Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2010, Article ID 381750, 22 pages
doi:10.1155/2010/381750
Research Article
Permanence and Positive Periodic Solutions of
a Discrete Delay Competitive System
Wenjie Qin and Zhijun Liu
Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China
Correspondence should be addressed to Zhijun Liu, zhijun liu47@hotmail.com
Received 30 December 2009; Accepted 1 March 2010
Academic Editor: Antonia Vecchio
Copyright q 2010 W. Qin and Z. Liu. 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 time non-autonomous two-species competitive system with delays is proposed, which
involves the influence of many generations on the density of species population. Sufficient
conditions for permanence of the system are given. When the system is periodic, by using the
continuous theorem of coincidence degree theory and constructing a suitable Lyapunov discrete
function, sufficient conditions which guarantee the existence and global attractivity of positive
periodic solutions are obtained. As an application, examples and their numerical simulations are
presented to illustrate the feasibility of our main results.
1. Introduction
In recent years, the application of theories of functional differential equations in mathematical
ecology has developed rapidly. Various delayed models have been proposed in the study
of population dynamics, ecology, and epidemic. In fact, more realistic population dynamics
should take into account the effect of delay. Also, delay differential equations may exhibit
much more complicated dynamic behaviors than ordinary differential equations since a
delay could cause a stable equilibrium to become unstable and cause the population to
fluctuate see 1. One of the famous models for dynamics of population is the delay LotkaVolterra competitive system. Owing to its theoretical and practical significance, various delay
competitive systems have been studied extensively see 2–8. Although much progress
has been seen for Lotka-Volterra competitive systems, such systems are not well studied in
the sense that most results are continuous time versions related. Many authors 9–11 have
argued that the discrete time models governed by difference equations are more appropriate
than the continuous ones when the populations have non-overlapping generations. Discrete
time models can also provide efficient computational models of continuous models for
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Discrete Dynamics in Nature and Society
numerical simulations. Therefore, the dynamic behaviors of population models governed
by difference equations have been studied by many authors, see 12–18 and the references
cited therein. Noting that some studies of the dynamics of natural populations indicate that
the density-dependent population regulation probably takes place over many generations
19, 20, many authors have discussed the influence of many past generations on the density
of species population and discussed the dynamic behaviors of competitive, predator-prey,
and cooperative systems see 21–24.
Motivated by the above work 19–24, in this paper we will investigate the following
discrete time non-autonomous two-species competitive system with delays:
m
m
c2l kx2 k − l
,
x1 k 1 x1 k exp r1 k − a1l kx1 k − l −
1 x2 k − l
l0
l0
x2 k 1 x2 k exp r2 k −
m
a2l kx2 k − l −
l0
m
1.1
c1l kx1 k − l
,
1 x1 k − l
l0
with the initial conditions
xi −l ≥ 0,
xi 0 > 0,
l 0, 1, . . . , m, i 1, 2,
1.2
where xi k represents the density of population xi at the kth generation, ri k is the intrinsic
growth rate of population xi at the kth generation, ail k measures the intraspecific influence
of the k − lth generation of population xi on the density of its own population, and cjl k
stands for the interspecific influence of the k −lth generation of population xj on population
j. The coefficients {ri k}, {ail k}, and {cil k} i 1, 2 are bounded
xi , i, j 1, 2 and i /
nonnegative sequences. The exponential form of the equations in system 1.1 ensures that
any forward trajectory {x1 k, x2 k } of system 1.1 with initial conditions 1.2 remains
positive for all k ∈ {0, 1, 2, . . .}. For the investigations of some continuous versions of 1.1 we
refer to 8, 25, 26 and the references cited therein.
The principle aim of this paper is to study the dynamic behaviors of system 1.1, such
as permanence, existence, and global attractivity of positive periodic solutions. To the best of
our knowledge, no work has been done for the discrete non-autonomous difference system
1.1. The paper is organized as follows. In Section 2, we obtain sufficient conditions which
guarantee the permanence of system 1.1. In Section 3, a good understanding of the existence
and global attractivity of positive periodic solutions of system 1.1 is gained by using the
method of coincidence degree theory and a Lyapunov discrete function. Some illustrative
examples are given to demonstrate the feasibility of the obtained results in Section 4. To do
this, we need to give the following notations and Definitions 1.1 and 1.2.
For the simplicity and convenience of exposition, throughout this paper we let
Z, Z , R , and R2 denote the sets of all integers, nonnegative integers, nonnegative real
numbers and two-dimensional Euclidian vector space, respectively. Meanwhile, we denote
that b∗ supk∈Z bk, b∗ infk∈Z bk for any bounded sequence bk.
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3
Definition 1.1. System 1.1 is said to be permanent if there exist positive constants mi and Mi
such that
mi ≤ lim inf xi k ≤ lim sup xi k ≤ Mi ,
k → ∞
k → ∞
i 1, 2
1.3
for any positive solution {x1 k, x2 k } of system 1.1.
