Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2011, Article ID 783136, 14 pages
doi:10.1155/2011/783136
Research Article
Stability Analysis of Three-Species
Almost Periodic Competition Models with
Grazing Rates and Diffusions
Chang-you Wang,1, 2 Rui-fang Wang,2, 3 Ming Yi,4 and Rui Li2, 3
1
Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
2
Key Laboratory of Network control & Intelligent Instrument, Chongqing University of
Posts and Telecommunications, Ministry of Education, Chongqing 400065, China
3
Automation Institute, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
4
College of Computer Science and Technology, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China
Correspondence should be addressed to Chang-you Wang, wangcy@cqupt.edu.cn
Received 10 May 2011; Accepted 2 June 2011
Academic Editor: Zhengqiu Zhang
Copyright q 2011 Chang-you Wang et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Almost periodic solution of a three-species competition system with grazing rates and diffusions is
investigated. By using the method of upper and lower solutions and Schauder fixed point theorem
as well as Lyapunov stability theory, we give sufficient conditions to ensure the existence and
globally asymptotically stable for the strictly positive space homogenous almost periodic solution,
which extend and include corresponding results obtained by Q. C. Lin 1999, F. D. Chen and X. X.
Chen 2003, Y. Q. Liu, S. L, Xie, and Z. D. Xie 1996.
1. Introduction
In this paper, we study the following three-species competition system with grazing rates
and diffusions:
∂v1 x, t
k1 tΔv1 x, t v1 x, ta1 t − b1 tv1 x, t − c1 tv2 x, t − d1 tv3 x, t
f1 t,
∂t
∂v2 x, t
k2 tΔv2 x, t v2 x, ta2 t − b2 tv1 x, t − c2 tv2 x, t − d2 tv3 x, t
f2 t,
∂t
∂v3 x, t
k3 tΔv3 x, t v3 x, ta3 t − b3 tv1 x, t − c3 tv2 x, t − d3 tv3 x, t
f3 t,
∂t
1.1
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Discrete Dynamics in Nature and Society
where x, t ∈ Ω × R , Ω ⊆ Rn is the bounded open subset of Rn with smooth boundary
∂Ω, which represent the habitat domain for three species. System 1.1 is supplement with
boundary conditions and initial conditions:
∂vi x, t
1.2
0, i 1, 2, 3, x, t ∈ ∂Ω × R ,
∂n
1.3
≡ 0, i 1, 2, 3, x ∈ Ω,
vi x, 0 vi0 x ≥ 0, vi0 x /
where ∂/∂n denotes the outward normal derivation on ∂Ω, and vi x, t represent the
density of ith species at point x x1 , . . . , xn and the time of t. Here, ki t, ai t, bi t,
ci t, di t, and fi t i 1, 2, 3 denote the diffusivity rates, competition rates, and grazing
rates, respectively. They are almost periodic functions in real number field R. Δ is a Laplace
operator on Ω.
System 1.1–1.3 describes the interaction among three species and is an important
model in biomathcmatics, which has been intensively investigated, and much attention is
carried to the problem 1–8
. When there is no diffusion, Jiang 1
and Lin 2
studied the
existence, uniqueness, and stability on periodic solution and almost periodic solution for
two-species competition system under the condition that the coefficients are the periodic
function and almost periodic function, respectively; F. D. Chen and X. X. Chen 3
extended
the results in 2
to n-species case. When there are no diffusion and grazing rates, Zhang
and Wang 4, 5
investigated the existence of a positive periodic solution for a two-species
nonautonomous competition Lotka-Volterra patch system with time delay and the existence
of multiple positive periodic solutions for a generalized delayed population model with
exploited term by using the continuation theorem of coincidence degree theory; Hu and
Zhang 6
established criteria for the existence of at least four positive periodic solutions for
a discrete time-delayed predator-prey system with nonmonotonic functional response and
harvesting by employing the continuation theorem of coincidence degree theory. When there
are no grazing rates, Pao and Wang 7
proved the stability for invariable coefficient case
by utilizing the method of upper and lower solutions. Liu et al. 8
showed the stability
on the periodic solution for n-species competition system with grazing rates and diffusions.
