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Permanence and global attractivity of a discrete multispecies Lotka–Volterra competition predator–prey systems
Applied Mathematics and Computation 182 (2006) 3–12 www.elsevier.com/locate/amc
Permanence and global attractivity of a discrete multispecies Lotka–Volterra competition predator–prey systems Fengde Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
Abstract In this paper, we propose a discrete multispecies Lotka–Volterra competition predator–prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Non-autonomous; Lotka–Volterra competition predator–prey systems; Positive periodic solution; Discrete; Permanence; Global attractivity
1. Introduction The aim of this paper is to investigate the dynamic behavior of the following discrete n + m-species Lotka– Volterra competition predator–prey systems " # n m X X xi ðk þ 1Þ ¼ xi ðkÞ exp bi ðkÞ ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ ; "
l¼1
y j ðk þ 1Þ ¼ y j ðkÞ exp rj ðkÞ þ
n X
l¼1
d jl ðkÞxl ðkÞ
l¼1
m X
#
ð1:1Þ
ejl ðkÞy l ðkÞ ;
l¼1
where i = 1, 2, . . . , n; j = 1, 2, . . . , m; xi(k) is the density of prey species i at kth generation. yj(k) is the density of predator species j at kth generation. ail(k) and ejl(k) measures the intensity of intraspecific competition or interspecific action of prey species and predator species, respectively. bi(k) representing the intrinsic growth rate of the prey species xi; rj(k) representing the death rate of the predator species yj. Dynamic behaviors of population models governed by difference equations had been studied by a number of papers, see [1–17] and the references cited therein. It has been found that the autonomous discrete systems can demonstrate quite rich and complicated dynamics, see [1–4,18,19]. Recently, more and more scholars paid E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.03.026
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
attention to the non-autonomous discrete population models, since such kind of model could be more appropriate. Zhou and Zou [15] had studied the dynamic behavior of the following non-autonomous single species discrete model xðkÞ xðk þ 1Þ ¼ xðkÞ exp rðkÞ 1 . ð1:2Þ KðkÞ Sufficient conditions on the persistence and the existence of a stable periodic solution of the system (1.2) are obtained. Recently, Chen and Zhou [14] further generalized the system (1.2) to the following two-species Lotka– Volterra competition system xðkÞ xðk þ 1Þ ¼ xðkÞ exp r1 ðkÞ 1 l2 ðkÞyðkÞ ; K 1 ðkÞ ð1:3Þ yðkÞ yðk þ 1Þ ¼ yðkÞ exp r2 ðkÞ 1 l1 ðkÞxðkÞ . K 2 ðkÞ They obtained the sufficient conditions which guarantee the persistence of the system (1.3). Also, for the periodic case, they obtained the sufficient conditions which guarantee the existence of a globally stable periodic solution of the system. Wang and Lu [12] proposed the following Lotka–Volterra model " # n X xi ðk þ 1Þ ¼ xi ðkÞ exp ri ðkÞ aij ðkÞxj ðkÞ ; i ¼ 1; 2; . . . ; n; ð1:4Þ j¼1
where xi(k) is the density of population i at kth generation, ri(k) is the growth rate of population i at kth generation, aij(k) measures the intensity of intraspecific competition or interspecific action of species. By constructing a suitable Lyapunov function and using the finite covering theorem of Mathematic Analysis, they obtained a set of sufficient conditions which ensure the system to be globally asymptotically stable. As was pointed out by Berryman [20], the dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Already, there are several scholars done works on the persistence, existence of positive periodic solution etc. of two species discrete predator–prey system, see [17,21–23] and the references cited therein. However, to the best of the author’s knowledge, to this day, still no scholar done works on multispecies competition predator–prey ecosystems. Indeed, as far as the continuous multispecies ecosystem is concerned, Yang and Xu [24] considered the following periodic n-prey and m-predator Lotka–Volterra system of differential equations " # n m X X x_ i ðtÞ ¼ xi ðtÞ bi ðtÞ aik ðtÞxk ðtÞ cik ðtÞy k ðtÞ ; i ¼ 1; 2; . . . ; n; "
k¼1
y_ j ðtÞ ¼ y j ðtÞ rj ðtÞ þ
n X k¼1
k¼1
d jk ðtÞxk ðtÞ
m X
# ejk ðtÞy k ðtÞ ;
ð1:5Þ j ¼ 1; 2; . . . ; m;
k¼1
where xi(t) denotes the density of prey species Xi at time t, yj(t) denotes the density of predator species Yj at time t; bi(t), rj(t), aik(t), cil(t), djk(t), and ejl(t) (i, k = 1, . . . , n; j, l = 1, . . . , m) are continuous periodic functions defined on [0, +1) with a common periodic T > 0; rj(t), aik(t), cil(t), djk(t) and ejl(t) are non-negative; aii(t), ejj(t) are strictly positive. Under the assumption that bi(t) are positive periodic functions they obtained a set of sufficient conditions for the existence and global attractivity of the periodic solution of system (1.5). For more works on this direction, one could refer to [24–36] and the references cited therein. Obviously, system (1.1) is the counterpart of continuous Lotka–Volterra competition predator–prey system (1.5). To the best of the author’s knowledge, this is the first time the multispecies discrete competition
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
5
predator–prey system is considered. The aim of this paper is, by developing the analysis technique of Huo and Li [16] and Chen and Zhou [14], to investigate the persistence and global stability property of the system (1.1). We say that system (1.1) is permanent if there are positive constants M and m such that for each positive solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) of system (1.1) satisfies m 6 lim inf xi ðkÞ 6 lim sup xi ðkÞ 6 M; k!þ1
k!þ1
m 6 lim inf y j ðkÞ 6 lim sup y j ðkÞ 6 M k!þ1
k!þ1
for all i = 1, 2, . . . , n; j = 1, 2, . . . , m. Throughout this paper, we assume that bi(k), ail(k), cil(n), rj(n), djl(n), ejl(n) are all bounded non-negative sequence, and use the following notations for any bounded sequence {x(n)}, xu ¼ sup xðnÞ;
xl ¼ inf xðnÞ. n2N
n2N
For biological reasons, we only consider solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) with xi ð0Þ > 0;
i ¼ 1; 2; . . . ; n;
y j ð0Þ > 0;
j ¼ 1; 2; . . . ; m. 1
Then system (1.1) has a positive solution ðx1 ðkÞ; . . . ; xn ðkÞ; y 1 ðkÞ; . . . ; y m ðkÞÞk¼0 passing through (x1(0), . . . , xn(0), y1(0), . . . , ym(0)). The organization of this paper is as follow: In Section 2, we obtain sufficient conditions which guarantee the permanence of the system (1.1). In Section 3, we obtain sufficient conditions which guarantee the global stability of the positive solution of system (1.1). As a consequence, for periodic case, we obtain sufficient conditions which ensure the existence of a globally stable positive solution of system (1.1). 2. Permanence In this section, we establish a permanence results for system (1.1). Proposition 1. For every solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) of system (1.1), we have lim sup xi ðkÞ 6 pi ;
ð2:1Þ
k!þ1
where pi ¼ a1l expðbui 1Þ. ii
Proof. To prove (2.1), we first assume that there exists an l0 2 N such that xi(l0 + 1) P xi(l0). Then n m X X bi ðl0 Þ ail ðl0 Þxl ðl0 Þ cil ðl0 Þy l ðl0 Þ P 0. l¼1
l¼1
Hence, xi ðl0 Þ 6
bi ðl0 Þ bui 6 . aii ðl0 Þ alii
ð2:2Þ bu
expðx1Þ x
expðbu 1Þ
P 1 it immediately follows that ali 6 ali . It follows from (2.2) that ii ii " # n m X X xi ðl0 þ 1Þ ¼ xi ðl0 Þ exp bi ðl0 Þ ail ðl0 Þxl ðl0 Þ cil ðl0 Þy l ðl0 Þ 6 xi ðl0 Þ exp½bui alii xi ðl0 Þ
By applying the fact
l¼1
6
l¼1
1 def expðbui 1Þ ¼ pi ; alii
for a, b > 0. here we had used the fact maxx2R x expðb axÞ ¼ expðb1Þ a We claim that xi ðkÞ 6 pi
for k P l0 .