Definition 1.2. A positive periodic solution {x1 k, x2 k } of system 1.1 is said to be
globally attractive if each other solution {x1 k, x2 k } of system 1.1 satisfies
lim |x1 k − x1 k| 0,
lim |x2 k − x2 k| 0.
k → ∞
k → ∞
1.4
2. Permanence
In this section, we will establish sufficient conditions for the permanence of system 1.1. To
do this, we first give two lemmas which will be useful for establishing our main result in this
section.
Lemma 2.1 see 27, Lemma 1. Assume that {xk} satisfies xk > 0 and
xk 1 ≤ xk exprk1 − αxk
2.1
for k ∈ k1 , ∞, where α is a positive constant and k1 ∈ Z . Then
lim sup xk ≤
k → ∞
1
expr ∗ − 1.
αr ∗
2.2
Lemma 2.2 see 27, Lemma 2. Assume that {xk} satisfies xk > 0 and
xk 1 ≥ xk exprk1 − αxk
2.3
for k ∈ k2 , ∞, lim supk → ∞ xk ≤ M, and xk2 > 0, where α is a constant such that αM > 1
and k2 ∈ Z . Then
lim inf xk ≥
n → ∞
1
expr ∗ 1 − αM.
α
2.4
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Discrete Dynamics in Nature and Society
Before stating Theorem 2.3, for the sake of convenience, we set
exp ri∗ − 1
Mi m
∗ ,
l0 ail∗ exp −ri l
def
i def
exp
G
lim Δε ,
ε→0 i
m
m
l0
l0
Δi Mi ε
def
Δεi def
a∗il m
i
a∗ G
mil ∗ ,
ri∗ − l0 cjl
l0
2.5
cjl∗ − ri∗ l ,
where i, j 1, 2, i /
j, ε > 0 is a sufficiently small constant.
We are now in a position to state our main result of this section on the permanence of
system 1.1.
Theorem 2.3. If the following assumptions:
m
m
∗
∗
min r1∗ − c2l
, r2∗ − c1l
l0
> 0,
l0
2.6
min{M1 Δ1 , M2 Δ2 } > 1
hold, then system 1.1 is permanent.
Proof. Clearly, any solution {x1 k, x2 k } of system 1.1 satisfies x1 k > 0, x2 k > 0. The
following two steps are considered.
Step 1. According to Definition 1.1, we will prove that any positive solution of system 1.1
satisfies lim supk → ∞ xi k ≤ Mi for i 1, 2.
It follows from the first equation of system 1.1 that
x1 k 1 ≤ x1 k expr1 k.
2.7
2
bi 1 if k1 > k2 for any bounded sequence bi.
We will make a convention that kik
1
For k ≥ m, l 0, 1, . . . , m, we can obtain
k−1
ik−l
x1 i 1 ≤
k−1
x1 i expr1 i,
2.8
ik−l
that is,
x1 k ≤ x1 k − l exp
k−1
r1 i ,
2.9
ik−l
in other words,
x1 k − l ≥ x1 k exp −
k−1
ik−l
r1 i .
2.10
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5
Consequently, we have
x1 k 1 ≤ x1 k exp r1 k −
m
a1l kx1 k exp −
l0
k−1
≤ x1 k exp r1 k 1 −
r1 i
ik−l
m
a1l∗ exp −r1∗ l x1 k
≤ x1 k exp r1 k −
l0
m
l0
2.11
a1l∗ exp −r1∗ l
x1 k .
r1∗
By Lemma 2.1, we can derive that
exp r1∗ − 1
lim sup x1 k ≤ m
∗ M1 .
k → ∞
l0 a1l∗ exp −r1 l
2.12
Similar to the above argument, we can verify that
exp r2∗ − 1
∗ M2 .
l0 a2l∗ exp −r2 l
lim sup x2 k ≤ m
k → ∞
2.13
Step 2. By a similar procedure to Step 1, we will prove that any positive solution of system
1.1 satisfies lim infk → ∞ xi k ≥ mi , where
∗
exp ri∗ − m
l0 cjl 1 − Δi Mi ,
mi Δi
def
i, j 1, 2, i / j.
2.14
For any sufficiently small ε > 0, according to 2.5, there exists a positive integer k0
such that x1 k ≤ M1 ε for all k ≥ k0 . Thus, for k ≥ k0 m, it follows from the first equation
of system 1.1 that
m
m
x1 k 1 ≥ x1 k exp r1 k − M1 ε a1l k − c2l k .
l0
2.15
l0
Therefore, for all l 0, 1, . . . , m and k ≥ k0 m, it follows that
k−1
x1 i 1 ≥
ik−l
k−1
m
m
x1 i exp r1 i − M1 ε a1l i − c2l i ,
l0
ik−l
2.16
l0
that is,
x1 k − l ≤ x1 k exp
k−1
ik−l
m
m
x1 kG1 ,
M1 ε a1l i c2l i − r1 i
l0
l0
2.17
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Discrete Dynamics in Nature and Society
where
def
G1 exp
k−1
m
m
M1 ε a1l i c2l i − r1 i .
