Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 605254, 15 pages
doi:10.1155/2009/605254
Research Article
Permanence of a Discrete Nonlinear
Prey-Competition System with Delays
Hongying Lu
School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics,
Dalian, Liaoning 116025, China
Correspondence should be addressed to Hongying Lu, hongyinglu543@163.com
Received 3 July 2009; Accepted 16 September 2009
Recommended by Xue-Zhong He
A discrete nonlinear prey-competition system with m-preys and n-m-predators and delays is
considered. Two sets of sufficient conditions on the permanence of the system are obtained. One
set is delay independent, while the other set is delay dependent.
Copyright q 2009 Hongying Lu. 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.
1. Introduction
In this paper, we investigate the following discrete nonlinear prey-competition system with
delays:
⎡
xi k 1 xi k exp⎣ri k −
n
j1
αij
aij kxj k
−
n
j1
⎤
βij bij kxj k
− τij k ⎦,
i 1, 2, . . . , m,
⎡
xi k 1 xi k exp⎣−ri k m
j1
n
αij
aij kxj k αij
−
aij kxj k
jm1
−
m
j1
n
βij bij kxj
k − τij k
βij bij kxj k
jm1
⎤
− τij k ⎦,
i m 1, 2, . . . , n,
1.1
2
Discrete Dynamics in Nature and Society
where xi k i 1, 2, . . . , m is the density of prey species i at kth generation, xi k i m 1, . . . , n is the density of predator species i at kth generation. In this system, the
competition among predator species and among prey species is simultaneously considered.
For more background and biological adjustments of system 1.1, we can see 1–5 and the
references cited therein.
Throughout this paper, we always assume that for all i, j 1, 2, . . . , n,
H1 ri k, aij k, bij k are all bounded nonnegative sequences and alii ≥ 0, biil ≥ 0, alii l
bii > 0. Here, for any bounded sequence f u supk∈N fk, f l infk∈N fk;
H2 τij k are bounded nonnegative integer sequences, and αij , βij are all positive
constants.
By a solution of system 1.1, we mean a sequence {x1 k, . . . , xn k} which defined for
N {0, 1, . . .} and which satisfies system 1.1 for N {0, 1, . . .}. Motivated by application
of system 1.1 in population dynamics, we assume that solutions of system 1.1 satisfy the
following initial conditions:
xi θ φi θ,
θ ∈ N−τ, 0 −τ, −τ 1, . . . , 0,
φi 0 > 0,
1.2
where τ max{τij k, i, j 1, 2, . . . , n}. The exponential forms of system 1.1 assure that the
solution of system 1.1 with initial conditions 1.2 remains positive.
Recently, Chen et al. in 1 proposed the following nonlinear prey-competition system
with delays:
⎡
ẋi t xi t⎣ri t −
n
j1
⎡
ẋi t xi t⎣−ri t αij
aij txj t
m
j1
n
⎤
n
βij − bij txj t − τij t ⎦,
αij
aij txj t αij
−
aij txj t
jm1
−
i 1, 2, . . . , m,
j1
n
m
βij bij txj t − τij t
1.3
j1
βij bij txj t
jm1
⎤
− τij t ⎦,
i m 1, 2, . . . , n.
By using Gaines and Mawhins continuation theorem of coincidence degree theory
and by constructing an appropriate Lyapunov functional, they obtained a set of sufficient
conditions which guarantee the existence and global attractivity of positive periodic solutions
of the system 1.3. In addition, sufficient conditions are obtained for the permanence of the
system 1.3 in 2.
On the other hand, though most population dynamics are based on continuous models
governed by differential equations, the discrete time models are more appropriate than
the continuous ones when the size of the population is rarely small or the population has
nonoverlapping generations 3–15. Therefore, it is reasonable to study discrete time preycompetition models governed by difference equations.
As we know, a more important theme that interested mathematicians as well as
biologists is whether all species in a multispecies community would survive in the long run,
that is, whether the ecosystems are permanent. In fact, no such work has been done for system
1.1.
Discrete Dynamics in Nature and Society
3
The main purpose of this paper is, by developing the analytical technique of 4, 8, 16,
to obtain two sets of sufficient conditions which guarantee the permanence of system 1.1.
2. Main Results
Firstly, we introduce a definition and some lemmas which will be useful in the proof of the
main results of this section.
