- Email: [email protected]
Permanence and global attractivity of a discrete Schoener’s competition model with delays
Mathematical and Computer Modelling 49 (2009) 1607–1617
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Permanence and global attractivity of a discrete Schoener’s competition model with delays Liping Wu a,b , Fengde Chen b,∗ , Zhong Li b a
Department of Mathematics, Minjiang University, Fuzhou, Fujian 350108, PR China
b
College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
article
info
Article history: Received 16 April 2008 Accepted 13 June 2008 Keywords: Discrete Schoener’s competitive model Delay Permanence Global attractivity
a b s t r a c t A discrete Schoener’s competitive model with delays is studied. Sufficient conditions which guarantee the permanence of the model and the global attractivity of positive solutions of the model are obtained. Numerical simulations show the feasibility of the main results. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this paper we study the permanence and global attractivity of positive solutions of the following discrete Schoener’s competition model:
a10 (n) x ( n + 1 ) = x ( n ) exp − a ( n ) x ( n − τ ) − a ( n ) x ( n − τ ) − c ( n ) , 1 11 1 11 12 2 12 1 1 x (n − τ10 ) + k1 (n) 1 a20 (n) x2 (n + 1) = x2 (n) exp − a21 (n)x1 (n − τ21 ) − a22 (n)x2 (n − τ22 ) − c2 (n) , x2 (n − τ20 ) + k2 (n)
(1.1)
where {ki (n)}, {aij (n)} and {ci (n)} are real positive bounded sequences, τij (i = 1, 2; j = 0, 1, 2) are positive integers. We consider system (1.1) together with the following initial condition xi (s) = φi (s) ≥ 0,
s ∈ [−τ , 0] ∩ Z ; φi (0) > 0, i = 1, 2,
(1.2)
where τ = max0≤i,j≤2 {τij }. Let (x1 (n), x2 (n))T be any solution of (1.1) and (1.2). One could easily see that xi (n) > 0, i = 1, 2 for all n ∈ N. The Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [1–6] and the references cited therein). However, few papers investigate the global stability of the pure-delay model. To the best of the authors’ knowledge, only recently did Liu, Xu and Wang [1] propose and study the global stability
∗
Corresponding author. E-mail addresses: [email protected] (L. Wu), [email protected], [email protected] (F. Chen), [email protected] (Z. Li).
0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.06.004
1608
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
of the following continuous Schoener’s competition model with delays
a10 (t ) x˙ 1 (t ) = x1 (t ) − a11 (t )x1 (t − τ11 ) − a12 (t )x2 (t − τ12 ) − c1 (t ) , x (t − τ10 ) + k1 (t ) 1 a20 (t ) ˙ − a ( t ) x ( t − τ ) − a ( t ) x ( t − τ ) − c ( t ) , x ( t ) = x ( t ) 21 1 21 22 2 22 2 2 2 x2 (t − τ20 ) + k2 (t ) xi (s) = φi (s), s ∈ [−τ , 0], φi (0) > 0, i = 1, 2
(1.3)
where τ = max0≤i,j≤2 {τij }, φi (s) > 0 (i = 1, 2) is continuous on [−τ , 0], ki (t ), aij (t ), ci (t ) are positive, bounded and continuous functions. Many authors [7–24] have argued that discrete models governed by difference equations are more appropriate than continuous ones when the populations have non-overlapping generations. Discrete models can also provide efficient computational models for numerical simulations of continuous models. In recent years, more and more theoretical and mathematical biologists have paid attention to studies of the dynamics of population models governed by difference equations with or without time delays (see [9–24] and the references cited therein). However, to the best of our knowledge, no work has been done for the permanence and global attractivity of system (1.1) and (1.2). The purpose of this paper is to develop the analysis techniques of [1,15–19], in order to study the permanence and global attractivity of system (1.1) and (1.2), which can be seen as a discrete analogue of system (1.3). For any bounded sequence {a(n)}, we set au = sup a(n),
al = inf a(n). n∈N
n∈N
The organization of this paper is as follows. In the next section we shall obtain sufficient conditions for the permanence of system (1.1) and (1.2). In Section 3, by constructing a non-negative Lyapunov-like discrete functional, we shall derive sufficient conditions for the global attractivity of positive solutions of system (1.1) and (1.2). In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results. 2. Permanence In order to establish a permanence result for system (1.1) and (1.2), we need some preparations. Definition 2.1. System (1.1) and (1.2) is said to be permanent if there exist positive constants m and M, such that m ≤ lim inf xi (n) ≤ lim sup xi (n) ≤ M , n→+∞
n→+∞
i = 1, 2,
for any solution (x1 (n), x2 (n)) of system (1.1) with the initial condition (1.2). Lemma 2.1 ([20]). Assume that {x(n)} satisfies x(n) > 0 and x(n + 1) ≤ x(n) exp {r (n)(1 − ax(n))} for n ∈ [n1 , +∞), where a is a positive constant. Then lim sup x(n) ≤ n→+∞
1 ar u
exp(r u − 1).
Lemma 2.2 ([20]). Assume that {x(n)} satisfies x(n + 1) ≥ x(n) exp {r (n)(1 − ax(n))} ,
n ≥ N0 ,
lim supn→+∞ x(n) ≤ x and x(N0 ) > 0, where a is a constant such that ax∗ > 1 and N0 ∈ N. Then ∗
lim inf x(n) ≥ n→+∞
1 a
exp{r u (1 − ax∗ )}.
Proposition 2.1. Any solution (x1 (n), x2 (n)) of system (1.1) with the initial condition (1.2) is positive and ultimately bounded, that is lim sup xi (n) ≤ Ni , n→+∞
where Ni =
1 alii
exp
aui0 kli
(τii + 1) − 1 ,
i = 1, 2.
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1609
Proof. Clearly, x1 (n) > 0 and x2 (n) > 0 for n ≥ 0. Then from (1.1) we have x1 (n + 1) ≤ x1 (n) exp
a10 (n)
k1 (n)
.
Let y1 (n) = ln x1 (n), then y1 (n + 1) − y1 (n) ≤ n −1 X
au10
(y1 (i + 1) − y1 (i)) ≤
kl1
i=n−τ11
au10 kl1
. It follows that
τ11 ,
which implies au10
y1 (n) − y1 (n − τ11 ) ≤
kl1
τ11 ,
and therefore
x1 (n − τ11 ) ≥ x1 (n) exp −
au10 kl1
τ11 .
Then from (1.1), we have x1 (n + 1) ≤ exp
au10
−
kl1
al11
exp −
au10 kl1
τ11
x 1 ( n) .
In view of Lemma 2.1, we have lim sup x1 (n) ≤ n→+∞
1 al11
exp
au10 kl1
def
(τ11 + 1) − 1 = N1 .
By arguments similar to those above, we can also show that lim supn→+∞ x2 (n) ≤ N2 . The proof of Proposition 2.1 is completed. Before stating Proposition 2.2, for the sake of convenience, we set Eiε =
ali0
(Ni + ε) + kui
− auii (Ni + ε) − auij (Nj + ε) − ciu ,
∆εi = auii exp{−Eiε (τii )} ε
∆i = lim ∆i , ε→0
ali0
(Ni + ε) + kui i 6= j; i, j = 1, 2.
− auij (Nj + ε) − ciu ,
Proposition 2.2. Assume that (H) min{∆1 N1 , ∆2 N2 } > 1 holds, where N1 , N2 are defined by Proposition 2.1 and ∆i , i = 1, 2 are defined as above. Then for any solution (x1 (n), x2 (n)) of system (1.1) with the initial condition (1.2), we have lim inf xi (n) ≥ mi , n→+∞
i = 1, 2,
where m1 = m2 =
1
∆1 1
∆2
exp
N1 + ku1
exp
al10 al20 N2 + ku2
−
au12 N2
−
au21 N1
−
c1u
−
c2u
(1 − ∆1 N1 ) ,
(1 − ∆2 N2 ) .
Proof. For any sufficiently small ε > 0, it follows from condition (H) that min{∆ε1 N1 , ∆ε2 N2 } > 1. For the above ε > 0, according to Proposition 2.1, there exists a positive integer l0 such that x1 (n) ≤ N1 + ε, x2 (n) ≤ N2 + ε for all n ≥ l0 . Thus, from (1.1) for n ≥ l0 + τ , x1 (n + 1) ≥ x1 (n) exp
al10
(N1 + ε) + ku1 = x1 (n) exp{E1 }. ε
− au11 (N1 + ε) − au12 (N2 + ε) − c1u
1610
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
Let y1 (n) = ln x1 (n), then y1 (n + 1) − y1 (n) ≥ E1ε . It follows that n −1 X
(y1 (i + 1) − y1 (i)) ≥ E1ε (τ11 ),
i=n−τ11
which implies y1 (n − τ11 ) ≤ y1 (n) − E1ε (τ11 ), and therefore x1 (n − τ11 ) ≤ x1 (n) exp{−E1ε (τ11 )}. Further from (1.1), x1 (n + 1) ≥ x1 (n) exp
= x1 (n) exp
al10
− au12 (N2 + ε) − c1u − au11 x1 (n) exp{−E1ε (τ11 )}
(N1 + ε) + ku1 l
− au12 (N2 + ε) − c1u (1 − ∆ε1 x1 (n)) .