ik−l
l0
2.18
l0
Combining 2.5 and 2.17 with the first equation of system 1.1 leads to
x1 k 1 ≥ x1 k exp r1∗ −
≥ x1 k exp r1∗ −
m
l0
m
l0
∗
c2l
∗
c2l
−
m
l0
−
m
l0
a∗1l x1 k
− l
a∗1l G1 x1 k
m
∗ l0 a1l G1
∗
1−
≥ x1 k exp
r1∗ − c2l
m ∗ x1 k
r1∗ − l0 c2l
l0
m
∗
ε
x1 k exp
r1∗ − c2l 1 − Δ1 x1 k .
m
2.19
l0
And hence, by applying Lemma 2.2 and letting ε → 0, it follows from 2.5-2.6 and
2.19 that
lim inf x1 k ≥
∗
exp r1∗ − m
l0 c2l 1 − Δ1 M1 k → ∞
Δ1
m1 .
2.20
Analogously, from the second equation of system 1.1, we can verify that
∗
exp r2∗ − m
l0 c1l 1 − Δ2 M2 lim inf x2 k ≥
m2 .
k → ∞
Δ2
2.21
The proof of Theorem 2.3 is completed by combining Steps 1 and 2.
3. Existence and Global Attractivity of Positive Periodic Solutions
In this section, we will give two main results. We first derive sufficient conditions for the
existence of positive periodic solutions of system 1.1. We further assume that ri , ail , cil :
Z → R are positive ω-periodic for system 1.1, that is,
ri k ω ri k,
cil k ω cil k,
ail k ω ail k,
l 0, 1, 2, . . . , m, i 1, 2
3.1
for any k ∈ Z, where ω, a fixed positive integer, denotes the prescribed common period of the
parameters in system 1.1.
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7
In order to obtain sufficient conditions for the existence of positive periodic solutions
of system 1.1, we will use the method of coincidence degree. For convenience, we will
summarize in the following a few concepts and results from 28 that will be useful in this
section.
Let X and Y be two Banach spaces. Consider an operator equation
Lx λNx,
λ ∈ 0, 1,
3.2
where L : Dom L ∩ X → Y is a linear operator and λ is a parameter. Let P and Q denote two
projectors such that
P : X ∩ Dom L −→ Ker L,
Q : Y −→
Y
.
Im L
3.3
Denote that J : Im Q → Ker L is an isomorphism of Im Q onto Ker L. Recall that a
linear mapping L : Dom L ∩ X → Y with Ker L L−1 0 and Im L LDom L will be called
a Fredholm mapping if the following two conditions hold:
i Ker L has a finite dimension;
ii Im L is closed and has a finite codimension.
Recall also that the codimension of Im L is the dimension of Y/ Im L, that is, the
dimension of the cokernel coker L of L.
When L is a Fredholm mapping, its index is the integer Ind L dim Ker L −
codim Im L.
We will say that a mapping N is L-compact on Ω if the mapping QN : Ω → Y is
continuous. QNΩ is bounded and Kp I − QN : Ω → X is compact, that is, it is continuous
and Kp I − QNΩ is relatively compact, where Kp : Im L − Dom L ∩ Ker P is an inverse of
the restriction Lp of L to Dom L ∩ Ker P , so that LKp I and Kp I − P .
Lemma 3.1 see 28, Continuation Theorem. Let X and Y be two Banach spaces and let L be a
Fredholm mapping of index zero. Assume that N : Ω → Y is L-compact on Ω with Ω open bounded
in X. Furthermore assume that
a for each λ ∈ 0, 1, x ∈ ∂Ω ∩ Dom L, Lx / λNx,
b QNx /
0 for each x ∈ ∂Ω ∩ Ker L,
c deg{JQNx, Ω ∩ Ker L, 0} /
0.
Then the equation Lx Nx has at least one solution in Dom L ∩ Ω.
In what follows, we will use the following notations:
Iω {0, 1, . . . , ω − 1},
f
1 ω−1
fk,
ω k0
f L minfk,
k∈Iω
f U maxfk,
where {fk} is an ω-periodic sequence of real numbers defined for k ∈ Z.
k∈Iω
3.4
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Discrete Dynamics in Nature and Society
Lemma 3.2 see 29, Lemma 3.2. Let f : Z → R be ω-periodic, that is, fk ω fk, then
for any fixed k1 , k2 ∈ Iω and any k ∈ Z, one has
fk ≤ fk1 ω−1
fs 1 − fs,
s0
fk ≥ fk2 −
ω−1
fs 1 − fs.
3.5
s0
Denote that
l2 x {xk} : xk ∈ R2 , k ∈ Z .
3.6
For a a1 , a2 ∈ R2 , define |a| max{a1 , a2 }. Let lω ⊂ l2 denote the subspace of all ωperiodic sequences equipped with the usual supremum norm · , that is,
x max|xk| for x {xk : k ∈ Z} ∈ lω .
k∈Iω
3.7
Then it follows that lω is a finite dimensional Banach space.