Definition 2.1. System 1.1 is said to be permanent, if there are positive constants m and M,
such that each positive solution x1 k, . . . , xn k of system 1.1 satisfies
m ≤ lim inf xi k ≤ lim sup xi k ≤ M,
k → ∞
k → ∞
i 1, 2, . . . , n.
2.1
Lemma 2.2 see 8. Assume that {xk} satisfies xk > 0 and
xk 1 ≤ xk exp{rk1 − axk}
2.2
for k ∈ k1 , ∞, where a is a positive constant. Then
lim sup xk ≤
k → ∞
1
expr u − 1.
ar u
2.3
Lemma 2.3 see 8. Assume that {xk} satisfies
xk 1 ≥ xk exp{rk1 − axk},
k ≥ K0 ,
2.4
lim supk → ∞ xk ≤ x∗ and xK0 > 0, where a is a constant such that ax∗ > 1 and K0 ∈ N. Then
lim inf xk ≥
k → ∞
1
expr u 1 − ax∗ .
a
2.5
For system 1.1, we will consider two cases, alii > 0, biil ≥ 0 and alii ≥ 0, biil > 0
respectively, and then we obtain Lemmas 2.4–2.6.
Lemma 2.4. Assume that alii > 0. Then for every positive solution x1 k, . . . , xn k of system 1.1
with initial condition 1.2, one has
lim sup xi k ≤ Mi ,
k → ∞
i 1, 2, . . . , n,
2.6
4
Discrete Dynamics in Nature and Society
where
1/αii
1
u
, i 1, 2, . . . , m,
exp
r
−
i
αii
αii alii
⎤
⎡
1/αii
n
n
1
1
αij
βij
⎦,
Mi exp⎣−ril auij Mj biju Mj −
αii
αii alii
j1
j1
Mi 1
2.7
i m 1, . . . , n.
Proof. Let xk x1 k, . . . , xn k be any positive solution of system 1.1 with initial
condition 1.2, for i 1, 2, . . . , m, it follows from system 1.1 that
xi k 1 ≤ xi k exp ri k − aii kxiαii k ,
2.8
xiαii k 1 ≤ xiαii k exp αii ri k − aii kxiαii k .
2.9
thus
Let ui k xiαii k, we can have
ui k 1 ≤ ui k expαii ri k − aii kui k
alii
≤ ui k exp αii ri k 1 − u ui k .
ri
2.10
By applying Lemma 2.2 to 2.10, we obtain
lim sup ui k ≤
k → ∞
1
αii alii
exp αii riu − 1 L
˙ i;
2.11
i 1, 2, . . . , m.
2.12
so, we immediately get
lim sup xi k ≤ Mi ,
k → ∞
For any ε > 0 small enough, it follows from 2.12 that there exists enough large K1 such that
for all i 1, 2, . . . , m and k ≥ K1
xi k ≤ Mi ε.
2.13
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5
For i m 1, . . . , n and k ≥ K1 τ, 2.13 combining with the i-th equation of system 1.1
leads to
⎤
m
m
αij βij
xi k 1 ≤ xi k exp⎣−ri k aij k Mj ε
bij k Mj ε
− aii kxiαii k⎦,
⎡
j1
j1
2.14
thus
⎛
⎡
xiαii k 1 ≤ xiαii k exp⎣αii ⎝−ri k m
m
α
aij k Mj ε ij
j1
bij k Mj ε
βij
j1
⎞⎤
2.15
− aii kxiαii k⎠⎦.
Similarly, let ui k xiαii k, we get
⎛
⎡
ui k 1 ≤ ui k exp⎣αii ⎝−ri k m
α
aij k Mj ε ij
j1
m
bij k Mj ε
βij
⎞⎤
− aii kui k⎠⎦
j1
⎛
⎞
m
m
αij βij
≤ ui k exp⎣αii ⎝−ri k aij k Mj ε
bij k Mj ε ⎠
⎡
j1
j1
⎛
× ⎝1 −
−ril u
j1 aij Mj ε
m
alii
αij
⎞⎤
βij ui k⎠⎦.
u
b
ε
M
j
j1 ij
m
2.16
By using 2.16, for i m 1, . . . , n, according to Lemma 2.2, it follows that
lim sup ui k ≤
k → ∞
1
αii alii
⎞
⎤
n
n
α
β
exp⎣αii ⎝−ril auij Mj ε ij biju Mj ε ij ⎠ − 1⎦;
⎡
⎛
j1
j1
2.17
6
Discrete Dynamics in Nature and Society
setting ε → 0 in above inequality, we have
lim sup ui k ≤
k → ∞
1
αii alii
⎛
⎞
⎤
n
n
α
β
ij
ij
˙ i,
exp⎣αii ⎝−ril auij Mj biju Mj ⎠ − 1⎦L
⎡
j1
2.18
j1
then
lim sup xi k ≤ Mi ,
i m 1, . . . , n.
k → ∞
2.19
This completes the proof.
