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Discrete Dynamics in Nature and Society
Volume 2011, Article ID 835321, 11 pages
doi:10.1155/2011/835321
Research Article
Global Attractivity and Periodic Solution
of a Discrete Multispecies Cooperation and
Competition Predator-Prey System
Zheyan Zhou
School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China
Correspondence should be addressed to Zheyan Zhou, zheyanzhou@126.com
Received 20 April 2011; Accepted 8 June 2011
Academic Editor: Ugurhan Mugan
Copyright q 2011 Zheyan Zhou. 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.
We propose a discrete multispecies cooperation and 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.
1. Introduction
In this paper, we consider the dynamic behavior of the following non-autonomous discrete
n m-species cooperation and competition predator-prey systems
xi k
− ci kxi k
xi k 1 xi k exp r1i k 1 −
n
ai k l1,l / i bil kxl k
⎤
−
m
⎥
dil kyl k⎥
⎦,
1.1
l1
yj k 1 yj k exp r2j k n
m
ejl kxl k − pjl kyl k ,
l1
l1
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.
2
Discrete Dynamics in Nature and Society
Dynamic behaviors of population models governed by difference equations had been
studied by a number of papers, see 1–4 and the references cited therein. It has been found
that the autonomous discrete systems can demonstrate quite rich and complicated dynamics,
see 5, 6 . Recently, more and more scholars paid attention to the non-autonomous discrete
population models, since such kind of model could be more appropriate.
May 7 suggested the following set of equations to describe a pair of mutualists:
u̇ r1 u 1 −
u
− c1 u ,
a1 b1 v
1.2
v
− c2 v ,
v̇ r2 v 1 −
a2 b2 u
where u, v are the densities of the species U, V at time t, respectively. ri , ai , bi , ci , i 1, 2
are positive constants. He showed that system 1.2 has a globally asymptotically stable
equilibrium point in the region u > 0, v > 0.
Bai et al. 8 argued that the discrete case of cooperative system is more appropriate,
and they proposed the following system:
x1 k 1 x1 k exp r1 k 1 −
x1 k
− c1 kx1 k ,
a1 k b1 kx2 k
x2 k
x2 k 1 x2 k exp r2 k 1 −
− c2 kx1 k .
a2 k b2 kx2 k
1.3
Chen 9 investigated the dynamic behavior of the following discrete n m-species
Lotka-Volterra competition predator-prey systems
xi k 1 xi k exp bi k −
yj k 1 yj k exp rj k n
ail kxl k −
m
cil kyl k ,
l1
l1
n
m
djl kxl k −
l1
1.4
ejl kyl k ,
l1
he investigated the dynamic behavior of the system 1.4.
The aim of this paper is, by further developing the analysis technique of Huo and Li
10 and Chen 9 , to obtain a set of sufficient condition which ensure the permanence and
the global stability of the system 1.1; for periodic case, sufficient conditions which ensure
the existence of a globally stable positive periodic solution of the system 1.1 are obtained.
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 ≤ lim inf xi k ≤ lim sup xi k ≤ M,
k → ∞
k → ∞
m ≤ lim inf yi k ≤ lim sup yi k ≤ M,
k → ∞
for all i 1, 2, . . . , n; j 1, 2, . . . , m.
k → ∞
1.5
Discrete Dynamics in Nature and Society
3
Throughout this paper, we assume that r1i k, bil k, ai k, ci k, r2j k, dil k, ejl k,
pjl k are all bounded nonnegative sequence, and use the following notations for any
bounded sequence {ak}
au supak,
al inf ak.
k∈N
k∈N
1.6
For biological reasons, we only consider solution x1 k, . . . , xn k, y1 k, . . . , ym k
with
xi 0 > 0;
i 1, 2, . . . , n,
yj 0 > 0,
j 1, 2, ..., m.
1.7
Then system 1.1 has a positive solution x1 k, . . . , xn k, y1 k, . . . , ym k∞
k0
passing through x1 0, . . . , xn 0, y1 0, . . . , ym 0.
The organization of this paper is as follows. 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 permanence results for system 1.1.
Lemma 2.1 see 11 . Let k ∈ Nk0 {k0 , k0 1, . . . , k0 l, . . .}, r ≥ 0. For any fixed k, gk, r is a
non-decreasing function with respect to r, and for k ≥ k0 , the following inequalities hold:
yk 1 ≤ g k, yk ,
uk 1 ≥ gk, uk.
2.1
If yk0 ≤ uk0 , then yk ≤ uk for all k ≥ k0 .