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
By way of contradiction, assume that there exists a q0 > l0 such that xi(q0) > pi. Then q0 P l0 + 2. Let ~ q0 P l0 þ 2 be the smallest integer such that xi ð~ q0 Þ > pi . Then xi ð~q0 1Þ < xi ð~q0 Þ. The above argument produces that xi ð~ q0 Þ 6 pi , a contradiction. This proves the claim. Now we assume that xi(k + 1) < xi(k) for all k 2 N. In particular, limk!+1xi(k) exists, denote by xi . We bu bu claim that xi 6 ali . By way of contradiction, assume that xi > ali . Taking limit in the ith equation in system (1.1) ii ii gives ! n m X X lim bi ðkÞ ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ ¼ 0; k!þ1
l¼1
l¼1
which is a contradiction since n m X X ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ 6 bi ðkÞ aii ðkÞxi ðkÞ 6 bui aliixi < 0 bi ðkÞ l¼1
l¼1 bu
for enough large k 2 N. This proves the claim. Note that ali 6 pi . It follows that (2.1) holds. This completes the ii proof of Proposition 1. h Proposition 2. Assume that n X rlj þ d ujl pl > 0
ðH1 Þ
l¼1
holds, where pl, l = 1, 2, . . . , n are defined by (2.1). Then for every solution (x1(k), . . . , xn(k), y1(k), . . . ,ym(k)) of (1.1), we have lim sup y j ðkÞ 6 qj ;
ð2:3Þ
k!þ1
where qj ¼ e1l expfrlj þ
Pn
u l¼1 d jl pl
jj
1g.
Proof. For any e > 0, according to Proposition 1, there exists a k1 2 N such that xi ðkÞ 6 pi þ e for all k P k 1 . Thus, by using n + jth equation in system (1.1), for k P k1, one has " # n m X X y j ðk þ 1Þ 6 y j ðkÞ exp rj ðkÞ þ d jl ðkÞðpl þ eÞ ejl ðkÞy l ðkÞ . l¼1
ð2:4Þ
l¼1
To prove (2.3), we first assume that there exists an l0 P k1 such that yj(l0 + 1) P yj(l0), then it follows from (2.4) that n m X X d jl ðl0 Þðpl þ eÞ ejl ðl0 Þy l ðl0 Þ P 0. rj ðl0 Þ þ l¼1
l¼1
And so, y j ðl0 Þ 6
rj ðl0 Þ þ
Pn
l¼1 d jl ðl0 Þðp l
þ eÞ
ejj ðl0 Þ
By using (2.5), one has
"
y j ðl0 þ 1Þ 6 y j ðl0 Þ exp rj ðl0 Þ þ rlj
þ
n X l¼1
Pn
u l¼1 d jl ðp l eljj
d jl ðl0 Þðpl þ eÞ
l¼1
" 6 y j ðl0 Þ exp
n X
6
rlj þ
d ujl ðpl
m X l¼1
#
þ eÞ
eljj y j ðl0 Þ
þ eÞ
ð2:5Þ
. #
ejl ðl0 Þy l ðl0 Þ
" # n X 1 u l 6 l exp rj þ d jl ðpl þ eÞ 1 . ejj l¼1
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
Let
7
" # n X 1 qje ¼ l exp rlj þ d ujl ðpl þ eÞ 1 . ejj l¼1
We claim that y j ðkÞ 6 qje
for k P l0 .
By way of contradiction, assume that there exists a q1 > l0 such that yj(q1) > qje. Then q1 P l0 + 2. Let ~ q1 P l0 þ 2 be the smallest integer such that y j ð~ q1 Þ > qje . Then y j ð~q1 1Þ < y j ð~q1 Þ. The above argument produces that y j ð~ q1 Þ 6 qje , a contradiction. This proves the claim. Now we assume that yj(k + 1) < yj(k) for all k P k1. In particular, limk!+1yj(k) exists, denote by y j . We claim that Pn rlj þ l¼1 d ujl ðpl þ eÞ y j 6 . eljj By way of contradiction, assume that Pn rlj þ l¼1 d ujl ðpl þ eÞ y j > . eljj Taking limit in the n + jth equation in system (1.1) gives ! n m X X d jl ðkÞxl ðkÞ ejl ðkÞy l ðkÞ ¼ 0; lim rj ðkÞ þ k!þ1
l¼1
l¼1
which is a contradiction since for k P k1, n m n X X X d jl ðkÞxl ðkÞ ejl ðkÞy l ðkÞ 6 rj ðkÞ þ d jl ðkÞxl ðkÞ ejj ðkÞy j ðkÞ rj ðkÞ þ l¼1
l¼1
l¼1
6 rlj þ Pn u l
rj þ
This proves the claim. Noting the fact that This completes the proof of Proposition 2.
h
n X
l¼1 d ðpl þeÞ
l¼1 jl eljj
d ujl ðpl þ eÞ eljj y j < 0.
6 qje and lime!0qje = q j. It follows that (2.3) holds.