For convenience, we introduce the following notation.
For i 1, 2, . . . , m
Ai Rui
ril −
riu
−
auii
n
αij
u
j1,j /
i aij Mj −
n
αij
alij Mj
j1,j /
i
−
n
n
j1
βij
u
j1 bij Mj
,
2.20
βij
bijl Mj .
For i m 1, . . . , n
Ai −riu Rui −ril m
m
j1
j1
αij
alij mj αij
auij mj
n
j1
βij
bijl mj
m
βij
biju mj −
j1
auii
,
αij
βij
− njm1,j / i auij Mj − njm1 biju Mj
n
jm1,j /
i
αij
alij Mj
−
n
2.21
βij
bijl Mj .
jm1
Lemma 2.5. Assume that alii > 0 and
min Li Ai > 1,
2.22
1≤i≤n
hold. Then for any positive solution x1 k, . . . , xn k of system 1.1 with initial condition 1.2,
one has
lim inf xi k ≥ mi ,
2.23
k → ∞
where
mi 1
Ai
1/αii
exp Rui 1 − Ai Li ,
i 1, 2, . . . , n.
2.24
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7
Proof. Let xk x1 k, . . . , xn k be any positive solution of system 1.1 with initial
condition 1.2. From Lemma 2.4, we know that there exists K2 > K1 τ, such that for
i 1, 2, . . . , n and k ≥ K2
xi k ≤ Mi ε.
2.25
For i 1, . . . , m and k ≥ K2 τ, 2.25 combining with the i-th equation of system 1.1 lead
to
⎡
n
xi k 1 ≥ xi k exp⎣ri k −
aij k Mj ε
αij
n
−
j1,j /
i
bij k Mj ε
βij
j1
⎤
− aii kxiαii k⎦,
2.26
thus
⎡
xiαii k
1 ≥
⎛
xiαii k exp⎣αii ⎝ri k
−
n
j1
n
−
α
aij k Mj ε ij
j1,j /
i
⎞⎤
2.27
β
bij k Mj ε ij − aii kxiαii k⎠⎦;
let ui k xiαii k, we can have
⎡
⎛
n
ui k 1 ≥ ui k exp⎣αii ⎝ri k −
α
aij k Mj ε ij
j1,j /
i
−
n
bij k Mj ε
βij
⎞⎤
− aii kui k⎠⎦
2.28
j1
≥ ui k expαii Riε k1 − Aiε ui k,
where
Riε k ri k −
n
j1,j /i
n
α
β
aij k Mj ε ij − bij k Mj ε ij ,
j1
auii
Aiε α
β .
ril − nj1,j / i auij Mj ε ij − nj1 biju Mj ε ij
2.29
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Discrete Dynamics in Nature and Society
According to Lemma 2.3, we obtain
lim inf ui k ≥
k → ∞
1
exp αii Ruiε 1 − Aiε Li ,
Aiε
2.30
where
Ruiε riu −
n
n
α
β
alij Mj ε ij − bijl Mj ε ij .
j1,j /
i
2.31
j1
Setting ε → 0 in 2.30 leads to
lim inf ui k ≥
k → ∞
1
exp αii Rui 1 − Ai Li ,
Ai
2.32
therefore
lim inf xi k ≥ mi ,
k → ∞
i 1, 2, . . . , m.