Nevertheless, generally speaking, the system does not always change strictly according to
periodic laws, sometimes it changes according to almost periodic laws, and it is important to
survey almost periodic solution for the multispecies competition system with grazing rates
and diffusions. To sum up, we pay more attention to almost periodic solution of a threespecies competition system 1.1–1.3 with grazing rates and diffusions; in this paper, by
using the method of upper and lower solutions and Schauder fixed point theorem as well as
Lyapunov stability theory, we obtain sufficient conditions which ensure the existence and
globally asymptotically stable for the strictly positive space homogenous almost periodic
solution, which extend and include corresponding results obtained in 2, 3, 8
. Many other
results on the periodic solution and almost periodic solution can be found in 9–16
.
2. Preliminary
Firstly, we give out some definitions and lemmas.
Definition 2.1. Suppose that ft is a continuous function in R. Then ft is said to be almost
periodic in t ∈ R if for every ε > 0 corresponds T ε > 0 such that for any interval I whose
length is equal to T ε there is at least one τ ∈ I such that
ft τ − ft ≤ ε, ∀t ∈ R.
2.1
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Definition 2.2. If a smooth function V t v1 t, v2 t, v3 t satisfies 1.1 in R , and
every component of V t is the almost periodic function, we called that V t is a spatial
homogeneity almost periodic solution for 1.1, which is denoted by V t, T ε.
Definition 2.3. For any nonnegative smooth initial data
V x, 0 v1 x, 0, v2 x, 0, v3 x, 0 v10 x, v20 x, v30 x ≥ 0,
≡ 0, x ∈ Ω,
V x, 0 /
2.2
if there exists a unique positive solution V x, t v1 x, t, v2 x, t, v3 x, t for the system
1.1 with boundary conditions 1.2, and limt → ∞ Vi x, t − Vi t, T ε 0, i 1, 2, 3,
uniformly for x ∈ Ω, we called that spatial homogeneity almost periodic solution V t, T ε
is globally asymptotically stable.
Definition 2.4. Suppose that V x, t ≡ v1 x, t, v2 x, t, v3 x, t, V x, t ≡ v1 x, t, v2 x, t,
v3 x, t; if V x, t ≥ V x, t and
∂v1 x, t
≥ k1 tΔv1 x, t v1 x, t a1 t − b1 tv1 x, t − c1 tv2 x, t − d1 tv3 x, t
∂t
f1 t, x, t ∈ Ω × R ,
∂v2 x, t
≥ k2 tΔv2 x, t v2 x, t a2 t − b2 tv1 x, t − c2 tv2 x, t − d2 tv3 x, t
∂t
f2 t, x, t ∈ Ω × R ,
∂v3 x, t
≥ k3 tΔv3 x, t v3 x, t a3 t − b3 tv1 x, t − c3 tv2 x, t − d3 tv3 x, t
∂t
f3 t, x, t ∈ Ω × R ,
∂vi x, t
≥ 0, i 1, 2, 3, x, t ∈ ∂Ω × R ,
∂n
vi x, 0 ≥ vi0 x, i 1, 2, 3, x ∈ Ω,
∂v1 x, t
≤ k1 tΔv1 x, t v1 x, t a1 t − b1 tv1 x, t − c1 tv2 x, t − d1 tv3 x, t
∂t
f1 t, x, t ∈ Ω × R ,
∂v2 x, t
≤ k2 tΔv2 x, t v2 x, t a2 t − b2 tv1 x, t − c2 tv2 x, t − d2 tv3 x, t
∂t
f2 t, x, t ∈ Ω × R ,
∂v3 x, t
≤ k3 tΔv3 x, t v3 x, t a3 t − b3 tv1 x, t − c3 tv2 x, t − d3 tv3 x, t
∂t
f3 t, x, t ∈ Ω × R ,
∂vi x, t
≤ 0, i 1, 2, 3, x, t ∈ ∂Ω × R ,
∂n
vi x, 0 ≤ vi0 x, i 1, 2, 3, x ∈ Ω,
2.3
we called V x, t and V x, t a pair of ordered upper and lower solutions for systems 1.1–
1.3.