Proposition 3. In addition to (H1), assume further that n m X X auil pl cuil ql > 0 bli l¼1;l6¼i
ðH2 Þ
l¼1
hold for all i = 1, 2, . . . , n, where pl, ql are defined by (2.1) and (2.3), respectively. Then for every solution (x1(k), . . . , xn(k),y1(k), . . . , ym(k)) of (1.1), we have lim inf xi ðkÞ P ai ; k!þ1 Pn Pm u bli au p c q Pn Pm l¼1;l6¼i il l l¼1 il l where ai ¼ expfbli l¼1 auil pl l¼1 cuil ql g. au
ð2:6Þ
ii
Proof. Condition (H2) implies that for enough small positive number e > 0, one has n m X X auil ðpl þ eÞ cuil ðql þ eÞ > 0. bli l¼1;l6¼i
ð2:7Þ
l¼1
For above e, from Propositions 1 and 2 we know that there exists enough large k2 such that xi ðkÞ 6 pi þ e;
y j ðkÞ 6 qj þ e
for all k P k 2 .
ð2:8Þ
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
Thus, by using ith equation in system (1.1), for k P k2, one has " # n m X X xi ðk þ 1Þ P xi ðkÞ exp bi ðkÞ ail ðkÞðpl þ eÞ cil ðkÞðql þ eÞ aii ðkÞxi ðkÞ . l¼1;l6¼i
ð2:9Þ
l¼1
To prove (2.6), we first assume that there exists an l0 P k2 such that xi(l0 + 1) 6 xi(l0). Then, from (2.9) it follows that n X
bi ðl0 Þ
ail ðl0 Þðpl þ eÞ
l¼1;l6¼i
m X
cil ðl0 Þðql þ eÞ aii ðl0 Þxi ðl0 Þ 6 0;
l¼1
which implies that P þ eÞ ml¼1 cil ðl0 Þðql þ eÞ aii ðl0 Þ Pn Pm l u bi l¼1;l6¼i ail ðpl þ eÞ l¼1 cuil ðql þ eÞ P : auii
xi ðl0 Þ P
bi ðl0 Þ
Pn
l¼1;l6¼i ail ðl0 Þðp l
It follows from (2.10) that
"
xi ðl0 þ 1Þ ¼ xi ðl0 Þ exp bi ðl0 Þ
ail ðl0 Þxl ðl0 Þ
l¼1
" P xi ðl0 Þ exp
n X
bli
n X
auil ðpl
m X
# cil ðl0 Þy l ðl0 Þ
l¼1
þ eÞ
l¼1
m X
ð2:10Þ
cuil ðql
# þ eÞ
l¼1
" # n m X X Ai def l u u P u exp bi ail ðpl þ eÞ cil ðql þ eÞ ¼ aie ; aii l¼1 l¼1 where n X
Ai ¼ bli
auil ðpl þ eÞ
l¼1;l6¼i
m X
cuil ðql þ eÞ.
l¼1
We claim that xi ðkÞ P aie
for k P l0 .
By way of contradiction, assume that there exists a q2 > l0 such that xi(q2) < aie. Then q2 P l0 + 2. Let ~ q2 P l0 þ 2 be the smallest integer such that xi ð~ q2 Þ < aie . Then xi ð~q2 1Þ > xi ð~q2 Þ. The above argument produces that xi ð~ q2 Þ P aie , a contradiction. This proves the claim. Now we assume that xi(k + 1) > xi(k) for all k P k2. In particular, limk!+1xi(k) exists, denote by xi. We claim that Pn Pm bli l¼1;l6¼i auil ðpl þ eÞ l¼1 cuil ðql þ eÞ xi P . auii By way of contradiction, assume that Pn Pm bli l¼1;l6¼i auil ðpl þ eÞ l¼1 cuil ðql þ eÞ xi < . auii Taking limit in the ith equation in system (1.1) gives ! n m X X lim bi ðkÞ ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ ¼ 0; k!þ1
l¼1
l¼1
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
which is a contradiction since lim inf k!þ1
bi ðkÞ
n X
ail ðkÞxl ðkÞ
l¼1
m X
! cil ðkÞy l ðkÞ
n X
P bli
auil ðpl þ eÞ
l¼1;l6¼i
l¼1
m X
9
cuil ðql þ eÞ auii xi > 0.
l¼1
bu
This proves the claim. Note that ali 6 pi implies that Pn Pm ii bli l¼1;l6¼i auil ðpl þ eÞ l¼1 cuil ðql þ eÞ P aie ; auii also noting the fact lim aie ¼ ai . e!0
It follows that (2.6) holds. This completes the proof of Proposition 3.
h
Proposition 4. In addition to (H1) and (H2), assume further that n n X X d ljl al eujl ql > 0 ruj þ
ðH3 Þ
l¼1;l6¼j
l¼1
holds, where pl, ql, al are defined by (2.1), (2.3) and (2.6), respectively. Then for every solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) of (1.1), we have lim inf y j ðkÞ P bj ;
ð2:11Þ
k!þ1
where bj ¼
ruj þ
Pn
l l¼1 d jl al eujj
Pn
u l¼1;l6¼j ejl ql
exp ruj þ
n X
d ljl al
l¼1
n X
! eujl ql .
l¼1
Proof. From (H3) one could take e > 0 enough small, such that ruj þ
n X
d ljl ðal eÞ
l¼1
n X
eujl ðql þ eÞ > 0.
l¼1;l6¼j
For above e > 0, according to Propositions 1–3, there exists a k3 P k2 such that for all k P k3, xi ðkÞ 6 pi þ e;
y j ðkÞ 6 qj þ e;
xi ðkÞ P ai e.
Thus, by using n + jth equation in system (1.1), for k P k3, one has " # n m X X y j ðk þ 1Þ P y j ðkÞ exp rj ðkÞ þ d jl ðkÞðai eÞ ejl ðkÞðqj þ eÞ ejj ðkÞy j ðkÞ . l¼1
ð2:12Þ
l¼1;l6¼j
By applying (2.12), the rest of the proof of Proposition 4 is similar to that of the proof of Proposition 3 and we omit the detail here. h Now we state the main result of this section.
Theorem 1. Assume that (H1)–(H3) hold, then system (1.1) is permanent. It should be noticed that, from the proofs of Propositions 1–4, we know that under the assumption of Theorem 1, the set ½a1 ; p1 ½an ; pn ½b1 ; q1 ½bm ; qm is an invariant set of system (1.1).
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
3. Global stability Now we study the stability property of the positive solution of system (1.1). Theorem 2. Assume that (H1)–(H3) hold. Assume further that n m X X auil pl þ cuil ql < 1; ki ¼ maxfj1 auii pi j; j1 alii ai jg þ l¼1;l6¼i n X
m X
l¼1
l¼1;l6¼j
l¼1
d ujl pl þ
dj ¼ maxfj1 eujj qi j; j1 eljj bj jg þ
ð3:1Þ eujl ql < 1.
Then for any two positive solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) and ð~x1 ðkÞ; . . . ; ~xn ðkÞ; ~y 1 ðkÞ; . . . ; ~y m ðkÞÞ of system (1.1), we have lim ð~xi ðkÞ xi ðkÞÞ ¼ 0;
k!þ1
lim ð~y j ðkÞ y j ðkÞÞ ¼ 0.