2.33
For any ε > 0 small enough, it follows from 2.33 that there exists enough large K3 > K2 τ
such that for all i 1, . . . , m and k ≥ K3
xi k ≥ mi − ε,
2.34
and so, for i m 1, . . . , n and k ≥ K3 τ, it follows from system 1.1 that
⎡
xi k 1 ≥ xi k exp⎣−ri k m
m
α
β
aij k mj − ε ij bij k mj − ε ij
j1
−
n
j1
aij k Mj ε
jm1,j /i
≥ xi k exp Riε k 1 − Aiε xiαii k ,
αij
−
n
bij k Mj ε
jm1
βij
⎤
− aii kxiαii k⎦
2.35
Discrete Dynamics in Nature and Society
9
where
Riε k −ri k m
m
α
β
aij k mj − ε ij bij k mj − ε ij
j1
n
−
j1
n
α
β
aij k Mj ε ij −
bij k Mj ε ij ,
jm1,j /
i
Aiε auii
jm1
⎧
⎨
m
n
α
β
−riu alij mj − ε ij bijl mj − ε ij
⎩
j1
2.36
j1
⎫
n
⎬
α
β
−
auij Mj ε ij −
biju Mj ε ij ,
⎭
jm1,j i
jm1
n
/
by using 2.35, similarly to the analysis of 2.33, for i m 1, . . . , n
lim inf ui k ≥
k → ∞
1
exp αii Rui 1 − Ai Li ,
Ai
2.37
and therefore, we easily get
lim inf xi k ≥ mi ,
k → ∞
i m 1, . . . , n.
2.38
This ends the proof of Lemma 2.5.
Denote for i 1, 2, . . . , m
Li 1
βii biil
Γi ril −
exp βii riu τ 1 − 1 ;
n
j1
αij
auij Mj −
n
j1
βij
biju Mj ;
biiu exp −βii Γi τ
Ai ;
αij
βij
ril − nj1 auij Mj − nj1,j / i biju Mj
u
Ri riu −
n
n
αij
βij
alij Mj −
bijl Mj .
j1
j1,j /
i
2.39
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Discrete Dynamics in Nature and Society
For i m 1, . . . , n
Υi −ril Li 1
βii biil
m
m
αij
βij
auij Mj biju Mj ;
j1
exp βii Υi τ 1 − 1 ;
m
Γi −riu Ai u
j1
−riu Ri −ril j1
m
αij
alij mj m
j1
j1
αij
alij mj
βij
bijl mj −
n
jm1
β
biju Mj ε ij ;
jm1
2.40
biiu exp −βii Γi τ
;
βij
αij
βij
nj1 bijl mj − njm1 auij Mj − njm1,j / i biju mj
m
n
n
αij
βij
αij
auij mj biju mj −
alij Mj −
j1
n
αij
auij Mj −
j1
jm1
n
βij
bijl mj .
jm1,j /
i
Lemma 2.6. Assume that biil > 0 and
H 3 min Li Ai > 1,
2.41
1≤i≤n
hold. Then for any positive solution x1 k, . . . , xn k of system 1.1 with initial condition 1.2,
one has
mi ≤ lim inf xi k ≤ lim sup xi k ≤ Mi ,
k → ∞
k → ∞
i 1, 2, . . . , n.
2.42
where
Mi 1/βii
Li ,
mi 1
Ai
1/βii
"
! u
exp Ri 1 − Ai Li .
2.43
Proof. Let xk x1 k, . . . , xn k be any positive solution of system 1.1 with initial
condition 1.2, for i 1, 2, . . . , m, it follows from system 1.1 that
xi k 1 ≤ xi k expri k ≤ xi k exp riu ,
"
!
β
xi k 1 ≤ xi k exp ri k − bii kxi ii k − τii k ,
2.44
2.45
It follows from 2.44that
k−1
k−1
#
#
xi j 1
exp riu ≤ exp riu τ ,
≤
x
j
i
jk−τii k
jk−τii k
2.46
Discrete Dynamics in Nature and Society
11
and hence
xi k − τii k ≥ xi k exp −riu τ ,
2.47
which, together with 2.45, produces
"
!
β
xi k 1 ≤ xi k exp ri k − bii k exp −βii riu τ xi ii k
biil exp −βii riu τ βii
≤ xi k exp ri k 1 −
xi k ,
riu
2.48
similar to the analysis of 2.11 and 2.12, for i 1, 2, . . . , m
lim sup ui k ≤ Li ,
2.49
k → ∞
and thus, we immediately get
lim sup xi k ≤ Mi ,
k → ∞
i 1, 2, . . . , m.
2.50
For any ε > 0 small enough, it follows from 2.50 that there exists enough large K 1 such that
for all i 1, 2, . . . , m and k ≥ K 1
xi k ≤ Mi ε.