Now let one consider the following single species discrete model:
Nk 1 Nk exp{ak − bkNk},
2.2
where {ak} and {bk} are strictly positive sequences of real numbers defined for k ∈ N {0, 1, 2, . . .} and 0 < al ≤ au , 0 < bl ≤ bu .
Lemma 2.2 see 12 . Any solution of system 2.5 with initial condition N0 > 0 satisfies
m ≤ lim inf Nk ≤ lim sup Nk ≤ M,
k → ∞
k → ∞
2.3
4
Discrete Dynamics in Nature and Society
where
M
1
exp{au − 1},
bl
m
al
l
u
exp
a
−
b
M
.
bu
2.4
Proposition 2.3. Assume that
u
−r2j
n
ejll Ml > 0,
j 1, 2, . . . , m
2.5
l1
holds, then
lim sup xi k ≤ Mi ,
i 1, 2, . . . , n,
lim sup yi k ≤ Qi ,
i 1, 2, ..., m,
k → ∞
k → ∞
2.6
where
1
exp r1iu − 1 ,
cil r1il
n
1
u
l
eil Ml − r2i − 1 .
Qi l exp
pii
l1
Mi 2.7
Proof. Let uk x1 k, . . . , xn k, y1 k, . . . , ym k be any positive solution of system 1.1,
from the ith equation of 1.1 we have
xi k 1 ≤ xi k exp{r1i k1 − ci kxi k }
2.8
xi k exp{r1i k − r1i kci kxi k}.
By applying Lemmas 2.1 and 2.2, it immediately follows that
lim sup xi k ≤
k → ∞
1
cil r1il
exp r1iu − 1 : Mi .
2.9
For any positive constant ε small enough, it follows from 2.9 that there exists enough large
K0 such that
xi k ≤ Mi ε,
i 1, 2, . . . , n,
∀k ≥ K0 .
2.10
From the n jth equation of the system 1.1 and 2.10, we can obtain
yj k 1 ≤ yj k exp −r2j k n
l1
ejl kMl ε − pjj kyj k .
2.11
Discrete Dynamics in Nature and Society
5
Condition 2.5 shows that Lemmas 2.1 and 2.2 could be applied to 2.11, and so, by applying
Lemmas 2.1 and 2.2, it immediately follows that
1
lim sup yj k ≤
l
pjj
k → ∞
n
u
l
exp
ejl Ml ε − r2j − 1 ,
1 ≤ j ≤ m.
2.12
1 ≤ j ≤ m.
2.13
l1
Setting ε → 0 in the above inequality leads to
n
u
l
exp
ejl Ml − r2j − 1 : Qj ,
1
lim sup yj k ≤
l
pjj
k → ∞
l1
This completes the proof of Proposition 2.3.
Now we are in the position of stating the permanence of the system 1.1.
Theorem 2.4. In addition to 2.5, assume further that
r1il −
m
dilu Ql > 0,
i 1, 2, . . . , n,
l1
n
m
u
−r2j
ejll ml −
pjlu Ql > 0,
l1
2.14
j 1, 2, . . . m,
l1,l /j
then system 1.1 is permanent, where
u
m
r1il − m
1
l1 dil Ql
l
u
u
u
mi ci Mi ,
exp r1i − dil Ql − r1i
ali
r1iu 1/ali ciu
l1
n
1
u
l
Qi l exp
eil Ml − r2i − 1 .
pii
l1
2.15
Proof. By applying Proposition 2.3, we see that to end the proof of Theorem 2.4, it is enough
to show that under the conditions of Theorem 2.4,
lim inf xj k ≥ mi ,
1 ≤ i ≤ n,
lim inf yj k ≥ qj ,
1 ≤ j ≤ m.
k → ∞
k → ∞
2.16
From Proposition 2.3, ∀ε > 0, there exists a K1 > 0, K1 ∈ N, ∀k > K1 ,
xi k ≤ Mi ε,
i 1, 2, . . . , n,
yj k ≤ Qj ε,
j 1, 2, . . . , m.
2.17
6
Discrete Dynamics in Nature and Society
From the ith equation of system 1.1 and 2.17, we have
xi k 1 ≥ xi k exp r1i k − r1i k
1
ali
ciu
m
xi k − dil kQl ε
l1
m
xi k exp r1i k − dil kQl ε − r1i k
l1
1
ali
2.18
ciu xi k ,
for all k > K1 .
By applying Lemmas 2.1 and 2.2 to 2.18, it immediately follows that
r1il −
lim inf xi k ≥
k → ∞
m
r1iu
dilu Ql ε
1/ali ciu
l1
m
× exp r1il − dilu Ql ε − r1iu
l1
1
ali
2.19
ciu Mi .