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Discrete Dynamics in Nature and Society
Lemma 2.5 see 12, 17
. Suppose that V x, t and V x, t are a pair of ordered upper and lower
solution for systems 1.1–1.3, then there exists a unique solution V x, t for systems 1.1–1.3.
Moreover, one has
2.4
V x, t ≥ V x, t ≥ V x, t.
For the almost periodic function Ft in R, one denotes F sup{Ft, t ∈ R}, F ∼
t
inf{Ft, t ∈ R}, and MF
limt−s → ∞ { s Fτdτ/t−s}. When Ft is T -periodic function,
T
one denotes MF
0 Fsds/T .
3. Main Results and Proofs
Now we are in a position to state our main results and give our proofs.
Theorem 3.1. If a , b , c , d , f are positive numbers, and
∼ i ∼ i ∼i
bi ci di
a
∼i
∼i
∼i
⎫
⎧
⎪
⎪
d
b ⎪
⎪
c
d
c
d
b
b
c
⎬
⎨
∼1
∼2
∼2
∼3
∼1
∼3
∼ 1 ∼2 ∼ 3
≤ L min
,
,
,
,
,
⎪
⎪
a1
a2
a3
⎪
⎪ f1
f2
f3
⎭
⎩
3.1
are satisfied for i 1, 2, 3, then there exists a strictly positive spatial homogeneity almost periodic
solution V t v1 t, v2 t, v3 t for 1.1.
Proof. By the conditions in Theorem 3.1, we have
0<
c1 d1
≤ 1,
La − b1
0<
b2 d2
≤ 1,
La − c2
0<
∼2
∼1
b3 c3
≤ 1.
La − d3
3.2
∼3
Let
⎧
⎫
⎪
⎪
⎨ c d
b2 d2 b3 c3 ⎬
1
1
.
,
,
m L · max
⎪
⎩ La − b1 La − c2 La − d3 ⎪
⎭
∼1
∼2
3.3
∼3
Then we have 0 < m ≤ L, and
L
c1 d1
≤ La − b1 ,
∼1
m
L
≤ La − c2 ,
b2 d2
∼2
m
L
≤ La − d3 .
b3 c3
∼3
m
3.4
Therefore
L
L
− f m2 ≤ La ,
− f m2 ≤ La ,
c2 b2 d2
b1 c1 d1
∼1
∼2
m ∼1
m ∼2
L
− f m2 ≤ La .
d3 b3 c3
∼3
m ∼3
3.5
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5
Furthermore, by the given conditions in Theorem 3.1, one has
b − f1 L2 ≥ 0,
∼1
c − f2 L2 ≥ 0,
∼2
b d
∼2
∼2
d − f3 L2 ≥ 0,
∼3
m
≥ m
a2 ,
L
b c
∼3
∼3
c d
∼1
∼1
m
≥ m
a1 ,
L
m
≥ m
a3 .
L
3.6
Thus
b c d
∼1
∼1
∼1
m 2
m 2
− f1 L ≥ m
− f2 L ≥ m
a1 ,
c b d
a2 ,
∼2
∼2 L
∼2
L
m 2
− f3 L ≥ m
d b c
a3 .
∼3
∼3
∼3 L
3.7
Combining 3.5 and 3.7, we have
m 2
− f1 L ≥ m
b c d
a1 ,
∼1
∼1 L
∼1
m 2
− f2 L ≥ m
c b d
a2 ,
∼2
∼2 L
∼2
m 2
− f3 L ≥ m
d b c
a3 .
∼3
∼3
∼3 L
L
− f m2 ≤ La ,
b1 c1 d1
∼1
m ∼1
L
− f m2 ≤ La ,
c2 b2 d2
∼2
m ∼2
L
− f m2 ≤ La ,
d3 b3 c3
∼3
m ∼3
3.8
Let
HLm φt, ϕt, γt : φ, ϕ, γ are almost periodic in t ∈ R, 0 < m ≤ φ, ϕ, γ ≤ L .
