
n
, 3.2
for all n ≥ n
0
, where n
0
is the initial time.
Consider the following auxiliary equation:
Δv
n 1
r
n
− e
1
n
v
n
, 3.3
from assumptions H
1
, ∀n ≥ n
0
. 3.5
From this, we further have
lim sup
n →∞
u
1
n
≤ M
1
. 3.6
6 Advances in Difference Equations
Then, we obtain that for any constant ε>0, there exists a constant n
1
>n
0
such that
u
1
n
<M
1
ε ∀n ≥ n
1
M
1
ε
}
, 3.8
for all n ≥ n
1
. Considering the following auxiliary equation:
z
n 1
z
n
exp
{
b
n
− a
11
n
z
. By the comparison
theorem, we have
x
n
≤ z
n
, ∀n ≥ n
1
. 3.11
From this, we further have
lim sup
n →∞
x
n
≤ M
2
. 3.12
Then, we obtain that for any constant ε>0, there exists a constant n
2
>n
1
such that
x
− a
22
n
y
n
, 3.14
for all n ≥ n
2
τ. Following a similar argument as above, we get that there exists a positive
constant M
3
such that
lim sup
n →∞
y
n
<M
3
. 3.15
By a similar argument of the above proof, we further obtain
lim sup
n →∞
<M,
lim sup
n →∞
u
1
n
<M, lim sup
n →∞
u
2
n
<M.
3.17
This completes the proof of Theorem 3.1.
In order t o obtain the permanence of system 1.3, we assume that
H
3
bnc
1
nu
∗
10
n
L
> 0, where u
lim inf
n →∞
x
n
>η
x
, 3.19
for any positive solution xn,yn,u
1
n,u
2
n of system 1.3.
Proof. According to assumptions H
1
and H
3
, we can choose positive constants ε
0
and ε
1
such that
b
n
− a
11
− ε
1
L
>ε
0
,
a
21
n
ε
0
1 β
n
ε
0
− d
n
M
< −ε
. 3.21
Let un be any positive solution of system 3.18 with initial value un
0
v
0
. By
assumptions H
1
–H
3
and Lemma 2.2,weobtainthatun is globally asymptotically stable
and converges to u
∗
10
n uniformly for n → ∞. Further, from Lemma 2.3,weobtainthat,
for any given ε
1
> 0 and a positive constant M>0 M is given in Theorem 3.1, there exist
constants δ
1
δ
1
ε
1
> 0andn
∗
1
n
∗
1
<ε
1
, ∀n ≥ n
0
n
∗
1
, 3.22
where vn, n
0
,v
0
is the solution of 3.21 with initial condition vn
0
,n
0
,v
0
v
0
.
8 Advances in Difference Equations
Let α
0
≤ min{ε
0
,δ
n
ε
1
c
1
n
u
∗
10
n
− ε
1
>α
0
,
a
21
n
α
0
n
≥ α
0
, 3.24
for any positive solution xn,yn,u
1
n,u
2
n of system 1.3. In fact, if 3.24 is not true,
then there exists a Φθφ
1
θ,φ
2
θ,ψ
1
θ,ψ
2
θ such that
lim sup
n →∞
x
n, Φ
<α
0
, 3.25
where xn, Φ,yn, Φ,u
1
− e
1
n
u
1
n
− f
M
1
α
0
, 3.27
for n>n
2
. Let vn be the solution of 3.21 with initial condition vn
2
u
1
n
2
, by the
comparison theorem, we have
u
1
n
v
n, n
2
,u
1
n
2
>u
∗
10
n
− ε
1
, 3.29