ð3:2Þ
k!þ1
Proof. Let xi ðkÞ ¼ ~xi ðkÞ expðui ðkÞÞ;
y j ðkÞ ¼ ~y j ðkÞ expðvj ðkÞÞ.
ð3:3Þ
Then system (1.1) is equivalent to n m X X ail ðkÞ~xl ðkÞðexpðul ðkÞÞ 1ÞÞ cil ðkÞ~y l ðkÞðexpðvl ðkÞÞ 1ÞÞ; ui ðk þ 1Þ ¼ ui ðkÞ vj ðk þ 1Þ ¼ vj ðkÞ
l¼1
l¼1
n X
m X
d jl ðkÞ~xl ðkÞðexpðul ðkÞÞ 1ÞÞ
l¼1
ð3:4Þ eil ðkÞ~y l ðkÞðexpðvl ðkÞÞ 1ÞÞ.
l¼1
And so, n X
ui ðk þ 1Þ ¼ ð1 aii ðkÞ~xi ðkÞ expðhi ðkÞui ðkÞÞÞui ðkÞ
m X
cil ðkÞ~y l ðkÞ expðnl ðkÞvl ðkÞÞvl ðkÞ;
l¼1
vj ðk þ 1Þ ¼ ð1 ejj ðkÞ~y j ðkÞ expðnj ðkÞvj ðkÞÞÞvj ðkÞ
m X
ail ðkÞ~xl ðkÞ expðhl ðkÞul ðkÞÞul ðkÞ
l¼1;l6¼i
n X
ð3:5Þ d jl ðkÞ~xl ðkÞ expðhl ðkÞul ðkÞÞul ðkÞ
l¼1
eil ðkÞ~y l ðkÞ expðnl ðkÞvl ðkÞÞÞvl ðkÞ;
l¼1
where hl(k), nl(k) 2 [0, 1]. To complete the proof, it suffices to show that lim ui ðkÞ ¼ 0;
k!þ1
lim vj ðkÞ ¼ 0.
ð3:6Þ
k!þ1
In view of (3.1), we can choose e > 0 small enough such that n m X X kei ¼ maxfj1 auii ðpi þ eÞj; j1 alii ðai eÞjg þ auil ðpl þ eÞ þ cuil ðql þ eÞ < 1; dej
¼ maxfj1
eujj ðqi
þ eÞj; j1
eljj ðbj
l¼1;l6¼i n X
d ujl ðpl
eÞjg þ
l¼1
l¼1
þ eÞ þ
m X
ð3:7Þ eujl ðqþ eÞ
< 1.
l¼1;l6¼j
For above e > 0, according to Propositions 1–4 in Section 2, there exists a k* 2 N such that ai e 6 ~xi ðkÞ; bj e 6 ~y j ðkÞ; *
for all k P k .
xi ðkÞ 6 pi þ e; y j ðkÞ 6 qj þ e
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
11
Noticing that hl(k), nl(k) 2 [0, 1] implies that ~xl ðkÞ expðhl ðkÞul ðkÞÞ lies between ~xl ðkÞ and xl(k), ~y l ðkÞ exp ðnl ðkÞvl ðkÞÞÞ lies between ~y l ðkÞ and yl(k). From (3.5), we get n m X X jui ðk þ 1Þj ¼ maxfj1 auii ðpi þ eÞj; j1 alii ðai eÞjgjui ðkÞj þ auil ðpl þ eÞjul ðkÞj þ cuil ðql þ eÞjvl ðkÞj; l¼1;l6¼i
jvj ðk þ 1Þj ¼ maxfj1 eujj ðqi þ eÞj; j1 eljj ðbj eÞjgjvj ðkÞj þ
n X
d ujl ðpl þ eÞjul ðkÞj þ
l¼1
l¼1 m X
euil ðql þ eÞjvl ðkÞj.
l¼1
ð3:8Þ Let c ¼
maxfkei ; dej g,
*
then c < 1. In view of (3.8), for k P k , we get
maxfjui ðk þ 1Þj; jvj ðk þ 1Þjg 6 c maxfjvj ðkÞj; jui ðkÞjg. i;j
This implies
maxfjui ðkÞj; jvj ðkÞjg 6 ckk maxfjvj ðk Þj; jui ðk Þjg. i;j
Therefore, (3.6) holds and the proof is complete.
h
4. Existence and stability of periodic solution In this section, we further assume that the coefficients of system (1.1) satisfies (H4). There exists a positive integer x such that for k 2 N, 0 < bi ðk þ xÞ ¼ bi ðkÞ; 0 < ail ðk þ xÞ ¼ ail ðkÞ; 0 < cil ðk þ xÞ ¼ cil ðkÞ; 0 < rj ðk þ xÞ ¼ rj ðkÞ; 0 < d jl ðk þ xÞ ¼ d jl ðkÞ; 0 < ejl ðk þ xÞ ¼ ejl .
ðH4 Þ
Our first result concerned with the existence of a positive periodic solution of system (1.1). Theorem 3. Assume that (H1)–(H4) hold, then system (1.1) admits at least one positive x-periodic solution which we denote by ð~x1 ðkÞ; . . . ; ~xn ðkÞ; ~y 1 ðkÞ; . . . ; ~y m ðkÞÞ. Proof. As noted at the end of Section 2, def
Dnþm ¼ ½a1 ; p1 ½an ; pn ½b1 ; q1 ½bm ; qm is an invariant set of system (1.1). Thus, we can define a mapping F on Dn+m by F ðx1 ð0Þ; . . . ; xn ð0Þ; y 1 ð0Þ; . . . ; y m ð0ÞÞ ¼ ðx1 ðxÞ; . . . ; xn ðxÞ; y 1 ðxÞ; . . . ; y m ðxÞÞ; for (x1(0), . . . , xn(0), y1(0), . . . , ym(0)) 2 Dn+m. Obviously, F depends continuously on (x1(0), . . . , xn(0), y1(0), . . . , ym(0)). Thus, F is continuous and maps the compact set Dn+m into itself. Therefore, F has a fixed point. It is easy to see that the solution ð~x1 ðkÞ; . . . ; ~xn ðkÞ; ~y 1 ðkÞ; . . . ; ~y m ðkÞÞ passing through this fixed point is an xperiodic solution of the system (1.1). This complete the proof of Theorem 3. h Theorem 4. Assume that (H1)–(H4) and (3.1) hold, then system (1.1) has a global stable positive x-periodic solution. Proof. Under the assumption of Theorem 4, it follows from Theorem 3 that system (1.1) admits at least one positive x-periodic solution. Also, Theorem 2 ensures the positive solution to be globally stable. This end the proof of Theorem 4. h Acknowledgements This work is supported by the National Natural Science Foundation of China (10501007), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian Education Bureau (JA04156).
