Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 656549, 19 pages
doi:10.1155/2009/656549
Research Article
Extinction in Nonautonomous Discrete
Lotka-Volterra Competitive System with
Pure Delays and Feedback Controls
Ling Zhang,1 Zhidong Teng,1 Tailei Zhang,1 and Shujing Gao2
1
2
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
College of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, China
Correspondence should be addressed to Zhidong Teng, zhidong1960@163.com
Received 25 June 2009; Accepted 12 October 2009
Recommended by Elena Braverman
The paper discusses a nonautonomous discrete time Lotka-Volterra competitive system with pure
delays and feedback controls. New sufficient conditions for which a part of the n-species is driven
to extinction are established by using the method of multiple discrete Lyapunov functionals.
Copyright q 2009 Ling Zhang 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.
1. Introduction
The coexistence and global stability of population models are of the interesting subjects in
mathematical biology. Many authors have argued that the discrete time models are governed
by differential equations which are more appropriate than the continuous ones to describe
the dynamics of population when the population has nonoverlapping generations, a lot has
been done on discrete Lotka-Volterra systems.
May in 1 firstly considered the following autonomous discrete two-species LotkaVolterra competitive system:
xn 1 xn exp r1 − a11 xn − a12 yn ,
yn 1 yn exp r2 − a21 xn − a22 yn
1.1
and studied the stable points, stable cycles, and the chaos behaivor. Further, Lu and Wang 2
studied the permanence and global attractivity of this system.
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Discrete Dynamics in Nature and Society
Chen and Zhou in 3 considered the following periodic discrete two-species LotkaVolterra competitive system:
xn
− μ2 nyn ,
xn 1 xn exp r1 1 −
K1 n
yn
yn 1 yn exp r2 1 − μ1 xn −
K2 n
1.2
and studied the permanence and existence of a periodic solution, and further, sufficient
conditions are established on the global stability of the periodic solution.
Zhang and Zhou in 4 investigated the following nonautonomous discrete twospecies Lotka-Volterra competitive system:
xn 1 xn exp r1 n − a11 nxn − k1 − a12 nyn − k2 ,
yn 1 yn exp r2 n − a21 nxn − l1 − a22 nyn − l2 .
1.3
Some sufficient conditions were obtained for the permanence of the system.
Wang et al. in 5 studied the following general nonautonomous discrete n-species
Lotka-Volterra systems:
⎧
⎨
⎫
m
n ⎬
l
xi k 1 xi k exp ri k −
aij kxj k − l − a12 nyn − k2 ,
⎩
⎭
j1 l0
i 1, 2, . . . , n.
1.4
By applying the linear method and constructing the appropriate Lyapunov functionals, the
author established the sufficient conditions which guarantee that any positive solution of this
system is stable and attracts others, and obtained some applications of main results.
Muroya in 6, 7 considered the following general nonautonomous discrete n-species
Lotka-Volterra systems:
⎧
⎫
m
n ⎨ ⎬
alij p Nj p − kl
Ni p 1 Ni p exp ci p − ai p Ni p −
,
⎩
⎭
j1 l0
Ni s Nis ≥ 0
∀s ≤ 0, Ni0 > 0,
p 0, 1, 2, . . . ,
i 1, 2, . . . , n,
1.5
and related pure delays models, that is, ai p 0 for all p ≥ 0. The author obtained the
permanence and the global asymptotically stable by applying mean-value conditions and the
method of constructing discrete Lyapunov functionals.
Discrete Dynamics in Nature and Society
3
Liao et al. in 8 discussed the following general discrete nonautonomous n-species
competitive system with feedback controls:
⎧
⎫
n
n
⎨
⎬
cij kxi kxj k−di kui k ,
xi k1 xi k exp bi k− aij kxj k−
⎩
⎭
j1
j1,j i
k 0, 1, 2, . . . ,
/
Δui k ri k − ei kui k fi kxi k,
i 1, 2, . . . , n.
1.6
Some sufficient conditions are established on the permanence and the global stability of the
system.
Recently, we see that in 9, 10 the authors studied the following nonautonomous
continuous Lotka-Volterra competitive system with pure delays and feedback controls:
⎡
ẋt xi t⎣ri t −
n
n
aij txj t − τij t −
j1
j1
0
−σij
bij t, sxj t sds
⎤
1.7
− ci tui t − di tui t − τi t⎦,
u̇t −ei tui t fi txi t gi txi t − δi t,
i 1, 2, . . . , n.