for all n ≥ n
2
n
∗
1
. Hence, from 3.28, we further have
u
1
n
α
0
1 β
n
α
0
− d
n
, 3.31
Advances in Difference Equations 9
for all n>n
2
τ. Obviously, we have yn → 0asn → ∞. Therefore, we get that there
exists an n
∗
2
such that
y
n
<ε
1
12
n
ε
1
1 γ
n
ε
1
c
1
n
u
∗
10
n
− ε
1
, 3.33
for any n>n
} of initial functions such that
lim inf
n →∞
x
n, Z
m
<
α
0
m 1
2
, ∀m 1, 2, 3.34
On the other hand, by 3.24, we have
lim sup
n →∞
x
n, Z
m
≥ α
0
. 3.35
Hence, there are two positive integer sequences {s
m
q
s
m
q
,Z
m
≥
α
0
m 1
,x
t
m
q
,Z
m
≤
α
0
m 1
2
,
3.37
α
0
<M, u
1
n, Z
m
<M,andu
2
n, Z
m
<Mfor all n>K
m
. Because of s
m
q
→ ∞
as q → ∞, there exists a positive integer K
m
1
such that s
m
q
>K
m
τ and s
m
q
>n
1
as
q>K
m
n
M −
a
12
n
M
1 γ
n
M
− c
1
n
M
≥ x
n, Z
m
exp
−θ
1
s
m
q
,Z
m
exp
−θ
1
t
m
q
− s
m
q
≥
α
0
m 1
exp
−θ
1
t
m
q
− s
m
q
≥ n
∗
τ 2, ∀m ≥ m
0
,q≥ K
m
1
. 3.42
From the t hird equation of system 1.3 and 3.38, we have
Δu
1
n 1,Z
m
≥ r
n
− e
1
n
u
1
α
0
,
3.43
for any m ≥ m
0
, q ≥ K
m
1
,andn ∈ s
m
q
1,t
m
q
. Assume that vn is the solution of 3.21
with the initial condition vs
m
q
1u
1
s
m
q
1, then from comparison theorem and the
above inequality, we have
u
1
1
s
m
q
1,since0<v
0
<Mand f
1
nα
0
<δ
1
,
then for all n ∈ s
m
q
1,t
m
q
, we have
v
n
v
n, s
m
q
1,u
n, Z
m
>u
∗
10
n
− ε
1
, 3.46
for all n ∈ s
m
q
1 n
∗
,t
m
q
, q ≥ K
m
1
, and m ≥ m
0
.
Advances in Difference Equations 11
From the second equation of system1.3, we have
y
, q ≥ K
m
1
,andn ∈ s
m
q
τ, t
m
q
. Therefore, we get that
y
n
<ε
1
, 3.48
for any n ∈ s
m
q
τ n
∗
,t
m
q
. Further, from the first equation of systems 1.3, 3.46 ,and
3.48,weobtain
x
n 1,Z
n
ε
1
c
1
n
u
∗
10
n
− ε
1
≥ x
n, Z
m
exp
α
0
m
exp
α
0
. 3.50
In view of 3.37 and 3.38, we finally have
α
0
m 1
2
≥ x
t
m
q
,Z
m
≥ x
t
m
q
− 1,Z
m
exp
x
n
exp
{
b
n
− a
11
n
x
n
c
1
n
u
1
n
.
3.52
For system 3.52, we further introduce the following assumption:
H
4
suppose λ max{|1 − a
M
11
x|, |1 − a
L
11
x|} c
M
1
< 1, δ 1 − e
L
1
f
M
1
x<1, where x, x
are given in the proof of Lemma 3.3.
For system3.52, we have the following result.
12 Advances in Difference Equations
Lemma 3.3. Suppose that assumptions H
1
–H
3
hold, then
a there exists a constant M>1 such that
is globally uniformly attractive on R
2
0
.
Proof. Based on assumptions H
1
–H
3
, conclusion a can be proved by a similar argument
as in Theorems 3.1 and 3.2.