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Permanence and global attractivity of a discrete multispecies Lotka–Volterra competition predator–prey systems Fengde Chen College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, PR China
Abstract In this paper, we propose a discrete multispecies Lotka–Volterra competition predator–prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Non-autonomous; Lotka–Volterra competition predator–prey systems; Positive periodic solution; Discrete; Permanence; Global attractivity
1. Introduction The aim of this paper is to investigate the dynamic behavior of the following discrete n + m-species Lotka– Volterra competition predator–prey systems " # n m X X xi ðk þ 1Þ ¼ xi ðkÞ exp bi ðkÞ ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ ; "
l¼1
y j ðk þ 1Þ ¼ y j ðkÞ exp rj ðkÞ þ
n X
l¼1
d jl ðkÞxl ðkÞ
l¼1
m X
#
ð1:1Þ
ejl ðkÞy l ðkÞ ;
l¼1
where i = 1, 2, . . . , n; j = 1, 2, . . . , m; xi(k) is the density of prey species i at kth generation. yj(k) is the density of predator species j at kth generation. ail(k) and ejl(k) measures the intensity of intraspecific competition or interspecific action of prey species and predator species, respectively. bi(k) representing the intrinsic growth rate of the prey species xi; rj(k) representing the death rate of the predator species yj. Dynamic behaviors of population models governed by difference equations had been studied by a number of papers, see [1–17] and the references cited therein. It has been found that the autonomous discrete systems can demonstrate quite rich and complicated dynamics, see [1–4,18,19]. Recently, more and more scholars paid E-mail addresses: [email protected], [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.03.026
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
attention to the non-autonomous discrete population models, since such kind of model could be more appropriate. Zhou and Zou [15] had studied the dynamic behavior of the following non-autonomous single species discrete model xðkÞ xðk þ 1Þ ¼ xðkÞ exp rðkÞ 1 . ð1:2Þ KðkÞ Sufficient conditions on the persistence and the existence of a stable periodic solution of the system (1.2) are obtained. Recently, Chen and Zhou [14] further generalized the system (1.2) to the following two-species Lotka– Volterra competition system xðkÞ xðk þ 1Þ ¼ xðkÞ exp r1 ðkÞ 1 l2 ðkÞyðkÞ ; K 1 ðkÞ ð1:3Þ yðkÞ yðk þ 1Þ ¼ yðkÞ exp r2 ðkÞ 1 l1 ðkÞxðkÞ . K 2 ðkÞ They obtained the sufficient conditions which guarantee the persistence of the system (1.3). Also, for the periodic case, they obtained the sufficient conditions which guarantee the existence of a globally stable periodic solution of the system. Wang and Lu [12] proposed the following Lotka–Volterra model " # n X xi ðk þ 1Þ ¼ xi ðkÞ exp ri ðkÞ aij ðkÞxj ðkÞ ; i ¼ 1; 2; . . . ; n; ð1:4Þ j¼1
where xi(k) is the density of population i at kth generation, ri(k) is the growth rate of population i at kth generation, aij(k) measures the intensity of intraspecific competition or interspecific action of species. By constructing a suitable Lyapunov function and using the finite covering theorem of Mathematic Analysis, they obtained a set of sufficient conditions which ensure the system to be globally asymptotically stable. As was pointed out by Berryman [20], the dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Already, there are several scholars done works on the persistence, existence of positive periodic solution etc. of two species discrete predator–prey system, see [17,21–23] and the references cited therein. However, to the best of the author’s knowledge, to this day, still no scholar done works on multispecies competition predator–prey ecosystems. Indeed, as far as the continuous multispecies ecosystem is concerned, Yang and Xu [24] considered the following periodic n-prey and m-predator Lotka–Volterra system of differential equations " # n m X X x_ i ðtÞ ¼ xi ðtÞ bi ðtÞ aik ðtÞxk ðtÞ cik ðtÞy k ðtÞ ; i ¼ 1; 2; . . . ; n; "
k¼1
y_ j ðtÞ ¼ y j ðtÞ rj ðtÞ þ
n X k¼1
k¼1
d jk ðtÞxk ðtÞ
m X
# ejk ðtÞy k ðtÞ ;
ð1:5Þ j ¼ 1; 2; . . . ; m;
k¼1
where xi(t) denotes the density of prey species Xi at time t, yj(t) denotes the density of predator species Yj at time t; bi(t), rj(t), aik(t), cil(t), djk(t), and ejl(t) (i, k = 1, . . . , n; j, l = 1, . . . , m) are continuous periodic functions defined on [0, +1) with a common periodic T > 0; rj(t), aik(t), cil(t), djk(t) and ejl(t) are non-negative; aii(t), ejj(t) are strictly positive. Under the assumption that bi(t) are positive periodic functions they obtained a set of sufficient conditions for the existence and global attractivity of the periodic solution of system (1.5). For more works on this direction, one could refer to [24–36] and the references cited therein. Obviously, system (1.1) is the counterpart of continuous Lotka–Volterra competition predator–prey system (1.5). To the best of the author’s knowledge, this is the first time the multispecies discrete competition
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
5
predator–prey system is considered. The aim of this paper is, by developing the analysis technique of Huo and Li [16] and Chen and Zhou [14], to investigate the persistence and global stability property of the system (1.1). We say that system (1.1) is permanent if there are positive constants M and m such that for each positive solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) of system (1.1) satisfies m 6 lim inf xi ðkÞ 6 lim sup xi ðkÞ 6 M; k!þ1
k!þ1
m 6 lim inf y j ðkÞ 6 lim sup y j ðkÞ 6 M k!þ1
k!þ1
for all i = 1, 2, . . . , n; j = 1, 2, . . . , m. Throughout this paper, we assume that bi(k), ail(k), cil(n), rj(n), djl(n), ejl(n) are all bounded non-negative sequence, and use the following notations for any bounded sequence {x(n)}, xu ¼ sup xðnÞ;
xl ¼ inf xðnÞ. n2N
n2N
For biological reasons, we only consider solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) with xi ð0Þ > 0;
i ¼ 1; 2; . . . ; n;
y j ð0Þ > 0;
j ¼ 1; 2; . . . ; m. 1
Then system (1.1) has a positive solution ðx1 ðkÞ; . . . ; xn ðkÞ; y 1 ðkÞ; . . . ; y m ðkÞÞk¼0 passing through (x1(0), . . . , xn(0), y1(0), . . . , ym(0)). The organization of this paper is as follow: In Section 2, we obtain sufficient conditions which guarantee the permanence of the system (1.1). In Section 3, we obtain sufficient conditions which guarantee the global stability of the positive solution of system (1.1). As a consequence, for periodic case, we obtain sufficient conditions which ensure the existence of a globally stable positive solution of system (1.1). 2. Permanence In this section, we establish a permanence results for system (1.1). Proposition 1. For every solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) of system (1.1), we have lim sup xi ðkÞ 6 pi ;
ð2:1Þ
k!þ1
where pi ¼ a1l expðbui 1Þ. ii
Proof. To prove (2.1), we first assume that there exists an l0 2 N such that xi(l0 + 1) P xi(l0). Then n m X X bi ðl0 Þ ail ðl0 Þxl ðl0 Þ cil ðl0 Þy l ðl0 Þ P 0. l¼1
l¼1
Hence, xi ðl0 Þ 6
bi ðl0 Þ bui 6 . aii ðl0 Þ alii
ð2:2Þ bu
expðx1Þ x
expðbu 1Þ
P 1 it immediately follows that ali 6 ali . It follows from (2.2) that ii ii " # n m X X xi ðl0 þ 1Þ ¼ xi ðl0 Þ exp bi ðl0 Þ ail ðl0 Þxl ðl0 Þ cil ðl0 Þy l ðl0 Þ 6 xi ðl0 Þ exp½bui alii xi ðl0 Þ
By applying the fact
l¼1
6
l¼1
1 def expðbui 1Þ ¼ pi ; alii
for a, b > 0. here we had used the fact maxx2R x expðb axÞ ¼ expðb1Þ a We claim that xi ðkÞ 6 pi
for k P l0 .