a10
(N1 + ε) + ku1
By applying Lemma 2.2, we obtain exp ∆ε1
n→+∞
al10
1
lim inf x1 (n) ≥
(N1 + ε) + ku1
− au12 (N2 + ε) − c1u (1 − ∆ε1 N1 ) ,
and letting ε → 0, we obtain lim infn→+∞ x1 (n) ≥ m1 . Similarly, we also can show that lim infn→+∞ x2 (n) ≥ m2 . This completes the proof of Proposition 2.2. Combining Proposition 2.1 with Proposition 2.2, we can easily obtain the following main result of this section. Theorem 2.1. Assume that (H) holds. Then system (1.1) and (1.2) is permanent. 3. Global attractivity In this section, by constructing a non-negative Lyapunov-like functional, we will obtain sufficient conditions for global attractivity of positive solutions of system (1.1) and (1.2). We first introduce a definition and prove a lemma which will be useful to obtain our main result. Definition 3.1. A solution (x1 (n), x2 (n)) of (1.1) and (1.2) is said to be globally attractive if for any other solution (x∗1 (n), x∗2 (n)) of (1.1) and (1.2), we have limn→+∞ (x∗i (n) − xi (n)) = 0, i = 1, 2. Lemma 3.1. For any two positive solutions (x1 (n), x2 (n)) and (x∗1 (n), x∗2 (n)) of system (1.1) and (1.2), we have ln(xi (n + 1)/x∗i (n + 1)) = ln(xi (n)/x∗i (n)) − aii (n) xi (n) − x∗i (n) − aij (n) xj (n − τij ) − x∗j (n − τij )
ai0 (n) xi (n − τi0 ) − x∗i (n − τi0 )
n−1 X
−
[xi (n − τi0 ) + ki (n)] xi (n − τi0 ) + ki (n)
∗
+ aii (n)
( Ai (s)
ai0 (s)
xi (s − τi0 ) + ki (s) ∗
s=n−τii
− aij (s)x∗j (s − τij ) − ci (s) xi (s) − x∗i (s) + Bi (s)xi (s)
− aii (s)x∗i (s − τii )
ai0 (s) x∗i (s − τi0 ) − xi (s − τi0 )
[xi (s − τi0 ) + ki (s)] x∗i (s − τi0 ) + ki (s)
!) − aii (s) xi (s − τii ) − xi (s − τii ) − aij (s) xj (s − τij ) − xj (s − τij ) ∗
∗
,
(3.1)
where
Ai (s) = exp θi (s)
Bi (s) = exp ϕi (s)
ai0 (s) x∗i (s − τi0 ) + ki (s)
− aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) ,
ai0 (s)
xi (s − τi0 ) + ki (s) ai0 (s)
+ (1 − ϕi (s))
∗
− aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s)
x∗i (s − τi0 ) + ki (s)
θi (s), ϕi (s) ∈ (0, 1), i 6= j; i, j = 1, 2.
∗
− aii (s)x∗i (s − τii ) − aij (s)x∗j (s − τij ) − ci (s)
,
(3.2)
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1611
Proof. For i 6= j; i, j = 1, 2, from (1.1), ln(xi (n + 1)/x∗i (n + 1)) − ln(xi (n)/x∗i (n)) = ln(xi (n + 1)/xi (n)) − ln(x∗i (n + 1)/x∗i (n)) ai0 (n)
=
xi (n − τi0 ) + ki (n)
− aii (n)xi (n − τii ) − aij (n)xj (n − τij ) − ci (n)
ai0 (n)
∗ ∗ − a ( n ) x ( n − τ ) − a ( n ) x ( n − τ ) − c ( n ) ii ii ij ij i i j x∗i (n − τi0 ) + ki (n) ∗ ai0 (n) xi (n − τi0 ) − xi (n − τi0 ) − aii (n) xi (n − τii ) − x∗i (n − τii ) − aij (n) xj (n − τij ) − x∗j (n − τij ) = [xi (n − τi0 ) + ki (n)] x∗i (n − τi0 ) + ki (n) ai0 (n) xi (n − τi0 ) − x∗i (n − τi0 ) ∗ − aii (n) xi (n) − x∗i (n) − aij (n) xj (n − τij ) − x∗j (n − τij ) =− [xi (n − τi0 ) + ki (n)] xi (n − τi0 ) + ki (n) + aii (n) xi (n) − x∗i (n) − aii (n) xi (n − τii ) − x∗i (n − τii ) ,
−
that is ln(xi (n + 1)/x∗i (n + 1)) = ln(xi (n)/x∗i (n)) − aij (n) xj (n − τij ) − x∗j (n − τij )
ai0 (n) xi (n − τi0 ) − x∗i (n − τi0 )
−
[xi (n − τi0 ) + ki (n)] x∗i (n − τi0 ) + ki (n)
− aii (n)
xi (n) − x∗i (n) − [xi (n) − xi (n − τii )] − x∗i (n) − x∗i (n − τii )
.
(3.3)
Since n −1 n−1 X X (xi (s + 1) − xi (s)) − (xi (n) − xi (n − τii )) − x∗i (n) − x∗i (n − τii ) = s=n−τii
x∗i (s + 1) − x∗i (s)
s=n−τii
n−1
=
X
xi (s + 1) − x∗i (s + 1) − xi (s) − x∗i (s)
,
(3.4)
s=n−τii
and xi (s + 1) − x∗i (s + 1) − xi (s) − x∗i (s)
ai0 (s) − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) = xi (s) exp xi (s − τi0 ) + ki (s) ai0 (s) − aii (s)x∗i (s − τii ) − aij (s)x∗j (s − τij ) − ci (s) − xi (s) − x∗i (s) − x∗i (s) exp ∗ x (s − τi0 ) + ki (s) i ai0 (s) = xi (s) exp − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) xi (s − τi0 ) + ki (s) ai0 (s) ∗ ∗ − exp ∗ − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) xi (s − τi0 ) + ki (s) ai0 (s) ∗ ∗ ∗ + xi (s) − xi (s) exp ∗ − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) − 1 , xi (s − τi0 ) + ki (s) by using the mean-value theorem, we have that xi (s + 1) − x∗i (s + 1) − xi (s) − x∗i (s)
ai0 (s) = xi (s) − x∗i (s) Ai (s) ∗ − aii (s)x∗i (s − τii ) − aij (s)x∗j (s − τij ) − ci (s) xi (s − τ10 ) + ki (s) ai0 (s) ai0 (s) + xi (s)Bi (s) − ∗ xi (s − τi0 ) + ki (s) xi (s − τi0 ) + ki (s) ∗ ∗ − aii (s) xi (s − τii ) − xi (s − τii ) − aij (s) xj (s − τij ) − xj (s − τij ) , here Ai (s), Bi (s) are defined by (3.2). Then from (3.3)–(3.5), we can easily obtain (3.1). The proof is completed.
(3.5)
1612
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
Theorem 3.1. Assume that in system (1.1) and (1.2), there exist positive constants α1 , α2 and η > 0 such that
αi Eii −
2 X
(αi Fij + αj Gji ) ≥ η,
i = 1, 2,
j=1,j6=i
where Eii , Fij , Gji are defined by (3.11). Then for any two positive solutions (x1 (n), x2 (n)) and (x∗1 (n), x∗2 (n)) of system (1.1) and (1.2), we have lim (x∗i (n) − xi (n)) = 0,
i = 1, 2.
n→+∞
Proof. Firstly, let V11 (n) = ln x1 (n) − ln x∗1 (n) . From (3.1), we have that
x1 (n + 1) x1 (n) ∗ ∗ ln ≤ ln x∗ (n + 1) x∗ (n) − a11 (n) x1 (n) − x1 (n) + a12 (n) x2 (n − τ12 ) − x2 (n − τ12 ) 1 1 n−1 X a10 (n) a10 (s) x1 (n − τ10 ) − x∗ (n − τ10 ) + a11 (n) + 2 + a11 (s)|x∗1 (s − τ11 )| 1 k ( s ) k1 (n) 1 s=n−τ11 a10 (s) ∗ ∗ x1 (s − τ10 ) − x∗ (s − τ10 ) + a12 (s)|x2 (s − τ12 )| + c1 (s) A1 (s) x1 (s) − x1 (s) + B1 (s)|x1 (s)| 1 2 k1 (n) + a11 (s) x1 (s − τ11 ) − x∗1 (s − τ11 ) + a12 (s) x2 (s − τ12 ) − x∗2 (s − τ12 ) .