The sufficient conditions for which a part of the n-species is driven to extinction and the
surplus part of the n-species remains permanence are established.
However, we see that for general discrete n-species population systems the results for
which a part of the n-species is driven to extinction and the surplus part of the n-species
remains the permanence, up to now, are still not obtained. Therefore, motivated by the
above works, in this paper we study the following discrete nonautonomous Lotka-Volterra
competitive system with pure delays and feedback controls
⎧
⎨
xi k 1 xi k exp ri k −
⎩
−
n
aij kxj k − τij
j1
⎫
⎬
σij
n bijl kxj k − l − di kui k − τi ,
⎭
j1 l0
ui k 1 1 − ei kui k gi kxi k − δi ,
1.8
i 1, 2, . . . , n.
The main purpose is to establish a criterion for which guarantee the part species
xr1 , xr2 , . . . , xn in system 1.8 is driven to extinction. The method used in this paper is to
constructing the multiple discrete Lyapunov functions. On the permanence of the surplus
species x1 , x2 , . . . , xr , owing to the length of this paper, we will give the discussion in another
paper.
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Discrete Dynamics in Nature and Society
This paper is organized as follows. In next section, as preliminaries, some assumptions
and useful lemmas are introduced. In Section 3, the main results of this paper on the
extinction of a part of the n-species of system 1.8 are established. In Section 4, an example
is presented to illustrate the feasibility of our results.
2. Preliminaries
Let Z denote the set of all nonnegative integers. For any bounded sequence yk, we denote
y supk∈Z {yk}, y infk∈Z {yk}. Throughout this paper, we introduce the following
assumptions.
H1 ri k is a bounded sequence defined on Z; aij k, di k, gi k and bijl k i, j 1, 2, . . . , n, l 0, 1, . . . , σij are nonnegative bounded sequences defined on Z; τij ,
τi , δi , and σij i, j 1, 2, . . . , n are nonnegative integers.
H2 Sequences ei k i 1, 2, . . . , n satisfy 0 < ei k ≤ 1 for all k ∈ Z.
H3 There exist positive integers ω and λ such that for each i 1, 2, . . . , n
lim inf
n→∞
σii
aii s biil s > 0.
nω−1
ri s > 0,
sn
lim inf
n→∞
nλ−1
sn
2.1
l0
H4 There exists positive integer β such that for each i 1, 2, . . . , n
lim sup
n→∞
nβ−1
2.2
1 − ei s < 1.
n
Let R 0, ∞ and Rn {x x1 , x2 , . . . , xn : xi ∈ R , i 1, 2, . . . , n}. We denote
by int Rn the interior of Rn . For any nonnegative constants a and b with a < b, we denote
by a, bZ the set of all integers in the interval a, b. For some integer m ≥ 0, we denote by
D −m, 0Z the space of all nonnegative discrete time function φ : −m, 0Z → Rn with norm
φ sup{|φs| : s ∈ −m, 0Z }.
Let τ max{τij , τi , δi , σij : i, j 1, 2, . . . , n}. Motivated by the biological background
of system 1.8, in this paper we only consider all solutions of system 1.8 that satisfy the
following initial conditions:
xi s φi s ≥ 0,
ui s ψi s ≥ 0,
s ∈ −m, 0Z , i 1, 2, . . . , n,
2.3
where φ φ1 , φ2 , . . . , φn ∈ D −m, 0Z and ψ ψ1 , ψ2 , . . . , ψn ∈ D −m, 0Z . For any
φ, ψ ∈ D −m, 0Z , let z φ, ψ, by the fundamental theory of difference equations, system
1.8 has a unique solution xs, z, s, z satisfying the initial condition 2.3, where xs, z x1 s, z, x2 s, z, . . . , xn s, z and us, z u1 s, z, u2 s, z, . . . , un s, z. It is obvious that
solution xs, z, s, z is positive, that is, xi s, z > 0 and ui s, z > 0 i 1, 2, . . . , n for all
s ∈ Z.