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
By way of contradiction, assume that there exists a q0 > l0 such that xi(q0) > pi. Then q0 P l0 + 2. Let ~ q0 P l0 þ 2 be the smallest integer such that xi ð~ q0 Þ > pi . Then xi ð~q0 1Þ < xi ð~q0 Þ. The above argument produces that xi ð~ q0 Þ 6 pi , a contradiction. This proves the claim. Now we assume that xi(k + 1) < xi(k) for all k 2 N. In particular, limk!+1xi(k) exists, denote by xi . We bu bu claim that xi 6 ali . By way of contradiction, assume that xi > ali . Taking limit in the ith equation in system (1.1) ii ii gives ! n m X X lim bi ðkÞ ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ ¼ 0; k!þ1
l¼1
l¼1
which is a contradiction since n m X X ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ 6 bi ðkÞ aii ðkÞxi ðkÞ 6 bui aliixi < 0 bi ðkÞ l¼1
l¼1 bu
for enough large k 2 N. This proves the claim. Note that ali 6 pi . It follows that (2.1) holds. This completes the ii proof of Proposition 1. h Proposition 2. Assume that n X rlj þ d ujl pl > 0
ðH1 Þ
l¼1
holds, where pl, l = 1, 2, . . . , n are defined by (2.1). Then for every solution (x1(k), . . . , xn(k), y1(k), . . . ,ym(k)) of (1.1), we have lim sup y j ðkÞ 6 qj ;
ð2:3Þ
k!þ1
where qj ¼ e1l expfrlj þ
Pn
u l¼1 d jl pl
jj
1g.
Proof. For any e > 0, according to Proposition 1, there exists a k1 2 N such that xi ðkÞ 6 pi þ e for all k P k 1 . Thus, by using n + jth equation in system (1.1), for k P k1, one has " # n m X X y j ðk þ 1Þ 6 y j ðkÞ exp rj ðkÞ þ d jl ðkÞðpl þ eÞ ejl ðkÞy l ðkÞ . l¼1
ð2:4Þ
l¼1
To prove (2.3), we first assume that there exists an l0 P k1 such that yj(l0 + 1) P yj(l0), then it follows from (2.4) that n m X X d jl ðl0 Þðpl þ eÞ ejl ðl0 Þy l ðl0 Þ P 0. rj ðl0 Þ þ l¼1
l¼1
And so, y j ðl0 Þ 6
rj ðl0 Þ þ
Pn
l¼1 d jl ðl0 Þðp l
þ eÞ
ejj ðl0 Þ
By using (2.5), one has
"
y j ðl0 þ 1Þ 6 y j ðl0 Þ exp rj ðl0 Þ þ rlj
þ
n X l¼1
Pn
u l¼1 d jl ðp l eljj
d jl ðl0 Þðpl þ eÞ
l¼1
" 6 y j ðl0 Þ exp
n X
6
rlj þ
d ujl ðpl
m X l¼1
#
þ eÞ
eljj y j ðl0 Þ
þ eÞ
ð2:5Þ
. #
ejl ðl0 Þy l ðl0 Þ
" # n X 1 u l 6 l exp rj þ d jl ðpl þ eÞ 1 . ejj l¼1
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
Let
7
" # n X 1 qje ¼ l exp rlj þ d ujl ðpl þ eÞ 1 . ejj l¼1
We claim that y j ðkÞ 6 qje
for k P l0 .
By way of contradiction, assume that there exists a q1 > l0 such that yj(q1) > qje. Then q1 P l0 + 2. Let ~ q1 P l0 þ 2 be the smallest integer such that y j ð~ q1 Þ > qje . Then y j ð~q1 1Þ < y j ð~q1 Þ. The above argument produces that y j ð~ q1 Þ 6 qje , a contradiction. This proves the claim. Now we assume that yj(k + 1) < yj(k) for all k P k1. In particular, limk!+1yj(k) exists, denote by y j . We claim that Pn rlj þ l¼1 d ujl ðpl þ eÞ y j 6 . eljj By way of contradiction, assume that Pn rlj þ l¼1 d ujl ðpl þ eÞ y j > . eljj Taking limit in the n + jth equation in system (1.1) gives ! n m X X d jl ðkÞxl ðkÞ ejl ðkÞy l ðkÞ ¼ 0; lim rj ðkÞ þ k!þ1
l¼1
l¼1
which is a contradiction since for k P k1, n m n X X X d jl ðkÞxl ðkÞ ejl ðkÞy l ðkÞ 6 rj ðkÞ þ d jl ðkÞxl ðkÞ ejj ðkÞy j ðkÞ rj ðkÞ þ l¼1
l¼1
l¼1
6 rlj þ Pn u l
rj þ
This proves the claim. Noting the fact that This completes the proof of Proposition 2.
h
n X
l¼1 d ðpl þeÞ
l¼1 jl eljj
d ujl ðpl þ eÞ eljj y j < 0.
6 qje and lime!0qje = q j. It follows that (2.3) holds.
Proposition 3. In addition to (H1), assume further that n m X X auil pl cuil ql > 0 bli l¼1;l6¼i
ðH2 Þ
l¼1
hold for all i = 1, 2, . . . , n, where pl, ql are defined by (2.1) and (2.3), respectively. Then for every solution (x1(k), . . . , xn(k),y1(k), . . . , ym(k)) of (1.1), we have lim inf xi ðkÞ P ai ; k!þ1 Pn Pm u bli au p c q Pn Pm l¼1;l6¼i il l l¼1 il l where ai ¼ expfbli l¼1 auil pl l¼1 cuil ql g. au
ð2:6Þ
ii
Proof. Condition (H2) implies that for enough small positive number e > 0, one has n m X X auil ðpl þ eÞ cuil ðql þ eÞ > 0. bli l¼1;l6¼i
ð2:7Þ
l¼1
For above e, from Propositions 1 and 2 we know that there exists enough large k2 such that xi ðkÞ 6 pi þ e;
y j ðkÞ 6 qj þ e
for all k P k 2 .