(3.6)
Since xi (n) − x∗i (n) = eln xi (n) − eln xi (n) = ξi (n) ln(xi (n)/x∗i (n)), i = 1, 2, ∗
where 0 < ξi (n) < max{xi (n), x∗i (n)}, i = 1, 2, it follows that
ln(x1 (n)/x∗ (n)) − a11 (n) x1 (n) − x∗ (n) 1 1 1 1 ∗ = ln(x1 (n)/x1 (n)) − − − a11 (n) x1 (n) − x∗1 (n) . ξ1 (n) ξ1 (n)
(3.7)
By Proposition 2.1, there are constants Mi > 0 (i = 1, 2), and a positive integer n0 such that for n ≥ n0 , 0 < xi (n), x∗i (n) ≤ Mi (i = 1, 2). Then from (3.6) and (3.7) we can obtain that for n ≥ n0 + τ ,
1 − − a11 (n) x1 (n) − x∗1 (n) + a12 (n) x2 (n − τ12 ) − x∗2 (n − τ12 ) ξ 1 ( n) ξ1 (n) n −1 X a10 (n) a10 (s) ∗ x1 (n − τ10 ) − x1 (n − τ10 ) + a11 (n) A1 (s) + 2 + a11 (s)M1 + a12 (s)M2 + c1 (s) k1 (s) k1 (n) s=n−τ11
∆V11 ≤ −
1
a10 (s) x1 (s − τ10 ) − x∗ (s − τ10 ) + B1 (s)a11 (s)M1 x1 (s − τ11 ) − x∗ (s − τ11 ) × x1 (s) − x∗1 (s) + B1 (s)M1 2 1 1 k1 (s) ∗ + B1 (s)a12 (s)M1 x2 (s − τ12 ) − x2 (s − τ12 ) . Secondly, let V12 (n) =
n −1 n−1 X X a10 (s + τ10 ) x1 (s) − x∗ (s) + a12 (s + τ12 ) x2 (s) − x∗2 (s) 1 2 k1 (s + τ10 ) s=n−τ12 s=n−τ10 n−X 1+τ11 n −1 X a10 (u) + a11 (s) A1 (u) + a11 (u)M1 + a12 (u)M2 + c1 (u) x1 (u) − x∗1 (u) k ( u ) 1 s =n u=s−τ11
+ B1 (u)M1
a10 (u) k21
( u)
x1 (u − τ10 ) − x∗ (u − τ10 ) + B1 (u)a11 (u)M1 x1 (u − τ11 ) − x∗ (u − τ11 ) 1
1
+ B1 (u)a12 (u)M1 x2 (u − τ12 ) − x∗2 (u − τ12 ) .
(3.8)
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1613
By a simple calculation, we can obtain
∆V12 =
x1 (n) − x∗ (n) − a10 (n) x1 (n − τ10 ) − x∗ (n − τ10 ) 1 1 2 (n + τ10 ) k (n) 1 ∗ + a12 (n + τ12 ) x2 (n) − x2 (n) − a12 (n) x2 (n − τ12 ) − x∗2 (n − τ12 ) nX +τ11 a10 (n) + a11 (n)M1 + a12 (n)M2 + c1 (n) x1 (n) − x∗1 (n) + a11 (s) A1 (n) k1 (n) s=n+1 a10 (n + τ10 ) k21
+ B1 (n)M1
a10 (n) k21
x1 (n − τ10 ) − x∗ (n − τ10 ) + B1 (n)a11 (n)M1 x1 (n − τ11 ) − x∗ (n − τ11 )
(n)
1
1
+ B1 (n)a12 (n)M1 x2 (n − τ12 ) − x∗2 (n − τ12 ) n−1 X
− a11(n)
A1 (u)
u=n−τ11
+ B1 (u)M1
a10 (u) k1 (u)
+ a11 (u)M1 + a12 (u)M2 + c1 (u) x1 (u) − x∗1 (u)
a10 (u) k21
(u)
x1 (u − τ10 ) − x∗ (u − τ10 ) + B1 (u)a11 (u)M1 x1 (u − τ11 ) − x∗ (u − τ11 ) 1
1
+ B1 (u)a12 (u)M1 x2 (u − τ12 ) − x∗2 (u − τ12 ) .
(3.9)
Thirdly, let V13 (n) = M1
n−1 10 +τ11 X l+τX a10 (l + τ10 ) x1 (l) − x∗ (l) a11 (s) B ( l + τ ) 1 10 1 k21 (l + τ10 ) l=n−τ10 s=l+τ10 +1 l+2τ11
n−1 X
+ M1
B1 (l + τ11 )a11 (l + τ11 ) x1 (l) − x∗1 (l)
l=n−τ11 n−1 X
+ M1
X
a11 (s)
s=l+τ11 +1 12 +τ11 l+τX
B1 (l + τ12 )a12 (l + τ12 ) x2 (l) − x∗2 (l)
l=n−τ12
a11 (s).
s=l+τ12 +1
Calculating ∆V13 , we can derive n+τ10 +τ11
X
∆V13 =
a11 (s)M1
s=n+τ10 +1 n+τ11
X
−
a11 (s)M1
s=n+1
a10 (n + τ10 ) k21
(n + τ10 )
a10 (n) k21 (n)
B1 (n + τ10 ) x1 (n) − x∗1 (n)
B1 (n) x1 (n − τ10 ) − x∗1 (n − τ10 )
n+2τ11
X
+
a11 (s)M1 B1 (n + τ11 )a11 (n + τ11 ) x1 (n) − x∗1 (n)
s=n+τ11 +1 n+τ11
X
−
a11 (s)M1 B1 (n)a11 (n) x1 (n − τ11 ) − x∗1 (n − τ11 )
s=n+1 n+τ12 +τ11
X
+
a11 (s)M1 B1 (n + τ12 )a12 (n + τ12 ) x2 (n) − x∗2 (n)
s=n+τ12 +1 n+τ11
X
−
a11 (s)M1 B1 (n)a12 (n) x2 (n − τ12 ) − x∗2 (n − τ12 ) .
s=n+1
Now we set V1 (n) = V11 (n) + V12 (n) + V13 (n). Then from (3.6)–(3.10), we have that for n ≥ n0 + τ ,
1 − − a11 (n) x1 (n) − x∗1 (n) + a12 (n + τ12 ) x2 (n) − x∗2 (n) ξ1 (n) ξ1 (n) a10 (n + τ10 ) x1 (n) − x∗ (n) + 2 1 k1 (n + τ10 )
∆ V1 ≤ −
1
(3.10)
1614
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617 n+τ11
+
X
a11 (s)A1 (n)
s=n+1 n+τ10 +τ11
X
+
a11 (s)M1
s=n+τ10 +1
a10 (n) k1 (n)
+ a11 (n)M1 + a12 (n)M2 + c1 (n) x1 (n) − x∗1 (n)
a10 (n + τ10 ) k21
(n + τ10 )
B1 (n + τ10 ) x1 (n) − x∗1 (n)
n+2τ11
X
+
a11 (s)M1 B1 (n + τ11 )a11 (n + τ11 ) x1 (n) − x∗1 (n)
s=n+τ11 +1 n+τ12 +τ11
X
+
a11 (s)M1 B1 (n + τ12 )a12 (n + τ12 ) x2 (n) − x∗2 (n) .
s=n+τ12 +1
By arguments similar to those above, we take V21 (n) = ln x2 (n) − ln x∗2 (n) ,
V22 (n) =
n −1 n −1 X X a20 (s + τ20 ) x2 (s) − x∗ (s) + a21 (s + τ21 ) x1 (s) − x∗1 (s) 2 2 k1 (s + τ20 ) s=n−τ20 s=n−τ21 n−X 1+τ22 n−1 X a20 (u) + a22 (u)M2 + a21 (u)M1 + c2 (u) x2 (u) − x∗2 (u) + a22 (s) A 2 ( u) k2 (u) s =n u=s−τ22 a20 (u) x2 (u − τ20 ) − x∗ (u − τ20 ) + B2 (u)a22 (u)M2 x2 (u − τ22 ) − x∗ (u − τ22 ) + B2 (u)M2 2 2 2 k 2 ( u) + B2 (u)a21 (u)M2 x1 (u − τ21 ) − x∗1 (u − τ21 ) ,
V23 (n) = M2
n −1 20 +τ22 X l+τX a20 (l + τ20 ) x2 (l) − x∗ (l) B ( l + τ ) a22 (s) 2 20 2 k22 (l + τ20 ) l=n−τ20 s=l+τ20 +1 n −1 X
+ M2
l+2τ22
B2 (l + τ22 )a22 (l + τ22 ) x2 (l) − x∗2 (l)
l=n−τ22 n −1 X
+ M2
X
a22 (s)
s=l+τ22 +1 21 +τ22 l+τX
B2 (l + τ21 )a21 (l + τ21 ) x1 (l) − x∗1 (l)
l=n−τ21
a22 (s).
s=l+τ21 +1
Similarly, we take V2 (n) = V21 (n) + V22 (n) + V23 (n). Then in the same way as obtaining ∆V1 , we can obtain for n ≥ n0 + τ ,
1 − − a22 (n) x2 (n) − x∗2 (n) + a21 (n + τ21 ) x1 (n) − x∗1 (n) ξ2 (n) ξ2 (n) a20 (n + τ20 ) x2 (n) − x∗ (n) + 2 2 k2 (n + τ20 ) nX +τ22 a20 (n) + a22 (s)A2 (n) + a22 (n)M2 + a21 (n)M1 + c2 (n) x2 (n) − x∗2 (n) k2 (n) s=n+1
∆ V2 ≤ −
1
n+τ20 +τ22
+
X
a22 (s)M2
s=n+τ20 +1
a20 (n + τ20 ) k22
(n + τ20 )
B2 (n + τ20 ) x2 (n) − x∗2 (n)
n+2τ22
+
X
a22 (s)M2 B2 (n + τ22 )a22 (n + τ22 ) x2 (n) − x∗2 (n)
s=n+τ22 +1 n+τ21 +τ22
+
X
a22 (s)M2 B2 (n + τ21 )a21 (n + τ21 ) x1 (n) − x∗1 (n) .
s=n+τ21 +1
Now we define a Lyapunov-like discrete functional V by V (n) = α1 V1 (n) + α2 V2 (n).