Setting ε → 0 in 2.19 leads to
u
r1il − m
l1 dil Ql
lim inf xi k ≥
k → ∞
r1iu 1/ali ciu
m
× exp r1il − dilu Ql − r1iu
l1
1
ali
ciu Mi
2.20
: mi .
Then, for any positive constant ε small enough, from 2.20 we know that there exists
an enough large K2 > K1 such that
xi k ≥ mi − ε, ∀ k ≥ k2 .
2.21
Equations 2.17, 2.21 combining with the n jth equation of the system 1.1 leads to
yj k 1 ≥ yj k exp −r2j k n
ejl kml − ε
l1
−
m
l1, l /
j
⎫
⎬
pjl kQl ε − pjj kyj k ,
⎭
2.22
Discrete Dynamics in Nature and Society
7
under the condition 2.14, by applying Lemmas 2.1 and 2.2 to 2.22, it immediately follows
that
u
−r2j
lim inf yj k ≥
n
l1
ejll ml − ε −
m
l1,l /
j
pjlu Ql ε
u
pjj
k → ∞
⎧
⎫
n
m
⎨
⎬
u
u
× exp −r2j
ejll ml − ε −
pjlu Ql ε − pjj
Qj .
⎩
⎭
l1
l1, l j
2.23
/
Setting ε → 0 in 2.23 leads to
lim inf yj k ≥
u
−r2j
n
l1
ejll ml −
m
l1,l /
j
pjlu Ql
u
pjj
k → ∞
⎧
⎨
u
× exp −r2j
⎩
n
ejll ml −
l1
m
l1, l /j
u
pjlu Ql − pjj
Qj
2.24
⎫
⎬
⎭
: qj .
This ends the proof of Theorem 2.4.
It should be noticed that, under the assumption of Theorem 2.4, the set
!
"
!
"
m1 , M1 × · · · mn , Mn × q1 , Q1 × · · · qm , Qm
2.25
is an invariant set of system 1.1.
3. Global Stability
Now we study the stability property of the positive solution of system 1.1.
Theorem 3.1. Assume that
# #
#
#
n
# #
#
#
r u Mi 1
1
# #
#
#
u
u
l
l
λi max #1 − ci l r1i Mi #, #1 − ci u r1i mi # 1i 2
bilu Ml
# #
#
#
ai
ai
ali l1, l / i
m
3.1
dilu Ql < 1,
l1
n
n
# #
# #
# #
#
#
u
l
δj max #1 − pjj
Qj #, #1 − pjj
qj # ejlu Ml pjlu Ql < 1.
l1
l1
Then for any two positive solution x1 k, . . . , xn k, y1 k, . . . , ym k and x$1 k, . . . , x$n k,
y$1 k, . . . , y$m k of system 1.1, one has
lim x$i k − xi k 0,
k → ∞
lim y$j k − yj k 0.
k → ∞
3.2
8
Discrete Dynamics in Nature and Society
Proof. Let
xi k x$i k exp{ui k},
yj k y$j k exp vj k .
3.3
Then system 1.1 is equivalent to
ui k 1 ui k ai k r1i kx$i k exp{ui k}
r1i kx$i k
−
n
n
b
a
x
$
$l k exp{ul k}
k
k
k
l
i
l1,l /
i il
l1,l /
i bil kx
m
− r1i kci kx$i k exp{ui k} − 1 − dil ky$l k exp{vl k} − 1 ,
l1
vj k 1 vj k −
n
ejl kx$l k exp{ul k} − 1
3.4
l1
−
m
pjl ky$l k exp{vl k} − 1 .
l1
So,
#
#
|ui k 1| ≤ ##1 − ci k #
#
1
r1i kx$i k exp{θi kui k}##|ui k|
ai k
n
r1i kx$i k bil kx$l k exp{θl kul k}|ul k|
a2i k l1, l / i
m
dil ky$l k exp{ξl kvl k}|vl k|
l1
#
#
##
|vi k 1| ≤ #1 − pjj ky$j k exp ξj kvj k ##vj k#
3.5
n
ejl kx$l k exp{θl kul k}|ul k|
l1
m
pil ky$l k exp{ξl kvl k}|vl k|,
l1
where θl k, ξl k ∈ 0, 1 , to complete the proof, it suffices to show that
lim ui k 0,
k → ∞
lim vj k 0.