We first consider the following nonautonomous difference inequality system:
yn 1 ≤ yn exp αn − βnyn ,
n ∈ Z,
2.4
Discrete Dynamics in Nature and Society
5
where αn and βn are bounded sequences defined on Z and βn ≥ 0 for all n ∈ Z. We
have the following result.
Lemma 2.1 see 11. Assume that there exists an integer λ > 0 such that
lim inf
n→∞
nλ
βk > 0,
2.5
kn
then there exists a constant M > 0 such that for any nonnegative solution yn of system 2.4 with
initial value yn0 y0 ≥ 0, where n0 ∈ Z is some integer,
lim sup yn < M.
2.6
n→∞
Next, we consider the following nonautonomous linear difference equation:
vn 1 γnvn ωn,
2.7
where γn and ωn are nonnegative bounded sequences defined on Z. We have the
following results.
Lemma 2.2 see 11. Assume that there exists an integer λ > 0 such that
lim sup
n→∞
nλ
2.8
γk < 1,
kn
then there exists a constant M > 0 such that for any nonnegative solution vn of system 2.7 with
initial value vn0 v0 ≥ 0, where n0 ∈ Z is some integer,
lim sup vn < M.
2.9
n→∞
Lemma 2.3 see 11. Assume that the conditions of Lemma 2.2 hold, then for any constants ε > 0
and M1 > 0 there exist positive constants δ δε
and n
n
ε, M1 such that for any n
0 ∈ Z and
0 ≤ v0 ≤ M1 , when ωn < δ for all n ≥ n
0 , one has
vn, n
0 , v0 < ε
,
∀n ≥ n
0 n
2.10
where vn, n
0 , v0 is the solution of 2.7 with initial value v
n0 v0 .
Lemma 2.4. Assume that assumptions H1 –H4 hold, then there exists a constant M0 > 0 such
that
lim sup xi k < M0 ,
k→∞
lim sup ui k < M0 ,
k→∞
i 1, 2, . . . , n
for any positive solution x1 k, x2 k, . . . , xn k, u1 k, u2 k, . . . , un k of system 1.8.
2.11
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Discrete Dynamics in Nature and Society
Proof. Let x1 k, x2 k, . . . , xn k, u1 k, u2 k, . . . , un k be any positive solution of system
1.8. For each i ∈ {1, 2, . . . , n}, we have
⎧
⎨
⎫
σij
n
n ⎬
bijl kxj k − l − di kui k − τi xi k 1 xi k exp ri k − aij kxj k − τij −
⎩
⎭
j1
j1 l0
≤ xi k exp{ri k},
2.12
then, for any integer k ≥ 0 and θ ≤ 0 with k θ ≥ 0, summing inequality 2.12 from k θ to
k − 1, we obtain
k−1
xi k θ ≥ xi k exp −
2.13
ri s .
skθ
Therefore, for any integer k ≥ τ, from 2.13 and the first equation of system 1.8 we obtain
xi k 1 ≤ xi k exp ri k − aii kxi k − τii −
σii
biil kxi k − l
l0
≤ xi k exp ri k − aii kxi k exp −
k−1
ri s
2.14
skθ
σii
k−1
.
biil kxi k exp −
ri s
−
l0
sk−l
Since for any k ≥ τ and l ∈ 0, τZ
k−1
2.15
ri s ≤ τriu ,
skθ
where riu supk∈Z ri k, we have from 2.14,
xi k i ≤ xi k exp ri k − hi
σii
aii k biil k xi k
2.16
l0
for any k ≥ τ, where hi exp{−τriu }.
We consider the following auxiliary equation:
yi k 1 ≤ yi k exp ri k − βi kyi k ,
k ∈ Z,
2.17
Discrete Dynamics in Nature and Society
7
ii
biil k, then by assumption H3 and applying Lemma 2.1 there
where βi k hi aii k σl0
exists a constant Ni > 0 such that
lim sup xi k < Ni
2.18
k→∞
for any positive solution yi k of 2.17. Therefore, from the comparison theorem of difference
equation, we finally obtain
lim sup xi k < Ni .
2.19
k→∞
Further form inequality 2.19, there exists a positive constant ki ≥ τ such that
xi k < Ni ,
xi k − δi < Ni
∀k ≥ ki .