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1615
It is easy to see that V (n0 + τ ) < +∞. Calculating the difference of V along the solution of (1.1) and (1.2), we have that for n ≥ n0 + τ ,
" n+τ 2 i0 +τii X X 1 1 ai0 (n + τi0 ) ai0 (n + τi0 ) ∆V ≤ − αi − − aii (n) − 2 aii (s)Mi 2 − Bi (n + τi0 ) ξ ( n ) ξ ( n ) k ( n + τ ) ki (n + τi0 ) i i i0 i i=1 s=n+τi0 +1 # nX +2τii nX +τii 2 X ai0 (n) αi aii (s)Mi Bi (n + τii )aii (n + τii ) − aii (s)Ai (n) − + aii (n)Mi + aij (n)Mj + ci (n) ki (n) s=n+τii +1 s=n+1 j=1,j6=i n+τji +τjj X + αj aji (n + τji ) + ajj (s)Mj Bj (n + τji )aji (n + τji ) xi (n) − x∗i (n) s=n+τji +1 ( 2 X 2 au au ≤− αi min(alii , − auii ) − 2i0 l − τii auii Mi 2i0 l Bui − τii (auii )2 Mi Bui Mi (ki ) (ki ) i=1 ) 2 X aui0 u u u u u u u u u − αi τii a A + a Mi + a Mj + c + αj (a + τjj a Mj B a ) xi (n) − x∗ (n) ii i
=−
2 X
(
2 X
αi Eii −
ii
kli
j=1,j6=i
ij
i
ji
jj
j
ji
i
) (αi Fij + αj Gji ) xi (n) − x∗i (n)
j=1,j6=i
i=1 2
≤ −η
X xi (n) − x∗ (n) , i
i =1
where Eii = min(alii ,
2
− auii ) −
aui0
au
− τii auii Mi 2i0 l Bui − τii (auii )2 Mi Bui , Mi ( ) (ki ) u ai0 + auii Mi + auij Mj + ciu , Fij = τii auii Aui l k2i l
(3.11)
ki
Gji = auji + τjj aujj Mj Buj auji . Then we have that n X
n 2 X X xi (p) − x∗ (p) ,
(V (p + 1) − V (p)) ≤ −η
i
p=n0 +τ i=1
p=n0 +τ
which implies V (n + 1) + η
n 2 X X xi (p) − x∗ (p) ≤ V (n0 + τ ). i
p=n0 +τ i=1
P2 P∞ P2 V (n0 +τ ) V (n0 +τ ) ∗ ∗ . Then < +∞, from which i=1 xi (p) − xi (p) ≤ n=n0 +τ i=1 xi (n) − xi (n) ≤ η η P2 ∗ we conclude that limn→+∞ i=1 xi (n) − xi (n) = 0. This means that (x1 (n), x2 (n)) is globally attractive. The proof is It follows
Pn
completed.
p=n0 +τ
4. Example and numerical simulation Consider the following system
0.3 + 0.2 sin n x1 (n + 1) = x1 (n) exp x1 (n − 3) + 10 − (2.52 + 0.02 sin n)x1 (n − 1) − ( 0 . 009 + 0 . 001 sin n ) x ( n − 3 ) − 0 . 01 , 2 0.4 + 0.1 sin n x2 (n + 1) = x2 (n) exp − (0.015 + 0.005 sin n)x1 (n − 4) x2 (n − 4) + 10 − (2.62 + 0.02 sin n)x2 (n − 1) − 0.02 .
(4.1)
1616
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
Fig. 1. Dynamic behavior of the first component x1 (n) of the solution (x1 (n), x2 (n)) to system (4.1) with the initial conditions (x1 (s), x2 (s))(s ∈ {−4, −3, −2, −1, 0}) = (0.013, 0.0113) and (0.017, 0.0115), respectively.
Fig. 2. Dynamic behavior of the second component x2 (n) of the solution (x1 (n), x2 (n)) to system (4.1) with the initial conditions (x1 (s), x2 (s))(s ∈ {−4, −3, −2, −1, 0}) = (0.013, 0.0113) and (0.017, 0.0115), respectively.
Here corresponding to system (1.1), we assume that a10 (n) = 0.3 + 0.2 sin n,
a11 (n) = 2.52 + 0.02 sin n,
a20 (n) = 0.4 + 0.1 sin n,
a21 (n) = 0.015 + 0.005 sin n,
c1 (n) = 0.01,
τ10 = 3,
a12 (n) = 0.009 + 0.001 sin n, a22 (n) = 2.62 + 0.02 sin n,
c2 (n) = 0.02,
τ11 = 1,
τ12 = 3,
τ20 = 4,
τ21 = 4,
τ22 = 1.
Corresponding to Theorem 3.1, by simple computation, we derive E11 ≈ 1.388,
F12 ≈ 1.27,
G21 ≈ 0.029,
E22 ≈ 1.45,
F21 ≈ 1.345,
G12 ≈ 0.014.
Then E11 − (F12 + G21 ) ≈ 0.089,
E22 − (F21 + G12 ) ≈ 0.091.
Also it is easy to see that the conditions in Theorem 3.1 are verified. Therefore, the positive solutions of (4.1) are globally attractive. Our numerical simulations support our result (see Figs. 1 and 2).
Acknowledgements This work was supported by the Program for New Century Excellent Talents in Fujian Province University (NCETFJ).
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1617
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Q.M. Liu, R. Xu, W.G. Wang, Global asymptotic stability of Schoener’s competitive model with delays, J. Biomath. 21 (1) (2006) 147–152 (in Chinese). Z.H. Lu, L.S. Chen, Analysis on a periodic Schoener model, Acta. Math. Sci. 12 (Suppl.) (1992) 105–109. L.J. Zhang, H.F. Huo, J.F. Chen, Asymptotic behavior of the nonautonomous competing system with feedback controls, J. Biomath. 16 (4) (2001) 405–410. Q.M. Liu, R. Xu, Periodic solutions of Schoener’s competitive model with delays, J. Biomath. 19 (4) (2004) 385–394 (in Chinese). C.D. Yuan, C.H. Wang, Permanence and periodic solutions of the nonautonomous Schoener’s competing system with diffusion, Biomath. 1 (2) (1997) 17–20 (in Chinese). H. Xiang, K.M. Yan, B.Y. Wang, Positive periodic solutions for discrete Schoener’s competitive model, J. Lanzhou Univ. Technol. 31 (5) (2005) 125–128 (in Chinese). R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000. J.D. Murray, Mathematical Biology, Springer, New York, 1989. Y. Saito, W.B. Ma, H. Hara, A necessary and sufficient condition for permanence of a Lotka–Volterra discrete system with delays, J. Math. Anal. Appl. 256 (2001) 162–174. Z.Y. Lu, W.D. Wang, Permanence and global attractivity for Lotka–Volterra difference system, J. Math. Biol. 22 (1999) 269–283. Z. Zhou, X. Zou, Stable periodic solutions in a discrete periodic logistic equation, Appl. Math. Lett. 16 (2) (2003) 165–171. Y.M. Chen, Z. Zhou, Stable periodic solution of a discrete periodic Lotka–Volterra competition system, J. Math. Anal. Appl. 277 (1) (2003) 358–366. F.D. Chen, Permanence and global attractivity of a discrete multi-species Lotka–Volterra competition predator–prey system, Appl. Math. Comput. 182 (1) (2006) 3–12. W.D. Wang, G. Mulone, F. Salemi, V. Salone, Global stability of discrete population models with delays and fluctuating environment, J. Math. Anal. Appl. 264 (2001) 147–167. F.D. Chen, Permanence of a single species discrete model with feedback control and delay, Appl. Math. Lett. 20 (7) (2007) 729–733. X.Y. Liao, S.F. Zhou, Y.M. Chen, Permanence for a discrete time Lotka–Volterra type food-chain model with delays, Appl. Math. Comput. 186 (1) (2007) 279–285. X.X. Chen, F.D. Chen, Periodicity and stability of a discrete time periodic n-species Lotka–Volterra competition system with feedback controls and deviating arguments, Soochow J. Math. 32 (3) (2006) 343–368. Y. Muroya, Persistence and global stability in discrete models of pure-delay nonautonomous Lotka–Volterra type, J. Math. Anal. Appl. 293 (2004) 446–461. Y. Muroya, Persistence and global stability in discrete models of Lotka–Volterra type, J. Math. Anal. Appl. 330 (1) (2007) 24–33. X.T. Yang, Uniform persistence and periodic solutions for a discrete predator–prey system with delays, J. Math. Anal. Appl. 316 (1) (2006) 161–177. F.D. Chen, L.P. Wu, Z. Li, Permanence and global attractivity of the discrete Gilpin-Ayala type population model, Comput. Math. Appl. 53 (8) (2007) 1214–1227. F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186 (1) (2007) 23–29. F.D. Chen, Permanence of a discrete n-species food-chain system with time delays, Appl. Math. Comput. 185 (1) (2007) 719–726. X.X. Chen, F.D. Chen, Stable periodic solution of a discrete periodic Lotka–Volterra competition system with a feedback control, Appl. Math. Comput. 181 (2) (2006) 1446–1454.