Let
l0ω
ω
x {xk ∈ l } :
ω−1
xk 0 ,
k0
lcω x {xk ∈ lω } : xk h ∈ R2 , k ∈ Z .
3.8
Then it follows that l0ω and lcω are both closed linear subspaces of lω and
lω l0ω ⊕ lcω ,
dim lcω 2.
3.9
We are now in a position to state one of the main results of this section on the existence
of positive periodic solutions of system 1.1.
Theorem 3.3. Assume that
r1 >
m
c2l ,
l0
r2 >
m
c1l .
3.10
l0
Then system 1.1 has at least one positive ω-periodic solution.
Proof. We first make the change of variables
x1 k exp y1 k ,
x2 k exp y2 k .
3.11
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9
By substituting 3.11 into system 1.1, we can get
m
m c
2l k exp y2 k − l
y1 k 1 − y1 k r1 k − a1l k exp y1 k − l −
,
l0
l0 1 exp y2 k − l
m
m c
1l k exp y1 k − l
y2 k 1 − y2 k r2 k − a2l k exp y2 k − l −
.
l0
l0 1 exp y1 k − l
3.12
It is easy to see that if system 3.12 has one ω-periodic solution, then system 1.1 has one
positive ω-periodic solution. Therefore, to complete the proof, it is only to show that system
3.12 has at least one ω-periodic solution.
Set X Y lω . Denote by L : X → X the difference operator given by Ly {Lyk}
with
Ly k yk 1 − yk
for y ∈ X, k ∈ Z,
3.13
and N : X → X as follows:
⎤
⎡
m
m c
2l k exp y2 k − l
⎥
⎢r1 k − a1l k exp y1 k − l −
1 exp y2 k − l ⎥
⎢
y1
l0
l0
⎢
⎥
⎢
N
⎥
⎢
⎥
m
m c
y2
1l k exp y1 k − l ⎦
⎣
r2 k − a2l k exp y2 k − l −
l0
l0 1 exp y1 k − l
3.14
for any y1 , y2 ∈ X and k ∈ Z. It is easy to see that L is a bounded linear operator and
Ker L lcω ,
Im L l0ω ,
dim Ker L 2 codim Im L,
3.15
then we get that L is a Fredholm mapping of index zero.
Define
⎡
⎤
1 ω−1
y1 s⎥
⎢
⎢ ω s0
⎥
y1
y1
⎢
⎥
P
Q
⎢
⎥,
⎢ ω−1
⎥
y2
y2
⎣1
⎦
y2 s
ω s0
y1
y2
∈ X Y.
3.16
It is not difficult to show that P and Q are continuous projectors such that
Im P Ker L,
Ker Q Im L ImI − Q.
3.17
Furthermore, the inverse to L Kp : Im L → Dom L ∩ Ker P exists and is given by
k−1
1 ω−1
ys −
Kp y ω − sys.
ω s0
s0
3.18
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Discrete Dynamics in Nature and Society
Then QN : X → Y and Kp I − QN : X → X are given by
QNy 1 ω−1
Nys ,
ω s0
k−1
1 ω−1
k 1 ω ω−1
Kp I − QNy −
Nys −
Nys.
−
sNys
−
ω
ω s0
ω
2ω
s0
s0
3.19
In order to apply Lemma 3.1, we need to search for an appropriately open, bounded
subset Ω.
Corresponding to the operator equation Ly λNy, λ ∈ 0, 1, we have
m c
2l k exp y2 k − l
y1 k 1 − y1 k λ r1 k − a1l k exp y1 k − l −
,
l0
l0 1 exp y2 k − l
m
m c
1l k exp y1 k − l
y2 k 1 − y2 k λ r2 k − a2l k exp y2 k − l −
.
l0
l0 1 exp y1 k − l
m
3.20
Suppose that y {yk} {y1 k, y2 k } ∈ X is a solution of 3.20 for a certain λ ∈ 0, 1.
Summing both sides of 3.20 from 0 to ω − 1 with respect to k, we can derive
m c
ω−1
2l k exp y2 k − l
r1ω a1l k exp y1 k − l ,
k0 l0
k0 l0 1 exp y2 k − l
m
ω−1
m c
ω−1
1l k exp y1 k − l
r2ω a2l k exp y2 k − l .
k0 l0
k0 l0 1 exp y1 k − l
m
ω−1
3.21
Since y {yk} ∈ X, there exist ξi ∈ Iω such that
yi ξi min yi k ,
k∈Iω
i 1, 2.
3.22
It follows from 3.21 that
m c k exp y ξ ω−1
jl
j i
r iω ≥
ail k exp yi ξi k0 l0
k0 l0 1 exp yj ξi ω−1
m
>
ω−1
m
ail k exp yi ξi k0 l0
ω
m
ail exp yi ξi ,
l0
i, j 1, 2, i /
j,
3.23
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which implies
ri
yi ξi < ln m
l0
def
where Ai m
l0
ail
ln
ri
,
Ai
i 1, 2,
3.24
ail , besides, from 3.20 and 3.21
ω−1
m
m c k exp y k − l
jl
j
yi k 1 − yi k ≤ λ
ri k ail k exp yi k − l k0
k0
l0
l0 1 exp yj k − l
ω−1
≤ 2r i ω,
i, j 1, 2, i /
j.