k → ∞
3.6
Discrete Dynamics in Nature and Society
9
In view of 3.1, we can choose ε > 0 small enough such that
λεi
# #
#
#
#
# #
#
1
1
#
# #
#
u
u
l
l
max #1 − ci l r1i Mi ε#, #1 − ci u r1i mi − ε#
#
#
#
#
a
ai
i
n
m
r1iu Mi ε bilu Ml ε dilu Ql ε < 1,
2
l1,l /
i
l1
ali
n
#
## ##
## #
u
l
δjε max #1 − pjj
Qj ε #, #1 − pjj
qj − ε # ejlu Ml ε
3.7
l1
n
pjlu Ql ε < 1.
l1
For the above ε > 0, according to Theorem 2.4 in Section 2, there exists a k ∗ ∈ N such that
mi − ε ≤ xi k,
x$i k ≤ Mi ε,
qj − ε ≤ yj k,
y$j k ≤ Qj ε,
3.8
for all k ≥ k∗ .
Noticing that θl k, ξl k ∈ 0, 1 implies that x$l k exp{θl kul k} lies between x$l k
and xl k, y$l k exp{ξl kvl k} lies between y$l k and yl k. From 3.5, we get
# #
#
#
# #
#
#
1
1
# #
#
#
u
u
l
l
|ui k 1| ≤ max #1 − ci l r1i Mi ε#, #1 − ci u r1i mi − ε# |ui k|
# #
#
#
ai
ai
n
m
r1iu Mi ε u
b
ε|u
dilu Ql ε|vl k|,
M
k|
l
l
2
il
l
l1,l /
i
l1
ai
#
#
#
#
# #
##
#vj k 1# ≤ max ##1 − pu Qj ε ##, ##1 − pl qj − ε ## #vj k#
jj
jj
n
l1
ejlu Ml ε|ul k| n
3.9
pjlu Ql ε|vl k|.
l1
Let γ max{λεi , δjε }, then γ < 1. In view of 3.9, for k ≥ k∗ , we get
#
#
#
#
max |ui k 1|, #vj k 1# ≤ γmax |ui k|, #vj k# .
3.10
#
#
#
#
∗
max |ui k|, #vj k# ≤ γ k−k max |ui k∗ |, #vj k∗ # .
3.11
This implies
10
Discrete Dynamics in Nature and Society
From 3.11, we have
lim ui k 0,
k → ∞
lim vj k 0.
k → ∞
3.12
This ends the proof of Theorem 3.1.
4. Existence and Stability of Periodic Solution
In this section, we further assume that the coefficients of the system 1.1 satisfies 4.1.
There exists a positive integer ω such that for k ∈ N,
0 < r1i k ω r1i k,
0 < ai k ω ai k,
0 < bil k ω b1i k,
0 < ci k ω ci k,
0 < r2j k ω r2j k,
0 < dil k ω dil k
0 < ejl k ω ejl k,
0 < pjl k ω pjl k.
4.1
Our first result concerned with the existence of a positive periodic solution of system
1.1.
Theorem 4.1. Assume that 2.5 and 2.14 hold, then system 1.1 admits at least one positive
ω-periodic solution which ones denotes by x$1 k, . . . , x$n k, y$1 k, . . . , y$m k.
Proof. As noted at the end of Section 2,
"
!
"
!
Dnm : m1 , M1 × · · · mn , Mn × q1 , Q1 × · · · qm , Qm
4.2
is an invariant set of system 1.1. Thus, we can define a mapping F on Dnm by
F x1 0, . . . , xn 0, y1 0, . . . , ym 0 x1 ω, . . . , xn ω, y1 ω, . . . , ym ω ,
4.3
for x1 0, . . . , xn 0, y1 0, . . . , ym 0 ∈ Dnm . Obviously, F depends continuously on
x1 0, . . . , xn 0, y1 0, . . . , ym 0. Thus, F is continuous and maps the compact set
Dnm into itself. Therefore, F has a fixed point. It is easy to see that the solution
x$1 k, . . . , x$n k, y$1 k, . . . , y$m k passing through this fixed point is an ω-periodic solution
of the system 1.1. This completes the proof of Theorem 4.1.
Theorem 4.2. Assume that 2.5, 2.14, and 3.1 hold, then system 1.1 has a global stable positive
ω-periodic solution.
Proof. Under the assumption of Theorem 4.2, it follows from Theorem 4.1 that system 1.1
admits at least one positive ω-periodic solution. Also, Theorem 3.1 ensures the positive
solution to be globally stable. This ends the proof of Theorem 4.2.
Discrete Dynamics in Nature and Society
11
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