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Permanence and global attractivity of a discrete Schoener’s competition model with delays Liping Wu a,b , Fengde Chen b,∗ , Zhong Li b a
Department of Mathematics, Minjiang University, Fuzhou, Fujian 350108, PR China
b
College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
article
info
Article history: Received 16 April 2008 Accepted 13 June 2008 Keywords: Discrete Schoener’s competitive model Delay Permanence Global attractivity
a b s t r a c t A discrete Schoener’s competitive model with delays is studied. Sufficient conditions which guarantee the permanence of the model and the global attractivity of positive solutions of the model are obtained. Numerical simulations show the feasibility of the main results. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this paper we study the permanence and global attractivity of positive solutions of the following discrete Schoener’s competition model:
a10 (n) x ( n + 1 ) = x ( n ) exp − a ( n ) x ( n − τ ) − a ( n ) x ( n − τ ) − c ( n ) , 1 11 1 11 12 2 12 1 1 x (n − τ10 ) + k1 (n) 1 a20 (n) x2 (n + 1) = x2 (n) exp − a21 (n)x1 (n − τ21 ) − a22 (n)x2 (n − τ22 ) − c2 (n) , x2 (n − τ20 ) + k2 (n)
(1.1)
where {ki (n)}, {aij (n)} and {ci (n)} are real positive bounded sequences, τij (i = 1, 2; j = 0, 1, 2) are positive integers. We consider system (1.1) together with the following initial condition xi (s) = φi (s) ≥ 0,
s ∈ [−τ , 0] ∩ Z ; φi (0) > 0, i = 1, 2,
(1.2)
where τ = max0≤i,j≤2 {τij }. Let (x1 (n), x2 (n))T be any solution of (1.1) and (1.2). One could easily see that xi (n) > 0, i = 1, 2 for all n ∈ N. The Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [1–6] and the references cited therein). However, few papers investigate the global stability of the pure-delay model. To the best of the authors’ knowledge, only recently did Liu, Xu and Wang [1] propose and study the global stability
∗
Corresponding author. E-mail addresses: [email protected] (L. Wu), [email protected], [email protected] (F. Chen), [email protected] (Z. Li).
0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.06.004
1608
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
of the following continuous Schoener’s competition model with delays
a10 (t ) x˙ 1 (t ) = x1 (t ) − a11 (t )x1 (t − τ11 ) − a12 (t )x2 (t − τ12 ) − c1 (t ) , x (t − τ10 ) + k1 (t ) 1 a20 (t ) ˙ − a ( t ) x ( t − τ ) − a ( t ) x ( t − τ ) − c ( t ) , x ( t ) = x ( t ) 21 1 21 22 2 22 2 2 2 x2 (t − τ20 ) + k2 (t ) xi (s) = φi (s), s ∈ [−τ , 0], φi (0) > 0, i = 1, 2
(1.3)
where τ = max0≤i,j≤2 {τij }, φi (s) > 0 (i = 1, 2) is continuous on [−τ , 0], ki (t ), aij (t ), ci (t ) are positive, bounded and continuous functions. Many authors [7–24] have argued that discrete models governed by difference equations are more appropriate than continuous ones when the populations have non-overlapping generations. Discrete models can also provide efficient computational models for numerical simulations of continuous models. In recent years, more and more theoretical and mathematical biologists have paid attention to studies of the dynamics of population models governed by difference equations with or without time delays (see [9–24] and the references cited therein). However, to the best of our knowledge, no work has been done for the permanence and global attractivity of system (1.1) and (1.2). The purpose of this paper is to develop the analysis techniques of [1,15–19], in order to study the permanence and global attractivity of system (1.1) and (1.2), which can be seen as a discrete analogue of system (1.3). For any bounded sequence {a(n)}, we set au = sup a(n),
al = inf a(n). n∈N
n∈N
The organization of this paper is as follows. In the next section we shall obtain sufficient conditions for the permanence of system (1.1) and (1.2). In Section 3, by constructing a non-negative Lyapunov-like discrete functional, we shall derive sufficient conditions for the global attractivity of positive solutions of system (1.1) and (1.2). In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results. 2. Permanence In order to establish a permanence result for system (1.1) and (1.2), we need some preparations. Definition 2.1. System (1.1) and (1.2) is said to be permanent if there exist positive constants m and M, such that m ≤ lim inf xi (n) ≤ lim sup xi (n) ≤ M , n→+∞
n→+∞
i = 1, 2,
for any solution (x1 (n), x2 (n)) of system (1.1) with the initial condition (1.2). Lemma 2.1 ([20]). Assume that {x(n)} satisfies x(n) > 0 and x(n + 1) ≤ x(n) exp {r (n)(1 − ax(n))} for n ∈ [n1 , +∞), where a is a positive constant. Then lim sup x(n) ≤ n→+∞
1 ar u
exp(r u − 1).
Lemma 2.2 ([20]). Assume that {x(n)} satisfies x(n + 1) ≥ x(n) exp {r (n)(1 − ax(n))} ,
n ≥ N0 ,
lim supn→+∞ x(n) ≤ x and x(N0 ) > 0, where a is a constant such that ax∗ > 1 and N0 ∈ N. Then ∗
lim inf x(n) ≥ n→+∞
1 a
exp{r u (1 − ax∗ )}.
Proposition 2.1. Any solution (x1 (n), x2 (n)) of system (1.1) with the initial condition (1.2) is positive and ultimately bounded, that is lim sup xi (n) ≤ Ni , n→+∞
where Ni =
1 alii
exp
aui0 kli
(τii + 1) − 1 ,
i = 1, 2.
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1609
Proof. Clearly, x1 (n) > 0 and x2 (n) > 0 for n ≥ 0. Then from (1.1) we have x1 (n + 1) ≤ x1 (n) exp
a10 (n)
k1 (n)
.
Let y1 (n) = ln x1 (n), then y1 (n + 1) − y1 (n) ≤ n −1 X
au10
(y1 (i + 1) − y1 (i)) ≤
kl1
i=n−τ11
au10 kl1
. It follows that
τ11 ,
which implies au10
y1 (n) − y1 (n − τ11 ) ≤
kl1
τ11 ,
and therefore
x1 (n − τ11 ) ≥ x1 (n) exp −
au10 kl1
τ11 .
Then from (1.1), we have x1 (n + 1) ≤ exp
au10
−
kl1
al11
exp −
au10 kl1
τ11
x 1 ( n) .
In view of Lemma 2.1, we have lim sup x1 (n) ≤ n→+∞
1 al11
exp
au10 kl1
def
(τ11 + 1) − 1 = N1 .
By arguments similar to those above, we can also show that lim supn→+∞ x2 (n) ≤ N2 . The proof of Proposition 2.1 is completed. Before stating Proposition 2.2, for the sake of convenience, we set Eiε =
ali0
(Ni + ε) + kui
− auii (Ni + ε) − auij (Nj + ε) − ciu ,
∆εi = auii exp{−Eiε (τii )} ε
∆i = lim ∆i , ε→0
ali0
(Ni + ε) + kui i 6= j; i, j = 1, 2.
− auij (Nj + ε) − ciu ,
Proposition 2.2. Assume that (H) min{∆1 N1 , ∆2 N2 } > 1 holds, where N1 , N2 are defined by Proposition 2.1 and ∆i , i = 1, 2 are defined as above. Then for any solution (x1 (n), x2 (n)) of system (1.1) with the initial condition (1.2), we have lim inf xi (n) ≥ mi , n→+∞
i = 1, 2,
where m1 = m2 =
1
∆1 1
∆2
exp
N1 + ku1
exp
al10 al20 N2 + ku2
−
au12 N2
−
au21 N1
−
c1u
−
c2u
(1 − ∆1 N1 ) ,
(1 − ∆2 N2 ) .
Proof. For any sufficiently small ε > 0, it follows from condition (H) that min{∆ε1 N1 , ∆ε2 N2 } > 1. For the above ε > 0, according to Proposition 2.1, there exists a positive integer l0 such that x1 (n) ≤ N1 + ε, x2 (n) ≤ N2 + ε for all n ≥ l0 . Thus, from (1.1) for n ≥ l0 + τ , x1 (n + 1) ≥ x1 (n) exp
al10
(N1 + ε) + ku1 = x1 (n) exp{E1 }. ε
− au11 (N1 + ε) − au12 (N2 + ε) − c1u
1610
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
Let y1 (n) = ln x1 (n), then y1 (n + 1) − y1 (n) ≥ E1ε . It follows that n −1 X
(y1 (i + 1) − y1 (i)) ≥ E1ε (τ11 ),
i=n−τ11
which implies y1 (n − τ11 ) ≤ y1 (n) − E1ε (τ11 ), and therefore x1 (n − τ11 ) ≤ x1 (n) exp{−E1ε (τ11 )}. Further from (1.1), x1 (n + 1) ≥ x1 (n) exp
= x1 (n) exp
al10
− au12 (N2 + ε) − c1u − au11 x1 (n) exp{−E1ε (τ11 )}
(N1 + ε) + ku1 l
− au12 (N2 + ε) − c1u (1 − ∆ε1 x1 (n)) .