2.51
For i m 1, . . . , n and k ≥ K 1 τ, 2.51 combining with the i-th equation of system 1.1
lead to
⎡
xi k 1 ≤ xi k exp⎣−ril m
j1
xi k expΥiε ,
⎡
xi k 1 ≤ xi k exp⎣−ri k ⎤
m
α
β
ij
ij
auij Mj ε
biju Mj ε ⎦
j1
2.52
m
α
aij k Mj ε ij
j1
⎤
m
βij
βii
bij k Mj ε
− bii kxi k − τii k⎦,
2.53
j1
from 2.53, similar to the argument of 2.44 and 2.47, for k ≥ K 1 τ, we have
xi k − τii k ≥ xi k exp−Υiε τ,
2.54
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Discrete Dynamics in Nature and Society
substituting 2.54 into 2.53, we get
⎡
xi k 1 ≤ xi k exp⎣−ri k m
α
aij k Mj ε ij
j1
⎤
m
βij
βii
bij k Mj ε
− bii k exp −βii Υiε τ xi k⎦,
2.55
j1
similar to the analysis of 2.18 and 2.19, for i m 1, . . . , n
lim sup ui k ≤ Li ,
2.56
lim sup xi k ≤ Mi , i m 1, . . . , n.
2.57
k → ∞
then,
k → ∞
For any ε > 0 small enough, it follows from 2.51 and 2.57 that there exists enough
large K 2 > K 1 τ such that for all i 1, 2, . . . , n and k ≥ K 2
xi k ≤ Mi ε.
2.58
Hence, for i 1, 2, . . . , m, and k ≥ K 2 τ, it follows from system 1.1 that
⎡
xi k 1 ≥
xi k exp⎣ril
⎤
n
n
αij βij
u
u
− aij Mj ε
− bij Mj ε ⎦
j1
xi k expΓiε ,
⎡
xi k 1 ≥ xi k exp⎣ri k −
2.59
j1
n
α
aij k Mj ε ij
j1
−
n
j1,j /
i
⎤
2.60
β
β
bij k Mj ε ij − bii kxi ii k − τii k⎦,
from2.59, similar to the argument of 2.44 and 2.47, for k ≥ K 2 τ, we have
xi k − τii k ≤ xi k exp−Γiε τ,
2.61
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13
and this combined with 2.60 gives
⎡
xi k 1 ≥ xi k exp⎣ri k −
n
α
aij k Mj ε ij
j1
n
−
bij k Mj ε
βij
j1,j /
i
⎤
βii
− bii k exp −βii Γiε τ xi k⎦.
2.62
Similar to the argument of 2.32 and 2.33, for k ≥ K 2 τ, we obtain
lim inf ui k ≥
k → ∞
1
Ai
"
!
u
exp βii Ri 1 − Ai Li ,
2.63
then
lim inf xi k ≥ mi ,
i 1, 2, . . . , m.
k → ∞
2.64
For any ε > 0 small enough, it follows from 2.63 that there exists enough large K 3 > K 2 τ
such that for all i 1, . . . , m and k ≥ K 3
xi k ≥ mi − ε,
2.65
and so, for i m 1, . . . , n and k ≥ K3 τ, it follows from system 1.1 that
⎡
xi k 1 ≥ xi k exp⎣−riu −
n
m
m
α
β
alij mj − ε ij bijl mj − ε ij
j1
j1
⎤
n
α
β
auij Mj ε ij −
biju Mj ε ij ⎦ xi k expΓiε ,
jm1
⎡
xi k 1 ≥ xi k exp⎣−ri k jm1
m
m
n
α
β
α
aij k mj − ε ij bij k mj − ε ij −
aij k Mj ε ij
j1
−
n
j1
bij k Mj ε
jm1,j /
i
βij
jm1
⎤
−
β
bii kxi ii k
− τii k⎦.
2.66
Similar to the argument of 2.61 and 2.62, for k ≥ K 3 τ, we have
lim inf ui k ≥
k → ∞
1
Ai
"
!
u
exp βii Ri 1 − Ai Li ,
2.67
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Discrete Dynamics in Nature and Society
then
lim inf xi k ≥ mi ,
k → ∞
i m 1, . . . , n.
2.68
This ends the proof of Lemma 2.6.
Denote H3 alii > 0,
1≤i≤n
biil > 0,
1≤i≤n
min Li Ai > 1,
2.69
min Li Ai > 1,
2.70
or
Our main result in this paper is the following theorem about the permanence of system
1.1.