3.25
By 3.24, 3.25, and Lemma 3.2, we have
yi k ≤ yi ξi ω−1
yi s 1 − yi s ≤ ln r i 2r i ω.
Ai
s0
3.26
On the other hand, there also exist ηi ∈ Iω such that
yi ηi max yi k ,
i 1, 2.
k∈Iω
3.27
In view of 3.21, we can obtain
r iω ≤
m
ω−1
m
ω−1
ail k exp yi ηi cjl k
k0 l0
m
l0
k0 l0
m
ail exp yi ηi cjl ω,
3.28
i, j 1, 2, i /
j.
l0
Therefore,
ri ≤
m
l0
m
ail exp yi ηi cjl ,
l0
i, j 1, 2, i / j.
3.29
Then
r i − m
l0 cjl def
exp yi ηi > m
Bi ,
l0 ail
i, j 1, 2, i / j.
3.30
That is,
yi ηi > ln Bi ,
i 1, 2.
3.31
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Discrete Dynamics in Nature and Society
By 3.25, 3.31, and Lemma 3.2, we have
ω−1
yi s 1 − yi s ≥ ln Bi − 2r i ω,
yi k ≥ yi ηi −
i 1, 2.
3.32
i 1, 2.
3.33
s0
Inequalities 3.26 and 3.32 imply
yi k ≤ max ln r i 2r i ω, |ln Bi − 2r i ω|
A
i
def
Hi ,
Obviously, Bi , Ai , and Hi in 3.33 are independent of λ, respectively. Denote H H1 H2 h0 , where h0 is taken sufficiently large such that any solution {y1 , y2 } of the
system of algebraic equations
m c exp y
2l
2
a1l exp y1 r1,
l0
l0 1 exp y2
m
m c exp y
1l
1
a2l exp y2 r2
1
exp
y
1
l0
l0
m
3.34
satisfies y1 , y2 max{|y1 |, |y2 |} < h0 If system 3.34 has at least one solution. Let
def
Ω {y : y1 , y2 ∈ X | y < H}, thus condition a in Lemma 3.1 holds. When y ∈
∂Ω ∩ Ker L ∂Ω ∩ R2 , y {y1 , y2 }, y1 , y2 is a constant vector in R2 with y H. If
system 3.34 has at least one solution, then
⎤
⎡
m
m c exp y
2l
2
⎥ ⎢r 1 − a1l exp y1 −
1
exp
y
⎢
2 ⎥
0
y1
l0
l0
⎢
⎥
.
⎢
QN
⎥/
⎢
⎥
m
m c exp y
0
y2
1l
1 ⎦
⎣
r 2 − a2l exp y2 −
l0
l0 1 exp y1
3.35
If system 3.34 does not have one solution, then it is obvious that
y1
0
QN
.
/
y2
0
3.36
This implies that condition b in Lemma 3.1 is satisfied.
Now we prove that condition c in Lemma 3.1 holds. Define Φ : Dom L × 0, 1 → X
as follows:
⎤
⎡ m
⎡
m
c2l exp y2
⎤
⎥
⎢−
⎢r 1 − a1l exp y1 ⎥
⎢ l0 1 exp y2 ⎥
⎥
⎢
l0
⎢
⎥
⎥ μ⎢
,
3.37
Φ y1 , y2 , μ ⎢
⎥
⎢
⎥
m
m
⎢
c1l exp y1 ⎥
⎦
⎣
⎣
⎦
r 2 − a2l exp y2
−
l0
l0 1 exp y1
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13
where μ is a parameter with μ ∈ 0, 1. When y {y1 , y2 } ∈ ∂Ω ∩ Ker L ∂Ω ∩ R2 ,
y1 , y2 is a constant vector in R2 with y H. We will show that {y1 , y2 } ∈ ∂Ω ∩ Ker L,
0. If the conclusion is not true, then there is a constant vector y1 , y2 ∈ R2 with
Φy1 , y2 , μ /
y H satisfying Φy1 , y2 , μ 0, that is,
m c exp y
2l
2
r 1 − a1l exp y1 − μ
0,
l0
l0 1 exp y2
m
m c exp y
1l
1
r 2 − a2l exp y2 − μ
0.