a10
(N1 + ε) + ku1
By applying Lemma 2.2, we obtain exp ∆ε1
n→+∞
al10
1
lim inf x1 (n) ≥
(N1 + ε) + ku1
− au12 (N2 + ε) − c1u (1 − ∆ε1 N1 ) ,
and letting ε → 0, we obtain lim infn→+∞ x1 (n) ≥ m1 . Similarly, we also can show that lim infn→+∞ x2 (n) ≥ m2 . This completes the proof of Proposition 2.2. Combining Proposition 2.1 with Proposition 2.2, we can easily obtain the following main result of this section. Theorem 2.1. Assume that (H) holds. Then system (1.1) and (1.2) is permanent. 3. Global attractivity In this section, by constructing a non-negative Lyapunov-like functional, we will obtain sufficient conditions for global attractivity of positive solutions of system (1.1) and (1.2). We first introduce a definition and prove a lemma which will be useful to obtain our main result. Definition 3.1. A solution (x1 (n), x2 (n)) of (1.1) and (1.2) is said to be globally attractive if for any other solution (x∗1 (n), x∗2 (n)) of (1.1) and (1.2), we have limn→+∞ (x∗i (n) − xi (n)) = 0, i = 1, 2. Lemma 3.1. For any two positive solutions (x1 (n), x2 (n)) and (x∗1 (n), x∗2 (n)) of system (1.1) and (1.2), we have ln(xi (n + 1)/x∗i (n + 1)) = ln(xi (n)/x∗i (n)) − aii (n) xi (n) − x∗i (n) − aij (n) xj (n − τij ) − x∗j (n − τij )
ai0 (n) xi (n − τi0 ) − x∗i (n − τi0 )
n−1 X
−
[xi (n − τi0 ) + ki (n)] xi (n − τi0 ) + ki (n)
∗
+ aii (n)
( Ai (s)
ai0 (s)
xi (s − τi0 ) + ki (s) ∗
s=n−τii
− aij (s)x∗j (s − τij ) − ci (s) xi (s) − x∗i (s) + Bi (s)xi (s)
− aii (s)x∗i (s − τii )
ai0 (s) x∗i (s − τi0 ) − xi (s − τi0 )
[xi (s − τi0 ) + ki (s)] x∗i (s − τi0 ) + ki (s)
!) − aii (s) xi (s − τii ) − xi (s − τii ) − aij (s) xj (s − τij ) − xj (s − τij ) ∗
∗
,
(3.1)
where
Ai (s) = exp θi (s)
Bi (s) = exp ϕi (s)
ai0 (s) x∗i (s − τi0 ) + ki (s)
− aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) ,
ai0 (s)
xi (s − τi0 ) + ki (s) ai0 (s)
+ (1 − ϕi (s))
∗
− aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s)
x∗i (s − τi0 ) + ki (s)
θi (s), ϕi (s) ∈ (0, 1), i 6= j; i, j = 1, 2.
∗
− aii (s)x∗i (s − τii ) − aij (s)x∗j (s − τij ) − ci (s)
,
(3.2)
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1611
Proof. For i 6= j; i, j = 1, 2, from (1.1), ln(xi (n + 1)/x∗i (n + 1)) − ln(xi (n)/x∗i (n)) = ln(xi (n + 1)/xi (n)) − ln(x∗i (n + 1)/x∗i (n)) ai0 (n)
=
xi (n − τi0 ) + ki (n)
− aii (n)xi (n − τii ) − aij (n)xj (n − τij ) − ci (n)
ai0 (n)
∗ ∗ − a ( n ) x ( n − τ ) − a ( n ) x ( n − τ ) − c ( n ) ii ii ij ij i i j x∗i (n − τi0 ) + ki (n) ∗ ai0 (n) xi (n − τi0 ) − xi (n − τi0 ) − aii (n) xi (n − τii ) − x∗i (n − τii ) − aij (n) xj (n − τij ) − x∗j (n − τij ) = [xi (n − τi0 ) + ki (n)] x∗i (n − τi0 ) + ki (n) ai0 (n) xi (n − τi0 ) − x∗i (n − τi0 ) ∗ − aii (n) xi (n) − x∗i (n) − aij (n) xj (n − τij ) − x∗j (n − τij ) =− [xi (n − τi0 ) + ki (n)] xi (n − τi0 ) + ki (n) + aii (n) xi (n) − x∗i (n) − aii (n) xi (n − τii ) − x∗i (n − τii ) ,
−
that is ln(xi (n + 1)/x∗i (n + 1)) = ln(xi (n)/x∗i (n)) − aij (n) xj (n − τij ) − x∗j (n − τij )
ai0 (n) xi (n − τi0 ) − x∗i (n − τi0 )
−
[xi (n − τi0 ) + ki (n)] x∗i (n − τi0 ) + ki (n)
− aii (n)
xi (n) − x∗i (n) − [xi (n) − xi (n − τii )] − x∗i (n) − x∗i (n − τii )
.
(3.3)
Since n −1 n−1 X X (xi (s + 1) − xi (s)) − (xi (n) − xi (n − τii )) − x∗i (n) − x∗i (n − τii ) = s=n−τii
x∗i (s + 1) − x∗i (s)
s=n−τii
n−1
=
X
xi (s + 1) − x∗i (s + 1) − xi (s) − x∗i (s)
,
(3.4)
s=n−τii
and xi (s + 1) − x∗i (s + 1) − xi (s) − x∗i (s)
ai0 (s) − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) = xi (s) exp xi (s − τi0 ) + ki (s) ai0 (s) − aii (s)x∗i (s − τii ) − aij (s)x∗j (s − τij ) − ci (s) − xi (s) − x∗i (s) − x∗i (s) exp ∗ x (s − τi0 ) + ki (s) i ai0 (s) = xi (s) exp − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) xi (s − τi0 ) + ki (s) ai0 (s) ∗ ∗ − exp ∗ − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) xi (s − τi0 ) + ki (s) ai0 (s) ∗ ∗ ∗ + xi (s) − xi (s) exp ∗ − aii (s)xi (s − τii ) − aij (s)xj (s − τij ) − ci (s) − 1 , xi (s − τi0 ) + ki (s) by using the mean-value theorem, we have that xi (s + 1) − x∗i (s + 1) − xi (s) − x∗i (s)
ai0 (s) = xi (s) − x∗i (s) Ai (s) ∗ − aii (s)x∗i (s − τii ) − aij (s)x∗j (s − τij ) − ci (s) xi (s − τ10 ) + ki (s) ai0 (s) ai0 (s) + xi (s)Bi (s) − ∗ xi (s − τi0 ) + ki (s) xi (s − τi0 ) + ki (s) ∗ ∗ − aii (s) xi (s − τii ) − xi (s − τii ) − aij (s) xj (s − τij ) − xj (s − τij ) , here Ai (s), Bi (s) are defined by (3.2). Then from (3.3)–(3.5), we can easily obtain (3.1). The proof is completed.
(3.5)
1612
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
Theorem 3.1. Assume that in system (1.1) and (1.2), there exist positive constants α1 , α2 and η > 0 such that
αi Eii −
2 X
(αi Fij + αj Gji ) ≥ η,
i = 1, 2,
j=1,j6=i
where Eii , Fij , Gji are defined by (3.11). Then for any two positive solutions (x1 (n), x2 (n)) and (x∗1 (n), x∗2 (n)) of system (1.1) and (1.2), we have lim (x∗i (n) − xi (n)) = 0,
i = 1, 2.
n→+∞
Proof. Firstly, let V11 (n) = ln x1 (n) − ln x∗1 (n) . From (3.1), we have that
x1 (n + 1) x1 (n) ∗ ∗ ln ≤ ln x∗ (n + 1) x∗ (n) − a11 (n) x1 (n) − x1 (n) + a12 (n) x2 (n − τ12 ) − x2 (n − τ12 ) 1 1 n−1 X a10 (n) a10 (s) x1 (n − τ10 ) − x∗ (n − τ10 ) + a11 (n) + 2 + a11 (s)|x∗1 (s − τ11 )| 1 k ( s ) k1 (n) 1 s=n−τ11 a10 (s) ∗ ∗ x1 (s − τ10 ) − x∗ (s − τ10 ) + a12 (s)|x2 (s − τ12 )| + c1 (s) A1 (s) x1 (s) − x1 (s) + B1 (s)|x1 (s)| 1 2 k1 (n) + a11 (s) x1 (s − τ11 ) − x∗1 (s − τ11 ) + a12 (s) x2 (s − τ12 ) − x∗2 (s − τ12 ) .