3.9
We consider the following system corresponding to 1.1:
v̇1 v1 a1 t − b1 tv1 − c1 tv2 − d1 tv3 f1 t,
t ∈ R ,
v̇2 v2 a2 t − b2 tv1 − c2 tv2 − d2 tv3 f2 t,
t ∈ R ,
v̇3 v3 a3 t − b3 tv1 − c3 tv2 − d3 tv3 f3 t,
t ∈ R .
3.10
Let zi 1/vi , i 1, 2, 3; then 3.10 becomes
ż1 b1 t − a1 tz1 c1 t
z1
z1
d1 t − f2 tz21 ,
z2
z3
ż2 c2 t − a2 tz2 b2 t
z2
z2
d2 t − f2 tz22 ,
z1
z3
ż3 d3 t − a3 tz3 b3 t
z3
z3
c3 t − f3 tz23 .
z1
z2
3.11
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For any φt, ϕt, γt ∈ HLm , by Mb1 > 0, Mc2 > 0, Md3 > 0, we observe 18
that
ż1 b1 t − a1 tz1 c1 t
φt
φt
d1 t
− f1 tφ2 t,
ϕt
γt
ż2 c2 t − a2 tz2 b2 t
ϕt
ϕt
d2 t
− f2 tϕ2 t,
φt
γt
ż3 d3 t − a3 tz3 b3 t
3.12
γt
γt
c3 t
− f3 tγ 2 t,
φt
ϕt
have almost periodic solution:
t
φs
φs
2
d1 s
− f1 sφ s ds,
z1 t e
b1 s c1 s
ϕs
γs
−∞
t
t
ϕs
ϕs
d2 s
− f2 sϕ2 s ds,
e− s a2 rdr c2 s b2 s
z2 t φs
γs
−∞
t
t
γs
γs
− s a2 rdr
2
c3 s
− f3 sγ s ds.
e
d3 s b3 s
z3 t φs
ϕs
−∞
−
t
s
a1 rdr
3.13
By the system 3.13, we define a mapping A:
A φt, ϕt, γt z1 t, z2 t, z3 t,
∀ φt, ϕt, γt ∈ HLm .
3.14
Combining 3.8 and 3.13, we have
t
m 2
− f1 L ds e−a1 t−s b c d
∼1
∼1 L
∼1
−∞
t
L
−a t−s
2
∼
z1 t ≤
− f m ds e 1
b1 c1 d1
m ∼1
−∞
z1 t ≥
t
1
m 2
− f1 L ≥ m,
b c d
∼1 L
∼1
a1 ∼ 1
L
1 2
− f m ≤ L,
b1 c1 d1
a
m ∼1
∼1
m 2
1
m 2
−
a2 t−s
z2 t ≥
− f2 L ds − f2 L ≥ m,
e
c b d
c b d
∼2
∼2 L
∼2
∼2 L
∼2
a2 ∼2
−∞
t
L
L
1
−a t−s
2
2
z2 t ≤
c2 b2 d2
c2 b2 d2
− f m ds − f m ≤ L,
e ∼2
m ∼2
a
m ∼2
−∞
t
∼2
m 2
1
m 2
z3 t ≥
− f3 L ds − f3 L ≥ m,
e−a3 t−s d b c
d b c
∼3
∼3
∼3
∼3 L
∼3 L
a3 ∼ 3
−∞
t
L
L
1 −a t−s
2
2
∼
z3 t ≤
− f m ds − f m ≤ L.
e 3
d3 b3 c3
d3 b3 c3
m ∼3
a
m ∼3
−∞
∼3
3.15
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Therefore, z1 t, z2 t, z3 t ∈ HLm , that is, AHLm ⊂ HLm . If A is uniformly boundness and
equicontinuous, by Ascoli-Arzela theorem, A is compact mapping.
It is obvious to obtain uniformly boundedness. In fact, for any φt, ϕt, γt ∈ HLm ,
by 3.15 we have z1 t, z2 t, z3 t Aφt, ϕt, γt ∈ HLm ; that is, it satisfies
0 < m, m, m ≤ A φt, ϕt, γt z1 t, z2 t, z3 t ≤ L, L, L.