2.20
Thus, from the second equation of system 1.8, we obtain
ui k 1 ≤ 1 − ei kui k fi k gi k Ni
2.21
for all k ≥ ki . We consider the following auxiliary equation:
vi k 1 1 − ei kvi k fi k gi k Ni ,
2.22
then by assumption H4 and applying Lemma 2.2, there exists a constant Ni∗ such that
lim sup vi k < Ni∗
k→∞
2.23
for any positive solution vi k of 2.22. Let vi∗ k be the solution of 2.22 with initial value
vi∗ ki ui ki , then from the comparison theorem of difference equation, we have ui k ≤
vi∗ ki for all k ≥ ki . Thus, we finally obtain
lim sup ui k < Ni∗ .
k→∞
2.24
Let N max1≤i≤n {Ni , Ni∗ }, then from 2.19 and 2.24 we finally see that the conclusions of
Lemma 2.4 hold.
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Discrete Dynamics in Nature and Society
3. Main Results
In this section, we discuss the extinction of the part of species xr1 , . . . , xn of system 1.8.
Define functions as follows:
Aij k aij k τij
σij
bijl k l,
l0
Di k di k τi ,
3.1
Ri k gi k δi ,
where i, j 1, 2, . . . , n.
Theorem 3.1. Assume that assumptions H1 –H4 hold and there exists an integer 1 ≤ r < n such
that for any h > r there exists an integer ih < h such that
kω−1
sk
lim sup kω−1
k→∞
sk
Dh k
> lim sup
lim inf
k → ∞ eh k
k→∞
rh s
rih s
Di k
lim sup h
< lim inf
k→∞
k → ∞ eih k
< lim inf
k→∞
Ahj k
Aih j k
∀j ≤ h,
kω−1
rh s Ahh k
Aih h k
sk
lim sup
,
−
Rh k k → ∞ kω−1 rih s
Rh k
3.2
sk
kω−1
rih s Aih ih k
Ahih k
sk
lim inf
,
−
Rih k k → ∞ kω−1 rh s
Rih k
sk
then for each i r 1, . . . , n one has
lim xi k 0,
lim ui k 0,
k→∞
k→∞
∞
xi k < ∞
3.3
k0
for any positive solution x1 k, x2 k, . . . , xn k, u1 k, u2 k, . . . , un k of system 1.8.
Proof. From assumption H2 , there exist constant η0 > 0 and integer K0 > 0 such that
kω−1
ri s ≥ η0
sk
∀k ≥ K0 .
3.4
Discrete Dynamics in Nature and Society
9
We first prove that limk → ∞ xn k 0. Let h n and ih p. From conditions 3.2 we can find
positive constants αn1 , αn2 , αn3 , αn4 , εn and integer Kn ≥ K0 such that
kω−1
sk
rn s
sk
rp s
kω−1
<
αn1 Anj k
αn1
− εn <
<
,
αn2
αn2 Apj k
Dn k αn4 αn1 Apn k − αn2 Ann k
>
>
,
en k
αn2
αn2 Rn k
3.5
Dp k αn3 αn2 Anp k − αn1 App k
<
<
ep k
αn1
αn1 Rp k
for all k ≥ Kn and j 1, 2, . . . , n. Consequently,
kω−1
−αn1 rp s αn2 rn s < −εn αn2 η0 ,
3.6
αn1 Apj k − αn2 Anj k < 0,
3.7
αn4 en k − αn2 Dn k < 0,
3.8
αn1 App k − αn2 Anp k αn3 Rp k < 0,
3.9
−αn3 ep k αn1 Dp k < 0
3.10
αn1 Apn k − αn2 Ann k − αn4 Rn k < 0
3.11
sk
for all k ≥ Kn and j 1, 2, . . . , n.
Let xk, uk x1 k, x2 k, . . . , xn k, u1 k, u2 k, . . . , un k be any positive
solution of system 1.8. Constructing the following discrete Lyapunov functional
Vn k xp−αn1 kxnαn2 k
⎧
⎨
× exp αn3 up k − αn4 un k
⎩
αn1
k−1
n pj k−1
n apj s τpj xj s αn1
bpjl s lxj s
σ
j1 sk−τpj
k−1
αn1
j1 l0 sk−l
k−1
n dp s τp up s − αn2
anj s τnj xj s
j1 sk−τnj
sk−τp
− αn2
σnj k−1
n j1 l0 sk−l
αn3
k−1
sk−δp
k−1
bnjl s lxj s − αn2
sk−τn
gp s δp xp s − αn4
k−1
sk−δn
dn s τn un s
⎫
⎬
gn s δn xn s .