(3.6)
Since xi (n) − x∗i (n) = eln xi (n) − eln xi (n) = ξi (n) ln(xi (n)/x∗i (n)), i = 1, 2, ∗
where 0 < ξi (n) < max{xi (n), x∗i (n)}, i = 1, 2, it follows that
ln(x1 (n)/x∗ (n)) − a11 (n) x1 (n) − x∗ (n) 1 1 1 1 ∗ = ln(x1 (n)/x1 (n)) − − − a11 (n) x1 (n) − x∗1 (n) . ξ1 (n) ξ1 (n)
(3.7)
By Proposition 2.1, there are constants Mi > 0 (i = 1, 2), and a positive integer n0 such that for n ≥ n0 , 0 < xi (n), x∗i (n) ≤ Mi (i = 1, 2). Then from (3.6) and (3.7) we can obtain that for n ≥ n0 + τ ,
1 − − a11 (n) x1 (n) − x∗1 (n) + a12 (n) x2 (n − τ12 ) − x∗2 (n − τ12 ) ξ 1 ( n) ξ1 (n) n −1 X a10 (n) a10 (s) ∗ x1 (n − τ10 ) − x1 (n − τ10 ) + a11 (n) A1 (s) + 2 + a11 (s)M1 + a12 (s)M2 + c1 (s) k1 (s) k1 (n) s=n−τ11
∆V11 ≤ −
1
a10 (s) x1 (s − τ10 ) − x∗ (s − τ10 ) + B1 (s)a11 (s)M1 x1 (s − τ11 ) − x∗ (s − τ11 ) × x1 (s) − x∗1 (s) + B1 (s)M1 2 1 1 k1 (s) ∗ + B1 (s)a12 (s)M1 x2 (s − τ12 ) − x2 (s − τ12 ) . Secondly, let V12 (n) =
n −1 n−1 X X a10 (s + τ10 ) x1 (s) − x∗ (s) + a12 (s + τ12 ) x2 (s) − x∗2 (s) 1 2 k1 (s + τ10 ) s=n−τ12 s=n−τ10 n−X 1+τ11 n −1 X a10 (u) + a11 (s) A1 (u) + a11 (u)M1 + a12 (u)M2 + c1 (u) x1 (u) − x∗1 (u) k ( u ) 1 s =n u=s−τ11
+ B1 (u)M1
a10 (u) k21
( u)
x1 (u − τ10 ) − x∗ (u − τ10 ) + B1 (u)a11 (u)M1 x1 (u − τ11 ) − x∗ (u − τ11 ) 1
1
+ B1 (u)a12 (u)M1 x2 (u − τ12 ) − x∗2 (u − τ12 ) .
(3.8)
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1613
By a simple calculation, we can obtain
∆V12 =
x1 (n) − x∗ (n) − a10 (n) x1 (n − τ10 ) − x∗ (n − τ10 ) 1 1 2 (n + τ10 ) k (n) 1 ∗ + a12 (n + τ12 ) x2 (n) − x2 (n) − a12 (n) x2 (n − τ12 ) − x∗2 (n − τ12 ) nX +τ11 a10 (n) + a11 (n)M1 + a12 (n)M2 + c1 (n) x1 (n) − x∗1 (n) + a11 (s) A1 (n) k1 (n) s=n+1 a10 (n + τ10 ) k21
+ B1 (n)M1
a10 (n) k21
x1 (n − τ10 ) − x∗ (n − τ10 ) + B1 (n)a11 (n)M1 x1 (n − τ11 ) − x∗ (n − τ11 )
(n)
1
1
+ B1 (n)a12 (n)M1 x2 (n − τ12 ) − x∗2 (n − τ12 ) n−1 X
− a11(n)
A1 (u)
u=n−τ11
+ B1 (u)M1
a10 (u) k1 (u)
+ a11 (u)M1 + a12 (u)M2 + c1 (u) x1 (u) − x∗1 (u)
a10 (u) k21
(u)
x1 (u − τ10 ) − x∗ (u − τ10 ) + B1 (u)a11 (u)M1 x1 (u − τ11 ) − x∗ (u − τ11 ) 1
1
+ B1 (u)a12 (u)M1 x2 (u − τ12 ) − x∗2 (u − τ12 ) .
(3.9)
Thirdly, let V13 (n) = M1
n−1 10 +τ11 X l+τX a10 (l + τ10 ) x1 (l) − x∗ (l) a11 (s) B ( l + τ ) 1 10 1 k21 (l + τ10 ) l=n−τ10 s=l+τ10 +1 l+2τ11
n−1 X
+ M1
B1 (l + τ11 )a11 (l + τ11 ) x1 (l) − x∗1 (l)
l=n−τ11 n−1 X
+ M1
X
a11 (s)
s=l+τ11 +1 12 +τ11 l+τX
B1 (l + τ12 )a12 (l + τ12 ) x2 (l) − x∗2 (l)
l=n−τ12
a11 (s).
s=l+τ12 +1
Calculating ∆V13 , we can derive n+τ10 +τ11
X
∆V13 =
a11 (s)M1
s=n+τ10 +1 n+τ11
X
−
a11 (s)M1
s=n+1
a10 (n + τ10 ) k21
(n + τ10 )
a10 (n) k21 (n)
B1 (n + τ10 ) x1 (n) − x∗1 (n)
B1 (n) x1 (n − τ10 ) − x∗1 (n − τ10 )
n+2τ11
X
+
a11 (s)M1 B1 (n + τ11 )a11 (n + τ11 ) x1 (n) − x∗1 (n)
s=n+τ11 +1 n+τ11
X
−
a11 (s)M1 B1 (n)a11 (n) x1 (n − τ11 ) − x∗1 (n − τ11 )
s=n+1 n+τ12 +τ11
X
+
a11 (s)M1 B1 (n + τ12 )a12 (n + τ12 ) x2 (n) − x∗2 (n)
s=n+τ12 +1 n+τ11
X
−
a11 (s)M1 B1 (n)a12 (n) x2 (n − τ12 ) − x∗2 (n − τ12 ) .
s=n+1
Now we set V1 (n) = V11 (n) + V12 (n) + V13 (n). Then from (3.6)–(3.10), we have that for n ≥ n0 + τ ,
1 − − a11 (n) x1 (n) − x∗1 (n) + a12 (n + τ12 ) x2 (n) − x∗2 (n) ξ1 (n) ξ1 (n) a10 (n + τ10 ) x1 (n) − x∗ (n) + 2 1 k1 (n + τ10 )
∆ V1 ≤ −
1
(3.10)
1614
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617 n+τ11
+
X
a11 (s)A1 (n)
s=n+1 n+τ10 +τ11
X
+
a11 (s)M1
s=n+τ10 +1
a10 (n) k1 (n)
+ a11 (n)M1 + a12 (n)M2 + c1 (n) x1 (n) − x∗1 (n)
a10 (n + τ10 ) k21
(n + τ10 )
B1 (n + τ10 ) x1 (n) − x∗1 (n)
n+2τ11
X
+
a11 (s)M1 B1 (n + τ11 )a11 (n + τ11 ) x1 (n) − x∗1 (n)
s=n+τ11 +1 n+τ12 +τ11
X
+
a11 (s)M1 B1 (n + τ12 )a12 (n + τ12 ) x2 (n) − x∗2 (n) .
s=n+τ12 +1
By arguments similar to those above, we take V21 (n) = ln x2 (n) − ln x∗2 (n) ,
V22 (n) =
n −1 n −1 X X a20 (s + τ20 ) x2 (s) − x∗ (s) + a21 (s + τ21 ) x1 (s) − x∗1 (s) 2 2 k1 (s + τ20 ) s=n−τ20 s=n−τ21 n−X 1+τ22 n−1 X a20 (u) + a22 (u)M2 + a21 (u)M1 + c2 (u) x2 (u) − x∗2 (u) + a22 (s) A 2 ( u) k2 (u) s =n u=s−τ22 a20 (u) x2 (u − τ20 ) − x∗ (u − τ20 ) + B2 (u)a22 (u)M2 x2 (u − τ22 ) − x∗ (u − τ22 ) + B2 (u)M2 2 2 2 k 2 ( u) + B2 (u)a21 (u)M2 x1 (u − τ21 ) − x∗1 (u − τ21 ) ,
V23 (n) = M2
n −1 20 +τ22 X l+τX a20 (l + τ20 ) x2 (l) − x∗ (l) B ( l + τ ) a22 (s) 2 20 2 k22 (l + τ20 ) l=n−τ20 s=l+τ20 +1 n −1 X
+ M2
l+2τ22
B2 (l + τ22 )a22 (l + τ22 ) x2 (l) − x∗2 (l)
l=n−τ22 n −1 X
+ M2
X
a22 (s)
s=l+τ22 +1 21 +τ22 l+τX
B2 (l + τ21 )a21 (l + τ21 ) x1 (l) − x∗1 (l)
l=n−τ21
a22 (s).
s=l+τ21 +1
Similarly, we take V2 (n) = V21 (n) + V22 (n) + V23 (n). Then in the same way as obtaining ∆V1 , we can obtain for n ≥ n0 + τ ,
1 − − a22 (n) x2 (n) − x∗2 (n) + a21 (n + τ21 ) x1 (n) − x∗1 (n) ξ2 (n) ξ2 (n) a20 (n + τ20 ) x2 (n) − x∗ (n) + 2 2 k2 (n + τ20 ) nX +τ22 a20 (n) + a22 (s)A2 (n) + a22 (n)M2 + a21 (n)M1 + c2 (n) x2 (n) − x∗2 (n) k2 (n) s=n+1
∆ V2 ≤ −
1
n+τ20 +τ22
+
X
a22 (s)M2
s=n+τ20 +1
a20 (n + τ20 ) k22
(n + τ20 )
B2 (n + τ20 ) x2 (n) − x∗2 (n)
n+2τ22
+
X
a22 (s)M2 B2 (n + τ22 )a22 (n + τ22 ) x2 (n) − x∗2 (n)
s=n+τ22 +1 n+τ21 +τ22
+
X
a22 (s)M2 B2 (n + τ21 )a21 (n + τ21 ) x1 (n) − x∗1 (n) .
s=n+τ21 +1
Now we define a Lyapunov-like discrete functional V by V (n) = α1 V1 (n) + α2 V2 (n).