3.16
Next we prove equicontinuous. For any φt, ϕt, γt ∈ HLm , we denote z1 t, z2 t,
z3 t Aφt, ϕt, γt, t1 < t2 , and then
t1
t
φs
φs
1
d1 s
− f1 sφ2 s
z1 t1 − z1 t2 | e− s a1 rdr b1 s c1 s
|
−∞
ϕs
γs
−
t2
−∞
e
−
t2
s
a1 rdr
φs
φs
2
d1 s
− f1 sφ s .
b1 s c1 s
ϕs
γs
3.17
Let h1 t b1 t c1 tφt/ϕt d1 tφt/γt − f1 tφ2 t; we obtain
t1
t2
t
t
− s2 a1 rdr
− s1 a1 rdr
e
h1 sds −
e
h1 sds
z1 t2 − z1 t1 | |
−∞
−∞
t
t2 t2
t1
t
− t 2 a1 rdr
− s a1 rdr
− s1 a1 rdr
1
≤ e
h1 sds e
− 1 h1 sds.
e
t1
−∞
3.18
Recalling φt, ϕt, γt ∈ HLm , we deduce that there exists a positive number M such that
|h1 s| ≤ M; then 3.18 becomes
z1 t2 − z1 t1 | ≤ Me
|
−
t2
ξ1
a1 rdr
|t2 − t1 | t
1 − 2 a rdr ,
M1 − e t1 1
a
3.19
∼1
where ξ1 ∈ t1 , t2 .
Similarly, we have
z2 t2 − z2 t1 | ≤ Ne
|
−
t2
ξ2
a2 rdr
|t1 − t2 | t2
1 N 1 − e− t1 a2 rdr ,
a
3.20
t2
1 P 1 − e− t1 a3 rdr ,
a
3.21
∼2
where ξ2 ∈ t1 , t2 , and N is a positive number.
By a completely analogous argument, we obtain
z3 t2 − z3 t1 | ≤ P e
|
−
t2
ξ3
a3 rdr
|t1 − t2 | ∼3
where ξ3 ∈ t1 , t2 , and P is a positive number.
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Discrete Dynamics in Nature and Society
By 3.19–3.21, for any φt, ϕt, γt ∈ HLm , we derive
lim sup A φ, ϕ, γ t1 − A φ, ϕ, γ t2 0,
ξ → 0|t −t |≤ξ
1 2
uniformly for x ∈ Ω.
3.22
Thus, A is a compact mapping which maps HLm into itself; by Schauder fixed point
theorem, there exists a fixed point φt, ϕt, γt ∈ HLm for A; namely, 3.11 has a
solution; therefore there exists a strictly positive almost periodic solution v1∗ t, v2∗ t, v3∗ t 1/φt, 1/ϕt, 1/γt, t ∈ R for system 3.10. It is obvious that v1∗ t, v2∗ t, v3∗ t, t ∈ R
is also the spatial homogeneity almost periodic solution for 1.1.
Theorem 3.2. Under the conditions of Theorem 3.1, suppose that system 1.1 satisfies the following
conditions:
supb3 t b2 t − b1 t −ε1 < 0,
t≥0
supc3 t c1 t − c2 t −ε2 < 0,
t≥0
3.23
supd1 t d2 t − d3 t −ε3 < 0.
t≥0
Then there exists a strictly positive spatial homogeneity almost periodic solution v1∗ t, v2∗ t, v3∗ t
for 1.1, and the corresponding solution for systems 1.1–1.3 is globally asymptotically stable; that
is, the solution v1 x, t, v2 x, t, v3 x, t, x, t ∈ Ω × R satisfies
lim vi x, t − v∗i t 0,
t→∞
i 1, 2, 3, uniformly for x ∈ Ω.
3.24
Proof. We have obtained the existence by Theorem 3.1; next we pay more attention to the
stability. Concerning 3.24, we have two cases on initial data vi0 x, i 1, 2, 3.
1 vi0 x > 0, x ∈ Ω.