ð2:8Þ
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
Thus, by using ith equation in system (1.1), for k P k2, one has " # n m X X xi ðk þ 1Þ P xi ðkÞ exp bi ðkÞ ail ðkÞðpl þ eÞ cil ðkÞðql þ eÞ aii ðkÞxi ðkÞ . l¼1;l6¼i
ð2:9Þ
l¼1
To prove (2.6), we first assume that there exists an l0 P k2 such that xi(l0 + 1) 6 xi(l0). Then, from (2.9) it follows that n X
bi ðl0 Þ
ail ðl0 Þðpl þ eÞ
l¼1;l6¼i
m X
cil ðl0 Þðql þ eÞ aii ðl0 Þxi ðl0 Þ 6 0;
l¼1
which implies that P þ eÞ ml¼1 cil ðl0 Þðql þ eÞ aii ðl0 Þ Pn Pm l u bi l¼1;l6¼i ail ðpl þ eÞ l¼1 cuil ðql þ eÞ P : auii
xi ðl0 Þ P
bi ðl0 Þ
Pn
l¼1;l6¼i ail ðl0 Þðp l
It follows from (2.10) that
"
xi ðl0 þ 1Þ ¼ xi ðl0 Þ exp bi ðl0 Þ
ail ðl0 Þxl ðl0 Þ
l¼1
" P xi ðl0 Þ exp
n X
bli
n X
auil ðpl
m X
# cil ðl0 Þy l ðl0 Þ
l¼1
þ eÞ
l¼1
m X
ð2:10Þ
cuil ðql
# þ eÞ
l¼1
" # n m X X Ai def l u u P u exp bi ail ðpl þ eÞ cil ðql þ eÞ ¼ aie ; aii l¼1 l¼1 where n X
Ai ¼ bli
auil ðpl þ eÞ
l¼1;l6¼i
m X
cuil ðql þ eÞ.
l¼1
We claim that xi ðkÞ P aie
for k P l0 .
By way of contradiction, assume that there exists a q2 > l0 such that xi(q2) < aie. Then q2 P l0 + 2. Let ~ q2 P l0 þ 2 be the smallest integer such that xi ð~ q2 Þ < aie . Then xi ð~q2 1Þ > xi ð~q2 Þ. The above argument produces that xi ð~ q2 Þ P aie , a contradiction. This proves the claim. Now we assume that xi(k + 1) > xi(k) for all k P k2. In particular, limk!+1xi(k) exists, denote by xi. We claim that Pn Pm bli l¼1;l6¼i auil ðpl þ eÞ l¼1 cuil ðql þ eÞ xi P . auii By way of contradiction, assume that Pn Pm bli l¼1;l6¼i auil ðpl þ eÞ l¼1 cuil ðql þ eÞ xi < . auii Taking limit in the ith equation in system (1.1) gives ! n m X X lim bi ðkÞ ail ðkÞxl ðkÞ cil ðkÞy l ðkÞ ¼ 0; k!þ1
l¼1
l¼1
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
which is a contradiction since lim inf k!þ1
bi ðkÞ
n X
ail ðkÞxl ðkÞ
l¼1
m X
! cil ðkÞy l ðkÞ
n X
P bli
auil ðpl þ eÞ
l¼1;l6¼i
l¼1
m X
9
cuil ðql þ eÞ auii xi > 0.
l¼1
bu
This proves the claim. Note that ali 6 pi implies that Pn Pm ii bli l¼1;l6¼i auil ðpl þ eÞ l¼1 cuil ðql þ eÞ P aie ; auii also noting the fact lim aie ¼ ai . e!0
It follows that (2.6) holds. This completes the proof of Proposition 3.
h
Proposition 4. In addition to (H1) and (H2), assume further that n n X X d ljl al eujl ql > 0 ruj þ
ðH3 Þ
l¼1;l6¼j
l¼1
holds, where pl, ql, al are defined by (2.1), (2.3) and (2.6), respectively. Then for every solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) of (1.1), we have lim inf y j ðkÞ P bj ;
ð2:11Þ
k!þ1
where bj ¼
ruj þ
Pn
l l¼1 d jl al eujj
Pn
u l¼1;l6¼j ejl ql
exp ruj þ
n X
d ljl al
l¼1
n X
! eujl ql .
l¼1
Proof. From (H3) one could take e > 0 enough small, such that ruj þ
n X
d ljl ðal eÞ
l¼1
n X
eujl ðql þ eÞ > 0.
l¼1;l6¼j
For above e > 0, according to Propositions 1–3, there exists a k3 P k2 such that for all k P k3, xi ðkÞ 6 pi þ e;
y j ðkÞ 6 qj þ e;
xi ðkÞ P ai e.
Thus, by using n + jth equation in system (1.1), for k P k3, one has " # n m X X y j ðk þ 1Þ P y j ðkÞ exp rj ðkÞ þ d jl ðkÞðai eÞ ejl ðkÞðqj þ eÞ ejj ðkÞy j ðkÞ . l¼1
ð2:12Þ
l¼1;l6¼j
By applying (2.12), the rest of the proof of Proposition 4 is similar to that of the proof of Proposition 3 and we omit the detail here. h Now we state the main result of this section.
Theorem 1. Assume that (H1)–(H3) hold, then system (1.1) is permanent. It should be noticed that, from the proofs of Propositions 1–4, we know that under the assumption of Theorem 1, the set ½a1 ; p1 ½an ; pn ½b1 ; q1 ½bm ; qm is an invariant set of system (1.1).
10
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
3. Global stability Now we study the stability property of the positive solution of system (1.1). Theorem 2. Assume that (H1)–(H3) hold. Assume further that n m X X auil pl þ cuil ql < 1; ki ¼ maxfj1 auii pi j; j1 alii ai jg þ l¼1;l6¼i n X
m X
l¼1
l¼1;l6¼j
l¼1
d ujl pl þ
dj ¼ maxfj1 eujj qi j; j1 eljj bj jg þ
ð3:1Þ eujl ql < 1.
Then for any two positive solution (x1(k), . . . , xn(k), y1(k), . . . , ym(k)) and ð~x1 ðkÞ; . . . ; ~xn ðkÞ; ~y 1 ðkÞ; . . . ; ~y m ðkÞÞ of system (1.1), we have lim ð~xi ðkÞ xi ðkÞÞ ¼ 0;
k!þ1
lim ð~y j ðkÞ y j ðkÞÞ ¼ 0.