⎭
3.12
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Discrete Dynamics in Nature and Society
By calculating, we obtain
Vn k 1
Vn k
exp
⎧
⎨
⎩
⎡
− αn1 ⎣rp k −
n
apj kxj k − τpj −
j1
σpj
n ⎤
bpjl kxj k l − dp kup k − τp ⎦
j1 l0
⎤
σnj
n
n αn2 ⎣rn k − anj kxj k − τnj −
bnjl kxj k l − dn kun k − τn ⎦
⎡
j1
j1 l0
αn3 −ep kup k gp kxp k − δp
αn1
− αn4 −en kun k gn kxn k − δn n
apj k τpj xj k − apj kxj k − τpj
j1
αn1
σpj
n bpjl k lxj k − bpjl kxj k − l
j1 l0
αn1 dp k τp up k − dp kup k − τp
− αn2
n
anj k τnj xj k − anj kxj k − τnj
j1
− αn2
σnj
n bnjl k lxj k − bnjl kxj k − l
j1 l0
− αn2 dn k τn un k − dn kun k − τn αn3 gp k δp xp k − gp kxp k − δp
⎫
⎬
− αn4 gn k δn xn k − gn kxn k − δn ⎭
exp
⎧
⎨
⎩
− αn1 rp k αn2 rn k − αn3 ep k − αn1 Dp k up k
− αn2 Dn k − αn4 en kun k n−1
αn1 Apj k − αn2 Anj k xj k
j/
p
αn1 App k − αn2 Anp k − αn3 Rp k xp k
⎫
⎬
αn1 Apn k − αn2 Ann k − αn4 Rn k xn k .
⎭
3.13
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11
From inequalities 3.7–3.11, we can obtain
Vn k 1 ≤ Vn k exp −αn1 rp k αn2 rn k
3.14
∀k ≥ Kn .
For any k > Kn , we choose an integer m ≥ 0, such that k ∈ Kn mω, Kn m 1ω, then
from 3.6 and 3.14 we further have
k−1
Vn k ≤ Vn Kn exp
−αn1 rp s αn2 rn s
sKn
K
≤ Vn Kn exp
n mω−1
sKn
k−1
−αn1 rp s αn2 rn s
sKn mω
3.15
≤ Vn Kn exp −εn αn2 η0 m Mn
≤ Vn Kn exp{−λn k Mn∗ },
where
λn Kn
Mn∗ Mn εn αn2 η0 1 ,
ω
Mn sup −αn1 rp k αn2 rn k ω.
εn αn2 η0
,
ω
3.16
k∈Z
On the other hand, from assumptions H1 , we have
Vn k ≥ xp−αn1 kxnαn2 k
⎧
n
k−1
⎨
× exp −αn4 un k − αn2
anj s τnj xj s
⎩
j1
sk−τ
nj
−
n
αn2
j1
− αn4
σnj k−1
bnjl s lxj s − αn2
l0 sk−l
k−1
k−1
dn s τn un s
sk−τn
gn s δn xn s
sk−δn
≥ xp−αn1 kxnαn2 k
⎧
n
⎨
× exp −αn4 M0 − αn2 τ anj M0
⎩
j1
⎫
σnj
n ⎬
− αn2
bnjl lM0 − αn2 τdn M0 − αn4 τgn M0
⎭
j1 l0
3.17
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Discrete Dynamics in Nature and Society
for all k ≥ τ, where constant M0 is given in Lemma 2.4. Hence, there exist a positive constant
Qn > 0 such that
Vn k ≥ xp−αn1 kxnαn2 kQn
∀k ≥ τ.
3.18
From 3.15 and 3.18, we obtain
"1/αn2
!
xn k ≤ Qn−1 xpαn1 kVn Kn exp{−λn k Mn∗ }
≤ Qn∗ exp{−λ∗n k}
3.19
for all k ≥ Kn τ, where
!
"1/αn2
Qn∗ sup xpαn1 kVn Kn Qn−1 exp Mn∗
,
λ∗n k∈Z
λn
.
αn2
3.20
From 3.19, we finally obtain
lim xn k 0,
k→∞
∞
xn k < ∞.