2 There exists a point x0 ∈ Ω, such that v10 x0 0, v20 x0 0 or v30 x0 0.
For the case 1, let li minx∈Ω vi0 x, ri maxx∈Ω vi0 x, i 1, 2, 3; then 0 < li ≤
vi0 x ≤ ri . Suppose that v1 t, v2 t, v3 t and v1 t, v2 t, v3 t are the solution for 3.10
corresponding to initial datum v1 0, v2 0, v3 0 r1 , r2 , r3 and v1 0, v2 0, v3 0 l1 , l2 , l3 , respectively; then there are a pair of ordered upper and lower solutions
v1 t, v2 t, v3 t and v1 t, v2 t, v3 t for 1.1–1.3; by Lemma 2.5, there exists a unique
solution v1 x, t, v2 x, t, v3 x, t, x, t ∈ Ω × R for system 1.1–1.3, which satisfies
v1 t, v2 t, v3 t ≤ v1 x, t, v2 x, t, v3 x, t ≤ v1 t, v2 t, v3 t.
3.25
If we have
lim vi t − vi∗ t lim vi t − vi∗ t 0,
t→∞
t→∞
i 1, 2, 3,
3.26
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9
then 3.24 holds. Therefore, if we want to obtain 3.26, we only need to prove that the
solution v1 t, v2 t, v3 t for 3.10 with arbitrary positive initial data v1 0, v2 0, v3 0 v10 , v20 , v30 satisfies
lim vi t − vi∗ t 0,
t→∞
i 1, 2, 3.
3.27
Because of the initial datum v10 , v20 , v30 > 0 and grazing rates f , f , f > 0, by
∼1
∼2
the practical meaning in biology, we know that v1 t, v2 t, v3 t > 0. Now let
Pi t ln vi t,
Qi t ln vi∗ t,
∼3
i 1, 2, 3.
3.28
Then one has
d
P1 t − Q1 t −b1 t eP1 t − eQ1 t − c1 t eP2 t − eQ2 t − d1 t eP3 t − eQ3 t
dt
1
1
− ∗
f1 t,
v1 t v1 t
d
P2 t − Q2 t −b2 t eP1 t − eQ1 t − c2 t eP2 t − eQ2 t − d2 t eP3 t − eQ3 t
dt
1
1
− ∗
f2 t,
v2 t v2 t
d
P3 t − Q3 t −b3 t eP1 t − eQ1 t − c3 t eP2 t − eQ2 t − d3 t eP3 t − eQ3 t
dt
1
1
−
f3 t.
v3 t v3∗ t
3.29
Namely,
f1 t
d
P1 t
Q1 t
P2 t
Q2 t
e
−
c
e
−
e
−
e
t
P1 t − Q1 t − b1 t 1
dt
v1 tv1∗ t
− d1 t eP3 t − eQ3 t ,
f2 t
d
P2 t
Q2 t
e
−
e
P2 t − Q2 t −b2 t eP1 t − eQ1 t − c2 t dt
v2 tv2∗ t
− d2 t eP3 t − eQ3 t ,
d
P3 t − Q3 t −b3 t eP1 t − eQ1 t − c3 t eP2 t − eQ2 t
dt
f3 t
P3 t
Q3 t
.
e
−
e
− d3 t v3 tv3∗ t
3.30
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Discrete Dynamics in Nature and Society
Consider the following Lyapunov function:
Ut 3
!
|Pi t − Qi t|,
t ≥ 0.
3.31
i1
Let D U represent the right derivation on function Ut; we have
D Ut
3
!
i1
D |Pi t − Qi t| 3
!
sgnPi t − Qi t
i1
sgnP1 t − Q1 t
f1 t
× − b1 t v1 tv1∗ t
e
P1 t
−e
Q1 t
sgnP2 t − Q2 t
× −b2 t eP1 t − eQ1 t − c2 t d
Pi t − Qi t
dt
− c1 t e
P2 t
−e
Q2 t
− d1 t eP3 t − eQ3 t
f2 t
P2 t
Q2 t
P3 t
Q3 t
e
−
d
−
e
−
e
e
t
2
v2 tv2∗ t
sgnP3 t − Q3 t
P1 t
Q1 t
P2 t
Q2 t
− c3 t e
− d3 t × −b3 t e
−e
−e
f3 t
P3 t
Q3 t
e
−e
v3 tv3∗ t
≤ b3 t b2 t − b1 teP1 t − eQ1 t c1 t c3 t − c2 teP2 t − eQ2 t d1 t d2 t − d3 teP3 t − eQ3 t ≤ −ε1 v1 t − v1∗ t − ε2 v2 t − v2∗ t − ε3 v3 t − v3∗ t.