1
exp
y
1
l0
l0
m
3.38
A similar argument to the above shows that y < H, which is a contradiction. Using the
property of topological degree and taking J I : Im Q → Ker L, y1 , y2 → y1 , y2 , we
have
deg JQN y1 , y2 , Ω ∩ Ker L, 0, 0
deg Φ y1 , y2 , 1 , Ω ∩ Ker L, 0, 0
deg Φ y1 , y2 , 0 , Ω ∩ Ker L, 0, 0
3.39
⎫
⎧
m
m
⎬
⎨
r 1 − a1l exp y1 , r 2 − a2l exp y2
, Ω ∩ Ker L, 0, 0 .
deg
⎭
⎩
l0
l0
Obviously, the following equations:
r1 −
m
a1l expu1 0,
r2 −
l0
m
a2l expu2 0
3.40
l0
have the unique solution u1 , u
2 ∈ R2 . Therefore, we have
m
m
0. 3.41
deg JQN y1 , y2 , Ω ∩ Ker L, 0, 0 sign
a1l exp
u1 a2l exp
u2 1 /
l0
l0
Finally, we will show that N is L-compact on Ω. For any y ∈ Ω, we have
'
'
ω−1
'
' '
'
1
'
'
'QNy' '
Nyx'
' ω s0
'
≤ max r1U m
l0
aU
1l expH1 m
l0
m
m
U U
U
c2l
, r2 aU
expH
c1l
2
2l
l0
Hence, QNΩ is bounded. Obviously, QNy : Ω → Y is continuous.
l0
3.42
def
E.
14
Discrete Dynamics in Nature and Society
It is easy to see that
ω−1
'
'
' 1 3ω ω−1
'
'
'
' ω−1
'
'Kp I − QNy' ≤
'Nys' 1
'Nys' ≤ 1 7ω E.
ω − s'Nys' ω s0
2ω s0
2
s0
3.43
For any y ∈ Ω, k1 , k2 ∈ Iω , without loss of generality, let k2 > k1 , then we have
Kp I − QNyk2 − Kp I − QNyk1 −1
k
k2 − k1 ω−1
2
Nys −
Nys
sk
ω s0
3.44
1
≤
k
2 −1
sk1
ω−1
Nys k2 − k1 Nys ≤ 2E|k2 − k1 |.
ω s0
Thus, the set {Kp I − QNy | y ∈ Ω} is equicontinuous and uniformly bounded. By
using the Arzela-Ascoli theorem, we see that Kp I −QN : Ω → X is compact. Consequently,
N is L-compact.
By now, we know that Ω verifies all the requirements in Lemma 3.1 and then system
3.12 has at least one ω-periodic solution. By the medium of 3.11, we derive that system
1.1 has at least one ω-periodic solution. This completes the proof of Theorem 3.3.
Next, by constructing a suitable Lyapunov-like discrete function, we further investigate the global attractivity of positive periodic solutions of system 1.1.
Theorem 3.4. In addition to 3.10, assume further that there exists a constant η > 0 such that
2
min ai0∗ ,
− a∗i0
Mi
M
− maM
i − m 1ci ≥ η,
i 1, 2,
3.45
where Mi (i=1,2) are defined in 2.5 and
∗
aM
i max ail : l 1, 2, 3, . . . , m ,
ciM max cil∗ : l 0, 1, 2, . . . , m ,
Then the positive periodic solution of system 1.1 is globally attractive.
i 1, 2.
3.46
Discrete Dynamics in Nature and Society
15
Proof. Let {x1 k, x2 k } be a positive periodic solution of system 1.1. To finish the proof
of Theorem 3.4, we will consider the following two steps.
Step 1. Let V11 k | ln x1 k − ln x1 k|, then it follows from the first equation of system 1.1
that
V11 k 1 |ln x1 k 1 − ln x1 k 1|
m
m
c2l kx2 k − l
ln x1 k r1 k − a1l kx1 k − l −
1 x2 k − l
l0
l0
m
m
c2l kx2 k − l
− ln x1 k r1 k − a1l kx1 k − l −
1 x2 k − l
l0
l0
3.47
≤ |ln x1 k − ln x1 k − a10 kx1 k − x1 k|
m
a1l k|x1 k − l − x1 k − l| l1
m
c2l k|x2 k − l − x2 k − l|.
l0
By the mean value theorem, we have
x1 k − x1 k expln x1 k − expln x1 k ξ1 k ln
x1 k
,
x1 k
3.48
that is,
ln
x1 k
1
x1 k − x1 k,
x1 k ξ1 k
3.49
where ξ1 k lies between x1 k and x1 k. Then we have
x1 k
ln
−
a
−
x
kx
k
k
10
1
1
x k
1
x1 k x1 k x1 k
− ln
ln
ln
−
a
−
x
kx
k
k
10
1
1
x1 k
x1 k
x1 k
x1 k − 1 |x1 k − x1 k| 1 x1 k − x1 k − a10 kx1 k − x1 k
ln
ξ1 k
ξ1 k
x1 k
x1 k − 1 |x1 k − x1 k| 1 − a10 k × |x1 k − x1 k|
ln
ξ1 k
ξ1 k
x1 k
(
)
1
x1 k 1
−
× |x1 k − x1 k|.