3.21
k0
Next, we consider the second equation of system 1.8, applying Lemma 2.3 we can
easily obtain un k → 0 as k → ∞.
Now, we suppose that for any t > r, we have obtained
lim xi k 0,
lim ui k 0,
k→∞
k→∞
∞
xn k < ∞
3.22
k0
for all i > t. We further will prove
lim xt k 0,
k→∞
∞
xt k < ∞.
3.23
k0
From conditions 3.2, we can choose positive constants αt1 , αt2 , αt3 , αt4 , εt and integer Kt ≥ Kn
such that
kω−1
sk
rt s
sk
rq s
kω−1
<
Atj k
αt1
αt1
,
− εt <
<
αt2
αt2 Aqj k
Dt k αt4 αt1 Aqn k − αt2 Att k
>
,
>
et k
αt2
αt2 Rt k
Dq k αt3 αt2 Atq k − αt1 Aqq k
<
>
eq k
αt1
αt1 Rq k
3.24
Discrete Dynamics in Nature and Society
13
for all k ≥ Kt and j 1, 2, . . . , t, where ih q. Consequently,
kω−1
−αt1 rq s αt2 rt s < −εt αt2 η0 ,
3.25
αt1 Aqj k − αt2 Atj k < 0,
3.26
αt4 et k − αt2 Dt k < 0,
3.27
αt1 Aqq k − αt2 Atq k αt3 Rq k < 0,
3.28
−αt3 eq k αt1 Dq k < 0
3.29
αt1 Aqt k − αt2 Att k − αt4 Rt k < 0
3.30
sk
for all k ≥ Kt and j 1, 2, . . . , t. Constructing the following discrete Lyapunov functional
Vt k xq−αt1 kxtαt2 k
⎧
⎨
× exp αt3 uq k − αt4 ut k
⎩
αt1
k−1
n qj k−1
n aqj k τqj xj s αt1
bqjl k lxj s
σ
j1 sk−τqj
k−1
αt1
j1 l0 sk−l
dq k τq uq s − αt2
σtj k−1
n αt3
sk−δq
3.31
atj k τtj xj s
k−1
btjl k lxj s − αt2
j1 l0 sk−l
k−1
j1 sk−τtj
sk−τq
− αt2
k−1
n dt k τt ut s
sk−τt
gq k δq xq s − αt4
k−1
sk−δt
⎫
⎬
gt k δt xt s .
⎭
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Discrete Dynamics in Nature and Society
Calculating Vt k 1/Vt k, similarly to Vn k, we can obtain
⎧
⎨
Vt k 1
exp − αt1 rq k αt2 rt k
⎩
Vt k
− αt3 eq k − αt1 Dq k uq k
−αt2 Dt k αt4 et kut k
n−1
αt1 Aqj k − αt2 Atj k xj k
3.32
j/
q,t
αt1 Aqq k − αt2 Atp k − αt2 Rq k xq k
⎫
⎬
αt1 Aqt k − αt2 Att k − αt4 Rt k xt k
⎭
for all k ≥ Kt . From inequalities 3.26–3.30, we further obtain
Vt k 1 ≤ Vt k exp −αt1 rq k αt2 rt k P k
3.33
for all k ≥ Kt , where
P k n
αt1 Aqj k αt2 Atj k xj k.
3.34
jt1
From 3.19 we have P k → 0 as k → ∞. Hence,
lim
k→∞
kω−1
P s 0.
3.35
sk
Thus, from 3.22 we can obtain that there exists an integer Kt∗ ≥ Kt such that
kω−1
1
−αt1 rq s αt2 rt s P s < − εt αn2 η0
2
sk
3.36
for all k > Kt∗ . By calculating, from 3.33 we obtain
Vt k ≤ Vt Kt∗ exp
⎧
k−1
⎨
⎩sK∗
t
⎫
⎬
−αt1 rq s αt2 rt s P s .
⎭
3.37
Discrete Dynamics in Nature and Society
15
From this, a similar argument as in the proof of 3.15–3.19, we further can obtain
xt k ≤ Qt∗ exp{−λ∗t k}
∀k ≥ Kt∗ ,
3.38
where Qt∗ and λ∗t are two positive constants. From 3.38, we finally obtain
∞
lim xt k 0,
k→∞
xt k < ∞.