Here, we prove conclusion b. Letting x
∗
10
n,u
∗
10
n be some solution of system
3.52, by conclusion a, there exist constants
x, x,andM>1, such that
x
− ε<x
n
,x
∗
10
n
n. Hence, system 3.52 is equivalent to
v
1
n 1
1 − a
11
n
x
∗
10
n
exp
{
θ
1
n
v
1
n
2
n
− f
1
n
x
∗
10
n
exp
{
θ
2
n
v
1
n
}
v
1
implies that x
∗
10
n expθ
i
nv
1
n i 1, 2 lie between x
∗
10
n and xn. Therefore, x − ε<
x
∗
10
n expθ
i
nv
1
n < x ε, i 1, 2. It follows from 3.55 that
|
v
1
n 1
|
≤
1 − a
11
n
v
2
n
,
|
v
2
n 1
|
≤
1 − e
1
n
v
2
n
− f
1
ε
,σ
ε
}, then 0 <μ<1. It follows easily from 3.56 that
max
{|
v
1
n 1
|
,
|
v
2
n 1
|}
≤ μ max
{|
v
1
n
|
,
|
n
exp
b
n
− a
11
n
x
n
− g
n
c
1
n
u
1
n
,
3.58
then we have the following result.
Lemma 3.4. Suppose that assumptions H
1
–H
4
hold, then there exists a positive constant δ
2
such
that for any positive solution xn,u
1
n of system 3.58, one has
lim
n →∞
|
x
n
− x
n
|
0, lim
n →∞
|
n
0
.
The proof of Lemma 3.4 is similar to Lemma 3.3, one omits it here.
Let x
∗
n,u
∗
1
n be a fixed solution of system 3.52 defined on R
2
0
, one assumes that
H
5
−dna
21
nx
∗
n − τ/1 βnx
∗
n − τ
L
> 0.
Theorem 3.5. Suppose that assumptions H
1
–H
5
hold, then there exists a constant η
y
n
a
21
n
x
∗
n − τ
− ε
3
1 β
n
x
∗
n − τ
− ε
3
γ
n
v
n
f
2
n
α
1
, 3.62
by Lemma 2.4, for given ε
3
> 0andM>0 M is given in Theorem 3.1., there exist constants
δ
3
δ
3
ε
3
> 0andn
∗
3
n
∗
3
. 3.63
14 Advances in Difference Equations
We choose α
1
< max{ε
2
,δ
3
/1 f
M
2
} if there exists a constant n
such that a
12
n −
δ
2
γn ≡ 0 for all n>n
, otherwise α
1
< max{ε
2
,δ
3
/1 f
M
2
,δ
n
α
1
<δ
3
, ∀n>n
2
. 3.64
Now, We prove that
lim sup
n →∞
y
n
≥ α
1
, 3.65
for any positive solution xn,yn,u
1
n,u
2
n of system 1.3. In fact, if 3.65 is not true,
then for α
1
, there exist a Φθφ
1
θ,φ
0 <
a
12
n
y
n
1 β
n
x
n
γ
n
y
n
<
a
12
u
1
n
− u
∗
n
|
0, 3.68
for any solution xn,yn,u
1
n,u
2
n of system 1.3. Therefore, for any small positive
constant ε
3
> 0, there exists an n
∗
4
such that for all n ≥ n
3
n
∗
4
, we have
x
α
1
. 3.70
In 3.63, we choose n
0
n
3
and v
0
un
3
. Since f
2
nα
1
<δ
3
, then for all n ≥ n
3
n
∗
3
,we
have
u
2
n
≤ ε
n
x
∗
10
n
− ε
3
γ
n
α
1
− d
n
− a
22
n
α
1
− c
m
2
,ψ
m
1
,ψ
m
2
} of initial functions, such that
lim inf
n →∞
y
n, Z
m
<
α
1
m 1
2
, ∀m 1, 2, , 3.73
where xn, Z
m
,yn, Z
m
,u
1
q
} satisfying
0 <s
m
1
<t
m
1
<s
m
2
<t
m