−
−
a
ln
k
10
ξ1 k
ξ1 k
x1 k
3.50
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Discrete Dynamics in Nature and Society
And hence it follows from 3.47 and 3.50 that
ΔV11 k V11 k 1 − V11 k
)
(
1
1
≤−
− − a10 k × |x1 k − x1 k|
ξ1 k
ξ1 k
3.51
m
m
a1l k|x1 k − l − x1 k − l| c2l k|x2 k − l − x2 k − l|.
l1
l0
Step 2. Let
V12 k k−1
m a1l s l|x1 s − x1 s| l1 sk−l
k−1
m c2l s l|x2 s − x2 s|.
3.52
l0 sk−l
2
For the sake of convenience, we will make a convention that kik
bi 1 if k1 > k2
1
for any bounded sequence bi. By a simple calculation, it derives that
ΔV12 k V12 k 1 − V12 k
k
m a1l s l|x1 s − x1 s| k
m l1 sk−l1
−
k−1
m a1l s l|x1 s − x1 s| −
l1 sk−l
c2l s l|x2 s − x2 s|
l0 sk−l1
k−1
m c2l s l|x2 s − x2 s|
l0 sk−l
3.53
m
m
a1l k l|x1 k − x1 k| − a1l k|x1 k − l − x1 k − l|
l1
m
l1
c2l k l|x2 k − x2 k| −
l0
m
c2l k|x2 k − l − x2 k − l|.
l0
Now, we are in a position to define V1 k by
V1 k V11 k V12 k.
3.54
Therefore, it follows from 3.51 and 3.53 that
ΔV1 k ΔV11 k ΔV12 k
1
m
1
−
− a10 k − a1l k l × |x1 k − x1 k|
≤−
ξ1 k ξ1 k
l1
m
c2l k l|x2 k − x2 k|.
l0
3.55
Discrete Dynamics in Nature and Society
17
By a similar argument, we can define V2 k by
V2 k V21 k V22 k,
3.56
where
V21 k |ln x2 k − ln x2 k|,
V22 k k−1
m a2l s l|x2 s − x2 s| l1 sk−l
k−1
m 3.57
c1l s l|x1 s − x1 s|.
l0 sk−l
Then it is easy to derive that
ΔV2 k ΔV21 k ΔV22 k
1
m
1
≤−
−
− a20 k − a2l k l × |x2 k − x2 k|
ξ2 k ξ2 k
l1
3.58
m
c1l k l|x1 k − x1 k|,
l0
where ξ2 k is between x2 k and x2 k.
Now we can define a Lyapunov-like discrete function V k by
V k V1 k V2 k.
3.59
It is easy to see that V k ≥ 0 for all k ∈ Z and V k0 m < ∞. For the arbitrariness of
ε and by 3.45, we can choose a small enough ε > 0 such that
min ai0∗ ,
2
− a∗i0
Mi ε
M
− maM
i − m 1ci ≥ η,
i 1, 2.
3.60
18
Discrete Dynamics in Nature and Society
Therefore, it follows from 3.55–3.60 that
ΔV k ΔV1 k ΔV2 k
2
m
1
m
1
−
− ai0 k − ail k l − cil k l × |xi k − xi k|
≤−
ξi k ξi k
i1
l1
l0
2
m
m
2
min ai0∗ ,
− a∗i0 − a∗il − cil∗ × |xi k − xi k|
≤−
Mi ε
3.61
i1
l1
l0
2 min ai0∗ ,
≤−
i1
≤ −η
2
− a∗i0
Mi ε
2
|xi k − xi k|
M
− maM
i − m 1ci
× |xi k − xi k|
for k ≥ k0 m.
i1
Thus, by 3.61 we obtain
V k 1 η
2 k
xi p − xi p ≤ V k0 m for any k ≥ k0 m,
pk0 m i1
2
∞ V k0 m
< ∞,
|xi k − xi k| ≤
η
kk m i1
3.62
0
from which we conclude that
2
i1
|xi k − xi k| 0 when k → ∞, that is,
lim |x1 k − x1 k| 0,
k → ∞
lim |x2 k − x2 k| 0.
k → ∞
3.63
According to Definition 1.2, this result implies that the positive periodic solution
{x1 k, x2 k } is globally attractive. This completes the proof of Theorem 3.4.
4. Example and Numerical Simulation
In this paper, a discrete time non-autonomous two-species competitive system with delays
is investigated. By using difference inequality technique, continuous theorem of coincidence
degree theory, and Lyapunov discrete function, sufficient conditions for the permanence of
system 1.1 and the existence and global attractivity of positive periodic solutions of system
1.1 are obtained, respectively.
Discrete Dynamics in Nature and Society
19
0.4
0.25
0.35
0.3
0.25
0.15
x2
x1
0.2
0.1
0.1
0.05
0
0.2
0.15
0.05
0
5
10
15
20
25
30
35
0
0
5
10
15
20
25
30
35
k
k
a
b
Figure 1: Permanence of system 4.1 with x1 −1 0.2, x1 0 0.018, x2 −1 0.3, and x2 0 0.017. a
Time series of x1 . b Time series of x2 .