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1615
It is easy to see that V (n0 + τ ) < +∞. Calculating the difference of V along the solution of (1.1) and (1.2), we have that for n ≥ n0 + τ ,
" n+τ 2 i0 +τii X X 1 1 ai0 (n + τi0 ) ai0 (n + τi0 ) ∆V ≤ − αi − − aii (n) − 2 aii (s)Mi 2 − Bi (n + τi0 ) ξ ( n ) ξ ( n ) k ( n + τ ) ki (n + τi0 ) i i i0 i i=1 s=n+τi0 +1 # nX +2τii nX +τii 2 X ai0 (n) αi aii (s)Mi Bi (n + τii )aii (n + τii ) − aii (s)Ai (n) − + aii (n)Mi + aij (n)Mj + ci (n) ki (n) s=n+τii +1 s=n+1 j=1,j6=i n+τji +τjj X + αj aji (n + τji ) + ajj (s)Mj Bj (n + τji )aji (n + τji ) xi (n) − x∗i (n) s=n+τji +1 ( 2 X 2 au au ≤− αi min(alii , − auii ) − 2i0 l − τii auii Mi 2i0 l Bui − τii (auii )2 Mi Bui Mi (ki ) (ki ) i=1 ) 2 X aui0 u u u u u u u u u − αi τii a A + a Mi + a Mj + c + αj (a + τjj a Mj B a ) xi (n) − x∗ (n) ii i
=−
2 X
(
2 X
αi Eii −
ii
kli
j=1,j6=i
ij
i
ji
jj
j
ji
i
) (αi Fij + αj Gji ) xi (n) − x∗i (n)
j=1,j6=i
i=1 2
≤ −η
X xi (n) − x∗ (n) , i
i =1
where Eii = min(alii ,
2
− auii ) −
aui0
au
− τii auii Mi 2i0 l Bui − τii (auii )2 Mi Bui , Mi ( ) (ki ) u ai0 + auii Mi + auij Mj + ciu , Fij = τii auii Aui l k2i l
(3.11)
ki
Gji = auji + τjj aujj Mj Buj auji . Then we have that n X
n 2 X X xi (p) − x∗ (p) ,
(V (p + 1) − V (p)) ≤ −η
i
p=n0 +τ i=1
p=n0 +τ
which implies V (n + 1) + η
n 2 X X xi (p) − x∗ (p) ≤ V (n0 + τ ). i
p=n0 +τ i=1
P2 P∞ P2 V (n0 +τ ) V (n0 +τ ) ∗ ∗ . Then < +∞, from which i=1 xi (p) − xi (p) ≤ n=n0 +τ i=1 xi (n) − xi (n) ≤ η η P2 ∗ we conclude that limn→+∞ i=1 xi (n) − xi (n) = 0. This means that (x1 (n), x2 (n)) is globally attractive. The proof is It follows
Pn
completed.
p=n0 +τ
4. Example and numerical simulation Consider the following system
0.3 + 0.2 sin n x1 (n + 1) = x1 (n) exp x1 (n − 3) + 10 − (2.52 + 0.02 sin n)x1 (n − 1) − ( 0 . 009 + 0 . 001 sin n ) x ( n − 3 ) − 0 . 01 , 2 0.4 + 0.1 sin n x2 (n + 1) = x2 (n) exp − (0.015 + 0.005 sin n)x1 (n − 4) x2 (n − 4) + 10 − (2.62 + 0.02 sin n)x2 (n − 1) − 0.02 .
(4.1)
1616
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
Fig. 1. Dynamic behavior of the first component x1 (n) of the solution (x1 (n), x2 (n)) to system (4.1) with the initial conditions (x1 (s), x2 (s))(s ∈ {−4, −3, −2, −1, 0}) = (0.013, 0.0113) and (0.017, 0.0115), respectively.
Fig. 2. Dynamic behavior of the second component x2 (n) of the solution (x1 (n), x2 (n)) to system (4.1) with the initial conditions (x1 (s), x2 (s))(s ∈ {−4, −3, −2, −1, 0}) = (0.013, 0.0113) and (0.017, 0.0115), respectively.
Here corresponding to system (1.1), we assume that a10 (n) = 0.3 + 0.2 sin n,
a11 (n) = 2.52 + 0.02 sin n,
a20 (n) = 0.4 + 0.1 sin n,
a21 (n) = 0.015 + 0.005 sin n,
c1 (n) = 0.01,
τ10 = 3,
a12 (n) = 0.009 + 0.001 sin n, a22 (n) = 2.62 + 0.02 sin n,
c2 (n) = 0.02,
τ11 = 1,
τ12 = 3,
τ20 = 4,
τ21 = 4,
τ22 = 1.
Corresponding to Theorem 3.1, by simple computation, we derive E11 ≈ 1.388,
F12 ≈ 1.27,
G21 ≈ 0.029,
E22 ≈ 1.45,
F21 ≈ 1.345,
G12 ≈ 0.014.
Then E11 − (F12 + G21 ) ≈ 0.089,
E22 − (F21 + G12 ) ≈ 0.091.
Also it is easy to see that the conditions in Theorem 3.1 are verified. Therefore, the positive solutions of (4.1) are globally attractive. Our numerical simulations support our result (see Figs. 1 and 2).
Acknowledgements This work was supported by the Program for New Century Excellent Talents in Fujian Province University (NCETFJ).
L. Wu et al. / Mathematical and Computer Modelling 49 (2009) 1607–1617
1617
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Q.M. Liu, R. Xu, W.G. Wang, Global asymptotic stability of Schoener’s competitive model with delays, J. Biomath. 21 (1) (2006) 147–152 (in Chinese). Z.H. Lu, L.S. Chen, Analysis on a periodic Schoener model, Acta. Math. Sci. 12 (Suppl.) (1992) 105–109. L.J. Zhang, H.F. Huo, J.F. Chen, Asymptotic behavior of the nonautonomous competing system with feedback controls, J. Biomath. 16 (4) (2001) 405–410. Q.M. Liu, R. Xu, Periodic solutions of Schoener’s competitive model with delays, J. Biomath. 19 (4) (2004) 385–394 (in Chinese). C.D. Yuan, C.H. Wang, Permanence and periodic solutions of the nonautonomous Schoener’s competing system with diffusion, Biomath. 1 (2) (1997) 17–20 (in Chinese). H. Xiang, K.M. Yan, B.Y. Wang, Positive periodic solutions for discrete Schoener’s competitive model, J. Lanzhou Univ. Technol. 31 (5) (2005) 125–128 (in Chinese). R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applications, in: Monographs and Textbooks in Pure and Applied Mathematics, vol. 228, Marcel Dekker, New York, 2000. J.D. Murray, Mathematical Biology, Springer, New York, 1989. Y. Saito, W.B. Ma, H. Hara, A necessary and sufficient condition for permanence of a Lotka–Volterra discrete system with delays, J. Math. Anal. Appl. 256 (2001) 162–174. Z.Y. Lu, W.D. Wang, Permanence and global attractivity for Lotka–Volterra difference system, J. Math. Biol. 22 (1999) 269–283. Z. Zhou, X. Zou, Stable periodic solutions in a discrete periodic logistic equation, Appl. Math. Lett. 16 (2) (2003) 165–171. Y.M. Chen, Z. Zhou, Stable periodic solution of a discrete periodic Lotka–Volterra competition system, J. Math. Anal. Appl. 277 (1) (2003) 358–366. F.D. Chen, Permanence and global attractivity of a discrete multi-species Lotka–Volterra competition predator–prey system, Appl. Math. Comput. 182 (1) (2006) 3–12. W.D. Wang, G. Mulone, F. Salemi, V. Salone, Global stability of discrete population models with delays and fluctuating environment, J. Math. Anal. Appl. 264 (2001) 147–167. F.D. Chen, Permanence of a single species discrete model with feedback control and delay, Appl. Math. Lett. 20 (7) (2007) 729–733. X.Y. Liao, S.F. Zhou, Y.M. Chen, Permanence for a discrete time Lotka–Volterra type food-chain model with delays, Appl. Math. Comput. 186 (1) (2007) 279–285. X.X. Chen, F.D. Chen, Periodicity and stability of a discrete time periodic n-species Lotka–Volterra competition system with feedback controls and deviating arguments, Soochow J. Math. 32 (3) (2006) 343–368. Y. Muroya, Persistence and global stability in discrete models of pure-delay nonautonomous Lotka–Volterra type, J. Math. Anal. Appl. 293 (2004) 446–461. Y. Muroya, Persistence and global stability in discrete models of Lotka–Volterra type, J. Math. Anal. Appl. 330 (1) (2007) 24–33. X.T. Yang, Uniform persistence and periodic solutions for a discrete predator–prey system with delays, J. Math. Anal. Appl. 316 (1) (2006) 161–177. F.D. Chen, L.P. Wu, Z. Li, Permanence and global attractivity of the discrete Gilpin-Ayala type population model, Comput. Math. Appl. 53 (8) (2007) 1214–1227. F.D. Chen, Permanence of a discrete N-species cooperation system with time delays and feedback controls, Appl. Math. Comput. 186 (1) (2007) 23–29. F.D. Chen, Permanence of a discrete n-species food-chain system with time delays, Appl. Math. Comput. 185 (1) (2007) 719–726. X.X. Chen, F.D. Chen, Stable periodic solution of a discrete periodic Lotka–Volterra competition system with a feedback control, Appl. Math. Comput. 181 (2) (2006) 1446–1454.