3.32
Integrated by the time, we have
t
3
!
Ut εi vi s − vi∗ sds ≤ U0.
i1
3.33
0
By the nonnegative of Ut and the boundedness of U0, we obtain that the Ut is bounded,
and
t
0
vi t − v∗ tds,
i
i 1, 2, 3,
3.34
convergences, by 3.32 we get D Ut < 0, then the limit
lim Ut l
t→∞
3.35
Discrete Dynamics in Nature and Society
11
exists, and Ut ≥ l. If l > 0, then at least one of the following three inequalities
|P1 t − Q1 t| >
l
,
4
|P2 t − Q2 t| >
l
,
4
|P3 t − Q3 t| >
l
4
3.36
holds. Without loss of generality, we assume that |P1 t−Q1 t| > l/4. Thus there is no point of
intersection between P1 t and Q1 t. Suppose that P1 t > Q1 t; then we have P1 t − Q1 t >
l/4. Thus
t
0
v1 t − v∗ tds 1
t
t
P1 s
Q1 s −e
e
ds eQ1 s eP1 s−Q1 s − 1ds
0
≥m
0
t
eP1 s−Q1 s − 1 ds > m
el/4 − 1 ds m el/4 − 1 t −→ ∞,
t
0
0
which contradicts with the convergence of
t
0
3.37
|vi s − vi∗ s|ds. Therefore l 0; consequently
lim vi t − vi∗ t 0,
t→∞
i 1, 2, 3.
3.38
Then we obtain 3.27.
For the case 2, firstly, choose three sufficient large positive numbers M1 , M2 , and M3 ,
such that
f1 t ≤ −M1 a1 t − b1 tM1 ,
t > 0,
f2 t ≤ −M2 a2 t − c2 tM2 ,
t > 0,
f3 t ≤ −M3 a3 t − d3 tM3 ,
t > 0,
3.39
and Mi ≥ maxx∈Ω vi0 x, i 1, 2, 3. Let v 0, vi Mi , i 1, 2, 3; then we have
∼i
∂
v1
− k1 tΔ
v1 − v1 a1 t − b1 t
v1 − c1 tv − d1 tv − f1 t ≥ 0,
∼2
∼3
∂t
∂v
∼1
∂t
− k1 tΔv − v a1 t − b1 tv − c1 t
v2 − d1 t
v3 − f1 t ≤ 0,
∼1
∼1
∼1
∂
v2
− k2 tΔ
v2 − v2 a2 t − b2 tv − c2 t
v2 − d2 tv − f2 t ≥ 0,
∼1
∼3
∂t
∂v
∼2
∂t
− k2 tΔv − v a2 t − b2 t
v1 − c2 tv − d2 t
v3 − f2 t ≤ 0,
∼2
∼2
∼2
∂
v3
− k3 tΔ
v3 − v3 a3 t − b3 tv − c3 tv − d3 t
v3 − f3 t ≥ 0,
∼1
∼2
∂t
∂v
∼3
∂t
− k3 tΔv − v a3 t − b3 t
v1 − c3 t
v2 − d3 tv
∼3
∼3
∼3
− f3 t ≤ 0.
3.40
12
Discrete Dynamics in Nature and Society
Namely, v 0, and vi Mi , i 1, 2, 3 are a pair of ordered upper and lower solutions for
∼i
systems 1.1–1.3. By Lemma 2.5, there exists a unique solution v1 x, t, v2 x, t, v3 x, t
for systems 1.1–1.3, which satisfy
0 ≤ vi x, t ≤ Mi ,
i 1, 2, 3; x, t ∈ Ω × 0, ∞.
3.41
Secondly, we choose positive numbers δ1 , δ2 , and δ3 such that
δ1 a1 t − b1 tv1 x, t − c1 tv2 x, t − d1 tv3 x, t > 0,
x, t ∈ Ω × 0, ∞,
δ2 a2 t − b2 tv1 x, t − c2 tv2 x, t − d2 tv3 x, t > 0,
x, t ∈ Ω × 0, ∞,
δ3 a3 t − b3 tv1 x, t − c3 tv2 x, t − d3 tv3 x, t > 0,
x, t ∈ Ω × 0, ∞.