2
< ··· <s
m
q
<t
m
q
< ··· 3.75
and lim
q →∞
s
m
q
∞, such that
y
s
2
≤ y
n, Z
m
≤
α
1
m 1
, ∀n ∈
s
m
q
1,t
m
q
− 1
. 3.77
By Theorem 3.1, for given positive integer m, there exists a K
m
such that xn, Z
m
<
M, yn, Z
m
<M, u
1
1
, for any n ∈ s
m
q
,t
m
q
, we have
y
n 1,Z
m
≥ y
n, Z
m
exp
−d
n
− a
21
n
M − a
22
nM c
2
nM}. Hence,
α
1
m 1
2
≥ y
t
m
q
,Z
m
≥ y
s
m
q
,Z
m
exp
−θ
2
q
− s
m
q
≥
ln
m 1
θ
2
, ∀q ≥ K
m
1
,m 1, 2, 3.80
Choosing a large enough m
1
, such that
t
m
q
− s
m
q
> n
∗∗
τ 2, ∀m ≥ m
1
,q≥ K
m
y
n
<
a
12
n
α
1
1 γ
n
α
1
<δ
2
, 3.82
for all n ∈ s
m
q
1,t
m
q
. Therefore, it follows from system 1.3 that
x
n
,
u
1
n 1
r
n
−
e
1
n
− 1
u
1
n
− f
1
, 3.84
for any m ≥ m
1
, q ≥ K
m
1
,andn ∈ s
m
q
1 n
∗∗
,t
m
q
. For any m ≥ m
1
, q ≥ K
m
1
,and
n ∈ s
m
q
1,t
m
q
, by the fi rst equation of systems 1.3 and 3.77, it follows that
Δu
2
n, Z
m
f
2
n
α
1
.
3.85
Assume that vn is the solution of 3.62 with the initial condition vs
m
q
1u
2
s
m
q
1,
then from comparison theorem and the above inequality, we have
u
2
n, Z
m
≤ v
0
<Mand f
2
nα
1
<δ
3
,
then we have
v
n
≤ ε
3
, ∀n ∈
s
m
q
1 n
∗∗
,t
m
q
. 3.87
Advances in Difference Equations 17
Equation 3.86 together with 3.87 lead to
u
m
q
τ 1 n
∗∗
,t
m
q
, from the second equation
of systems 1.3, 3.61, 3.77, 3.84,and3.88, it follows that
y
n 1,Z
m
y
n, Z
m
exp
−d
n
a
21
n
n, Z
m
− c
2
n
u
2
n, Z
m
≥ y
n, Z
m
exp
−d
n
a
21
α
1
−a
22
n
α
1
− c
2
n
ε
3
≥ y
n, Z
m
exp
{
α
1
}
.
3.89
t
m
q
,Z
m
≥ y
t
m
q
− 1,Z
m
exp
α
1
≥
α
1
m 1
2
exp
α
1
a
21
n
x
∗
n − τ
1 β
n
x
∗
n − τ
M
< 0 3.92
18 Advances in Difference Equations
holds, then
lim
n →∞
y
n
ε
1
1 β
n
x
∗
n − τ
ε
1
− a
22
n
ε<−ε
1
, 3.94
for n>n
1
. First, we show that there exists an n
2
>n
1
exp
b
n
− a
11
n
x
n
−
a
12
n
ε
1 γ
n
ε
c
1
1
n
x
n
.
3.96
Therefore, from Lemma 3.3 and comparison theorem, it follows that for the above ε
1
, there
exists an n
∗
2
> 0, such that
x
n
<x
∗
n
ε
1
, ∀n>n
1
x
∗
n − τ
ε
1
1 β
n
x
∗
n − τ
ε
1
− a
22
n
ε
≤ y
n
<εexp
μ
, ∀n>n
2
, 3.99
where
μ max
n∈Z
d
n
a
21
n
x
∗
n − τ
ε
1
∈ n
2
,n
3
− 1 such that yn
4
<ε, yn
4
1 ≥ ε,andyn ≥ ε for n ∈ n
4
1,n
3
.
Let P
1
be a nonnegative integer, such that
n
3
n
4
P
1
1. 3.101
It follows from 3.101 that
ε exp
μ
≤ y
s − τ
ε
1
1 β
s
x
∗
s − τ
ε
1
− a
22
s
ε
≤ y
n
4
1 β
n
4
P
1
x
∗
n
4
P
1
− τ
ε
1
− a
22
n
4
P
1
ε
<εexp
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