3.39
k0
Next, we consider the second equation of system 1.8, applying Lemma 2.3 we can
easily obtain ut k → 0 as k → ∞.
Finally, according to the induction principle, we have
lim xi k 0,
lim ui k 0,
k→∞
k→∞
∞
xi k < ∞ ∀i > r.
3.40
k0
This completes the proof of Theorem 3.1.
As consequences of Theorem 3.1, we consider the following two special cases of
system 1.8.
Case 1. Nondelayed nonautonomous discrete n-species Lotka-Volterra competitive systems
with feedback controls
xi k 1 xi k exp ri k −
n
aij kxj k − di kui k ,
j1
ui k 1 1 − ei kui k gi kxi k,
3.41
i 1, 2, . . . , n.
For system 3.41, assumptions H1 and H3 become into the following form
H1
ri k is a bounded sequence defined on Z; aij k, di k and gi k i, j 1, 2, . . . , n
are nonnegative bounded sequences defined on Z.
H3
There exist positive integers ω and λ such that for each i 1, 2, . . . , n
lim inf
n→∞
nω−1
ri s > 0,
sn
lim inf
n→∞
nλ−1
aii s > 0.
sn
Directly from Theorem 3.1, we have the following corollary.
3.42
16
Discrete Dynamics in Nature and Society
Corollary 3.2. Assume that assumptions H1
, H2 , H3
and H4 hold and there exists a integer
1 ≤ r < n such that for any h > r there exists an integer ih < h such that
kω−1
sk
lim sup kω−1
rh s
ahj k
aih j k
∀j ≤ h,
rih s
kω−1
rh s ahh k
aih h k
dh k
sk
lim inf
> lim sup
lim sup
,
−
k → ∞ eh k
gh k k → ∞ kω−1 rih s
gh k
k→∞
k→∞
k→∞
sk
di k
lim sup h
< lim inf
k→∞
k → ∞ eih k
< lim inf
3.43
sk
kω−1
rih s aih ih k
ahih k
sk
lim inf
,
−
gih k k → ∞ kω−1 rh s
gih k
sk
then for each i r 1, . . . , n one has
lim xi k 0,
k→∞
lim ui k 0,
k→∞
∞
xi k < ∞
3.44
k0
for any positive solution x1 k, x2 k, . . . , xn k, u1 k, u2 k, . . . , un k of system 3.41.
Case 2. Pure delayed nonautonomous discrete n-species Lotka-Volterra competitive systems
without feedback controls
xi k 1 xi k exp ri k−
n
σij
n aij kxj k−τij −
bijl kxj k−l ,
j1
i 1, 2, . . . , n.
j1 l0
3.45
For system 3.41, assumption H1 becomes into the following form.
H1
rik is a bounded sequence defined on Z; aij k and bijl k i, j 1, 2, . . . , n, l 0, 1, . . . , σij are nonnegative bounded sequences defined on Z; τij and σij i, j 1, 2, . . . , n are nonnegative integers.
Directly from Theorem 3.1, we have the following corollary.
Corollary 3.3. Assume that assumptions H1
and H3 hold and there exists an integer 1 ≤ r < n
such that for any h > r there exists an integer ih < h such that
kω−1
Ahj k
rh s
sk
< lim inf
lim sup kω−1
k → ∞ Aih j k
rih s
k→∞
sk
∀j ≤ h,
3.46
then for each i r 1, . . . , n one has
lim xi k 0,
k→∞
∞
xi k < ∞
k0
for any positive solution x1 k, x2 k, . . . , xn k of system 3.45.
3.47
Discrete Dynamics in Nature and Society
17
Remark 3.4. By comparison, we easily see that the results obtained in this paper are a very
good extension of the corresponding results obtained in 9 on the extinction of species
for nonautonomous continuous Lotka-Volterra competitive system with pure-delays and
feedback controls to discrete ones.