To substantiate our analytical results, we construct the following example:
)
(
c20 kx2 k c21 kx2 k − 1
−
,
x1 k 1 x1 k exp r1 k − a10 kx1 k − a11 kx1 k − 1 −
1 x2 k
1 x2 k − 1
)
(
c10 kx1 k c11 kx1 k − 1
x2 k 1 x2 k exp r2 k − a20 kx2 k − a21 kx2 k − 1 −
−
.
1 x1 k
1 x1 k − 1
4.1
We first verify the sufficient conditions for permanence of system 4.1 and choose the
coefficients
r1 k 0.85 0.04 sin k,
r2 k 0.87 0.01 sin k,
a10 k a11 k 1.83 0.02 sin k,
a20 k a21 k 1.20 0.03 sin k,
c10 k c11 k 0.05 0.01 sin k,
c20 k c21 k 0.04 0.01 sin k
4.2
and the initial conditions
x1 −1 0.2,
x1 0 0.018,
x2 −1 0.3,
x2 0 0.017.
4.3
It is easy to see that system 4.1 satisfies assumption 2.6 of Theorem 2.3, and hence system
4.1 is permanent see Figure 1.
20
Discrete Dynamics in Nature and Society
0.0685
0.0612
0.068
x2
x1
0.0611
0.0675
0.061
0.067
0.0609
0.0665
0.066
0
20
40
60
80
100
0.0608
120
0
20
40
k
60
80
100
120
k
a
b
0.0614
0.0613
0.0613
0.0613
0.0612
0.0612
0.0612
x2
x2
0.0611
0.061
0.0612
0.0612
0.0609
0.0612
0.0608
0.0612
0.0607
0.066 0.0665 0.067 0.0675 0.068 0.0685 0.069
0.0612
0.0681
0.0682
0.0683
x1
c
0.0684
0.0685
0.0686
x1
d
Figure 2: Dynamic behaviors of system 4.1 with x1 −1 0.02, x1 0 0.0660, x2 −1 0.03, and x2 0 0.0608. a Time series of x1 . b Time series of x2 . c Phase portrait of x1 and x2 with k over 0, 120. d
Phase portrait of x1 and x2 with k over 110, 120.
In the following, we will consider the existence and global attractivity of positive
periodic solutions of system 4.1. We assume that
r1 k 0.0695 0.0055 cos πk,
r2 k 0.0665 − 0.0005 cos πk,
a10 k 1.0080 0.0010 cos πk,
a20 k 1.0425 − 0.0005 cos πk,
a11 k 0.0080 − 0.0010 cos πk,
a21 k 0.0425 − 0.0005 cos πk,
c10 k 0.00035 0.00015 cos πk,
c20 k 0.00035 − 0.00005 cos πk,
c11 k 0.00045 0.00015 cos πk,
c21 k 0.00045 − 0.00005 cos πk.
4.4
It is easy to verify that assumptions 3.10 of Theorem 3.3 are satisfied. Figure 2 shows
that system 4.1 has a 2-periodic solution {x1 k, x2 k }, which implies that the two
species x1 and x2 can coexist. Furthermore, a calculation can show that the assumptions
3.34 of Theorem 3.4 are satisfied, so {x1 k, x2 k } is globally attractive, that is, any
Discrete Dynamics in Nature and Society
21
0.0613
0.0685
0.0612
0.0611
0.068
0.061
0.0675
0.0609
0.067
0.0608
0.0665
0.066
0.0607
0
20
40
60
80
100
120
0.0606
0
20
40
60
k
80
100
120
k
x2∗
x2
x1∗
x1
a
b
Figure 3: Time series of x1∗ , x1 and x2∗ , x2 . a Time series of x1∗ , x1 with x1∗ −1 0.03, x1∗ 0 0.0662,
x1 −1 0.02, and x1 0 0.0668. b Time series of x2∗ , x2 with x2∗ −1 0.04, x2∗ 0 0.0606, x2 −1 0.05,
and x2 0 0.0607.
positive solution {x1∗ k, x2∗ k } of system 4.1 tends to {x1 k, x2 k } see Figure 3.
From Figure 3a, we see that x1∗ with x1∗ −1 0.03 and x1∗ 0 0.0662 will tend to x1 with
x1 −1 0.02 and x1 0 0.0668. Similarly, from Figure 3b, we see that x2∗ with x2∗ −1 0.04
and x2∗ 0 0.0606 will tend to x2 with x2 −1 0.05 and x2 0 0.0607.
Acknowledgments
The work is supported by the Key Project of Chinese Ministry of Education, the Research
Program for Outstanding Groups Science Foundation of Educational Department of Hubei
Province in China T200804, the National Science Foundation of Hubei Province of China
2008CDB068, the Innovation Term of Hubei University for Nationalities MY2009T001,
and the Fund of Cultivating Excellent Master Dissertation of Hubei University for
Nationalities. The authors would like to thank Editor Professor A. Vecchio and the referee
for careful reading of the original manuscript and valuable comments and suggestions that
greatly improved the presentation of this work.
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