Theorem 2.7. Assume that H1 , H2 , and H3 hold, then system 1.1 is permanent.
Proof. Let xk x1 k, . . . , xn k be any positive solution of system 1.1 with initial
condition 1.2. Suppose M maxi1,...,n {Mi , Mi }, m mini1,...,n {mi , mi }. By Lemmas 2.4–
2.6, if system 1.1 satisfies H1 , H2 , and H3 , then we have
m ≤ lim inf xi k ≤ lim sup xi k ≤ M,
k → ∞
k → ∞
i 1, 2, . . . , n.
2.71
The proof is completed.
In this paper, we study a discrete nonlinear predator-prey system with m-preys and
n-m-predators and delays, which can be seen as the modification of the traditional LotkaVolterra prey-competition model. From our main results, Theorem 2.7 gives two sets of
sufficient conditions on the permanence of the system 1.1. One set is delay independent,
while the other set is delay dependent.
Acknowledgments
The author is grateful to the Associate Editor, Professor Xue-Zhong He, and two referees for
a number of helpful suggestions that have greatly improved her original submission. This
research is supported by the Nation Natural Science Foundation of China no. 10171010,
Key Project on Science and Technology of Education Ministry of China no. 01061, and
Innovation Group Program of Liaoning Educational Committee No. 2007T050.
Discrete Dynamics in Nature and Society
15
References
1 F. Chen, X. Xie, and J. Shi, “Existence, uniqueness and stability of positive periodic solution for a
nonlinear prey-competition model with delays,” Journal of Computational and Applied Mathematics, vol.
194, no. 2, pp. 368–387, 2006.
2 A. Huang, “Permanence of a nonlinear prey-competition model with delays,” Applied Mathematics
and Computation, vol. 197, no. 1, pp. 372–381, 2008.
3 X. Liao, S. Zhou, and Y. N. Raffoul, “On the discrete-time multi-species competition-predation system
with several delays,” Applied Mathematics Letters, vol. 21, no. 1, pp. 15–22, 2008.
4 X. Liao, S. Zhou, and Y. Chen, “On permanence and global stability in a general Gilpin-Ayala
competition predator-prey discrete system,” Applied Mathematics and Computation, vol. 190, no. 1, pp.
500–509, 2007.
5 F. Chen, L. Wu, and Z. Li, “Permanence and global attractivity of the discrete Gilpin-Ayala type
population model,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1214–1227, 2007.
6 R. P. Agarwal and P. J. Y. Wong, Advanced Topics in Difference Equations, vol. 404 of Mathematics and Its
Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
7 R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of
Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd
edition, 2000.
8 X. Yang, “Uniform persistence and periodic solutions for a discrete predator-prey system with
delays,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 161–177, 2006.
9 F. Chen, “Permanence in a discrete Lotka-Volterra competition model with deviating arguments,”
Nonlinear Analysis: Real World Applications, vol. 9, no. 5, pp. 2150–2155, 2008.
10 W. Yang and X. Li, “Permanence for a delayed discrete ratio-dependent predator-prey model with
monotonic functional responses,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1068–
1072, 2009.
11 Y. Muroya, “Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra
systems,” Tokyo Journal of Mathematics, vol. 28, no. 1, pp. 189–200, 2005.
12 Y. Saito, W. Ma, and T. Hara, “A necessary and sufficient condition for permanence of a Lotka-Volterra
discrete system with delays,” Journal of Mathematical Analysis and Applications, vol. 256, no. 1, pp. 162–
174, 2001.
13 L. Chen and L. Chen, “Permanence of a discrete periodic volterra model with mutual interference,”
Discrete Dynamics in Nature and Society, vol. 2009, Article ID 205481, 9 pages, 2009.
14 M. Fazly and M. Hesaaraki, “Periodic solutions for a discrete time predator-prey system with
monotone functional responses,” Comptes Rendus Mathématique, vol. 345, no. 4, pp. 199–202, 2007.
15 W. Wang and Z. Lu, “Global stability of discrete models of Lotka-Volterra type,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 35, no. 8, pp. 1019–1030, 1999.
16 X. Meng and L. Chen, “Periodic solution and almost periodic solution for a nonautonomous LotkaVolterra dispersal system with infinite delay,” Journal of Mathematical Analysis and Applications, vol.
339, no. 1, pp. 125–145, 2008.