4. An Example
We consider the following periodic discrete three-species Lotka-Volterra competitive system
with pure-delays and feedback controls:
xi k 1 xi k exp ri k −
3
aij kxj k − τij − di kui k − τi ,
j1
ui k 1 1 − ei kui k gi kxi k − δi ,
4.1
i 1, 2, 3.
where
1 π
1
sin
k ,
2
3 2
1 π
1
r3 k 3
sin
k
,
4
5 2
π
1
2 1
a12 k sin
k ,
6
3 2
1 π
a21 k 1 cos2
k ,
3 2
1 π
a31 k 4 sin2
k ,
3 2
1 π
a33 k 8 sin2
k ,
3 2
1 π
1
d2 k sin2
k ,
2
3 2
1 π
1
e1 k 1 cos2
k
,
3
3 2
π
1
2 1
e3 k k
,
1 sin
3
3 2
π
2 1
g2 k 1 cos
k ,
3 2
r1 k 1 π
1
sin
k ,
2
4 2
π
1
2 1
a11 k k
,
1 cos
3
3 2
π
1
2 1
a13 k cos
k ,
4
3 2
1 π
1
a22 k cos2
k ,
2
3 2
1 π
9
a32 k sin2
k ,
2
3 2
1 π
1
d1 k 1 sin2
k
,
10
3 2
1 π
d3 k 2 cos2
k ,
3 2
1 π
1
e2 k 1 sin2
k
,
4
3 2
π
1
2 1
g1 k k
,
1 cos
3
3 2
π
2 1
g3 k 1 cos
k .
3 2
r2 k 4.2
18
Discrete Dynamics in Nature and Society
Clearly, in system 4.2, r 2, h 3, and ih 1. By calculating, we obtain
k3
k3
31/4 sin1/5 π/2s 3
sk r3 s
lim sup k3
lim sup sk
,
k3
2
k→∞
k→∞
sk r1 s
sk 1/2 sin1/3 π/2s
k3
sk r1 s
lim inf k3
k→∞
sk r3 s
k3
2
sk 1/2 sin1/3 π/2s
,
lim inf k3
k→∞
3
sk 31/4 sin1/5 π/2s
lim inf
A33 k
8 sin2 1/3 π/2k τ33 32
,
lim inf
≥
2
k → ∞ 1/4 cos 1/3 π/2k τ13 A13 k
5
lim inf
A32 k
9/2 sin2 1/3 π/2k τ32 54
≥
,
lim inf
k → ∞ 1/6 sin2 1/3 π/2k τ13 A12 k
14
k→∞
k→∞
A31 k
4 sin2 1/3 π/2k τ31 lim inf
≥ 6,
k → ∞ A11 k
k → ∞ 1/31 cos2 1/3 π/2k τ11 k3
A13 k
A33 k
sk r3 s
lim sup
lim sup
−
R3 k k → ∞ k3 r1 s
R3 k
k→∞
lim inf
sk
lim sup
$
#
3/2 1/4 cos2 1/3 π/2k τ13 − 8 sin2 1/3 π/2k τ33 1 cos2 1/3 π/2k δ3 k→∞
3/21/4 1 − 8
< 0,
2
k3
A31 k
A11 k
sk r1 s
lim inf
lim inf
−
k→∞
R1 k k → ∞ k3 r3 s
R1 k
sk
$ #
2/3 4 sin2 1/3 π/2k τ31 − 1/3 1 cos2 1/3 π/2k τ11 lim inf
k→∞
1/31 cos2 1/3 π/2k δ1 ≤
2/34 − 1/32
3,
2/3
≥
lim inf
k→∞
D3 k
2 cos2 1/3 π/2k τ3 lim inf
$ ≥ 3,
#
k→∞
e3 k
1/3 1 sin2 1/3 π/2k
lim sup
k→∞
D1 k
lim sup
e1 k
k→∞
$
#
1/10 1 sin2 1/3 π/2k τ1 1/31 cos2 1/3 π/2k
≤
3
.
5
4.3
From these inequalities we see that all conditions 3.2 in Theorem 3.1 hold. In addition, we
also see that assumptions H1 –H4 obviously hold. Therefore, by Theorem 3.1 we obtain
that species x3 in system 4.3 is extinct.
Discrete Dynamics in Nature and Society
19
Acknowledgment
This work was supported by the National Natural Science Foundation of China 10961022,
10901130, the Scientific Research Programmes of Colleges in Xinjiang XJEDU2007G01,
XJEDU2006I05, XJEDU2008S10, and the National Key Technologies R & D Program of China
2008BAI68B01 and the Foundation of Education Committee of Jiangxi GJJ0931.
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