ð3:2Þ
k!þ1
Proof. Let xi ðkÞ ¼ ~xi ðkÞ expðui ðkÞÞ;
y j ðkÞ ¼ ~y j ðkÞ expðvj ðkÞÞ.
ð3:3Þ
Then system (1.1) is equivalent to n m X X ail ðkÞ~xl ðkÞðexpðul ðkÞÞ 1ÞÞ cil ðkÞ~y l ðkÞðexpðvl ðkÞÞ 1ÞÞ; ui ðk þ 1Þ ¼ ui ðkÞ vj ðk þ 1Þ ¼ vj ðkÞ
l¼1
l¼1
n X
m X
d jl ðkÞ~xl ðkÞðexpðul ðkÞÞ 1ÞÞ
l¼1
ð3:4Þ eil ðkÞ~y l ðkÞðexpðvl ðkÞÞ 1ÞÞ.
l¼1
And so, n X
ui ðk þ 1Þ ¼ ð1 aii ðkÞ~xi ðkÞ expðhi ðkÞui ðkÞÞÞui ðkÞ
m X
cil ðkÞ~y l ðkÞ expðnl ðkÞvl ðkÞÞvl ðkÞ;
l¼1
vj ðk þ 1Þ ¼ ð1 ejj ðkÞ~y j ðkÞ expðnj ðkÞvj ðkÞÞÞvj ðkÞ
m X
ail ðkÞ~xl ðkÞ expðhl ðkÞul ðkÞÞul ðkÞ
l¼1;l6¼i
n X
ð3:5Þ d jl ðkÞ~xl ðkÞ expðhl ðkÞul ðkÞÞul ðkÞ
l¼1
eil ðkÞ~y l ðkÞ expðnl ðkÞvl ðkÞÞÞvl ðkÞ;
l¼1
where hl(k), nl(k) 2 [0, 1]. To complete the proof, it suffices to show that lim ui ðkÞ ¼ 0;
k!þ1
lim vj ðkÞ ¼ 0.
ð3:6Þ
k!þ1
In view of (3.1), we can choose e > 0 small enough such that n m X X kei ¼ maxfj1 auii ðpi þ eÞj; j1 alii ðai eÞjg þ auil ðpl þ eÞ þ cuil ðql þ eÞ < 1; dej
¼ maxfj1
eujj ðqi
þ eÞj; j1
eljj ðbj
l¼1;l6¼i n X
d ujl ðpl
eÞjg þ
l¼1
l¼1
þ eÞ þ
m X
ð3:7Þ eujl ðqþ eÞ
< 1.
l¼1;l6¼j
For above e > 0, according to Propositions 1–4 in Section 2, there exists a k* 2 N such that ai e 6 ~xi ðkÞ; bj e 6 ~y j ðkÞ; *
for all k P k .
xi ðkÞ 6 pi þ e; y j ðkÞ 6 qj þ e
F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
11
Noticing that hl(k), nl(k) 2 [0, 1] implies that ~xl ðkÞ expðhl ðkÞul ðkÞÞ lies between ~xl ðkÞ and xl(k), ~y l ðkÞ exp ðnl ðkÞvl ðkÞÞÞ lies between ~y l ðkÞ and yl(k). From (3.5), we get n m X X jui ðk þ 1Þj ¼ maxfj1 auii ðpi þ eÞj; j1 alii ðai eÞjgjui ðkÞj þ auil ðpl þ eÞjul ðkÞj þ cuil ðql þ eÞjvl ðkÞj; l¼1;l6¼i
jvj ðk þ 1Þj ¼ maxfj1 eujj ðqi þ eÞj; j1 eljj ðbj eÞjgjvj ðkÞj þ
n X
d ujl ðpl þ eÞjul ðkÞj þ
l¼1
l¼1 m X
euil ðql þ eÞjvl ðkÞj.
l¼1
ð3:8Þ Let c ¼
maxfkei ; dej g,
*
then c < 1. In view of (3.8), for k P k , we get
maxfjui ðk þ 1Þj; jvj ðk þ 1Þjg 6 c maxfjvj ðkÞj; jui ðkÞjg. i;j
This implies
maxfjui ðkÞj; jvj ðkÞjg 6 ckk maxfjvj ðk Þj; jui ðk Þjg. i;j
Therefore, (3.6) holds and the proof is complete.
h
4. Existence and stability of periodic solution In this section, we further assume that the coefficients of system (1.1) satisfies (H4). There exists a positive integer x such that for k 2 N, 0 < bi ðk þ xÞ ¼ bi ðkÞ; 0 < ail ðk þ xÞ ¼ ail ðkÞ; 0 < cil ðk þ xÞ ¼ cil ðkÞ; 0 < rj ðk þ xÞ ¼ rj ðkÞ; 0 < d jl ðk þ xÞ ¼ d jl ðkÞ; 0 < ejl ðk þ xÞ ¼ ejl .
ðH4 Þ
Our first result concerned with the existence of a positive periodic solution of system (1.1). Theorem 3. Assume that (H1)–(H4) hold, then system (1.1) admits at least one positive x-periodic solution which we denote by ð~x1 ðkÞ; . . . ; ~xn ðkÞ; ~y 1 ðkÞ; . . . ; ~y m ðkÞÞ. Proof. As noted at the end of Section 2, def
Dnþm ¼ ½a1 ; p1 ½an ; pn ½b1 ; q1 ½bm ; qm is an invariant set of system (1.1). Thus, we can define a mapping F on Dn+m by F ðx1 ð0Þ; . . . ; xn ð0Þ; y 1 ð0Þ; . . . ; y m ð0ÞÞ ¼ ðx1 ðxÞ; . . . ; xn ðxÞ; y 1 ðxÞ; . . . ; y m ðxÞÞ; for (x1(0), . . . , xn(0), y1(0), . . . , ym(0)) 2 Dn+m. Obviously, F depends continuously on (x1(0), . . . , xn(0), y1(0), . . . , ym(0)). Thus, F is continuous and maps the compact set Dn+m into itself. Therefore, F has a fixed point. It is easy to see that the solution ð~x1 ðkÞ; . . . ; ~xn ðkÞ; ~y 1 ðkÞ; . . . ; ~y m ðkÞÞ passing through this fixed point is an xperiodic solution of the system (1.1). This complete the proof of Theorem 3. h Theorem 4. Assume that (H1)–(H4) and (3.1) hold, then system (1.1) has a global stable positive x-periodic solution. Proof. Under the assumption of Theorem 4, it follows from Theorem 3 that system (1.1) admits at least one positive x-periodic solution. Also, Theorem 2 ensures the positive solution to be globally stable. This end the proof of Theorem 4. h Acknowledgements This work is supported by the National Natural Science Foundation of China (10501007), the Foundation of Science and Technology of Fujian Province for Young Scholars (2004J0002), the Foundation of Fujian Education Bureau (JA04156).
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F. Chen / Applied Mathematics and Computation 182 (2006) 3–12
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