3.42
Accordingly, we have
∂v1
− k1 tΔv1 δ1 v1 v1 δ1 a1 t − b1 tv1 − c1 tv2 − d1 tv3 f1 t ≥ 0,
∂t
∂v2
− k2 tΔv2 δ2 v2 v2 δ2 a2 − b2 tv1 − c2 tv2 − d2 tv3 f2 t ≥ 0,
∂t
3.43
∂v3
− k3 tΔv3 δ3 v3 v3 δ3 a3 − b3 tv1 − c3 tv2 − d3 tv3 f3 t ≥ 0.
∂t
Next, we prove vi x, t > 0 in Ω×0, ∞ for i 1, 2, 3. Firstly, we show vi x, t > 0 in Ω×0, ∞.
If there exists one point x0 , t0 ∈ Ω×0, ∞ such that vi x0 , t0 0, by extremum principle, we
have vi x, t ≡ 0 in Ω × 0, t0 . However vi x, 0 vi0 x ≥ 0, and not being constant zero, we
obtain a contradiction. Therefore we have vi x, t > 0 in Ω × 0, ∞. Then we show vi x, t > 0
in ∂Ω × 0, ∞. If there exists a point x0 , t0 ∈ ∂Ω × 0, ∞ such that vi x0 , t0 0, by the
extremum principle, we have ∂vi x, t/∂n < 0, where x, t ∈ ∂Ω × 0, ∞, which is contrary
with boundary conditions 1.2. Thus we have vi x, t > 0 in Ω × 0, ∞.
For a fixed number λ > 0, by 3.41, we have
0 < vi x, λ ≤ Mi ,
i 1, 2, 3, x ∈ Ω.
3.44
Because vi x, t λ satisfy system 1.1 in Ω × 0, ∞ and the conditions 1.2 in ∂Ω × 0, ∞,
thereby v1 x, t λ, v2 x, t λ, v3 x, t λ is regarded as a solution for system 1.1 under
initial data v10 x, v20 x, v30 x v1 x, λ, v2 x, λ, v3 x, λ, nevertheless, we have vi0 > 0
in Ω, i 1, 2, 3; combining the conclusions in case 1, we have
lim vi x, t λ − vi∗ t 0,
t−→∞
i 1, 2, 3, uniformly for x ∈ Ω.
3.45
By the arbitrariness of λ, we obtain
lim vi x, t − vi∗ t 0,
t−→∞
i 1, 2, 3, uniformly for x ∈ Ω.
3.46
Discrete Dynamics in Nature and Society
13
If ki t, ai t, bi t, ci t, di t, and fi t i 1, 2, 3 of 1.1 are periodic functions in
real number field R, respectively, then we have the following results.
Corollary 3.3. If a , b , c , d , f are positive numbers, and
∼ i ∼ i ∼i ∼ i ∼
i
bi ci di
a
∼i
⎫
⎧
⎪
⎪
b ⎪
⎪
c
d
c
d
b
b
⎬
⎨
c d
∼1
∼2
∼2
∼3
∼1
∼3
∼ 1 ∼2 ∼ 3
≤ L min
,
,
,
,
,
⎪
⎪
a1
a2
a3
⎪
⎪
f2
f3
⎭
⎩ f1
3.47
are satisfied for i 1, 2, 3, then there exists a strictly positive spatial homogeneity periodic solution
V t v1 t, v2 t, v3 t for 1.1.
Corollary 3.4. Under the conditions of Corollary 3.3, suppose that system 1.1 satisfies the following
conditions:
supb3 t b2 t − b1 t −ε1 < 0,
t≥0
supc3 t c1 t − c2 t −ε2 < 0,
t≥0
supd1 t d2 t − d3 t −ε3 < 0.
3.48
t≥0
Then there exists a strictly positive spatial homogeneity periodic solution v1∗ t, v2∗ t, v3∗ t for 1.1,
and the corresponding solution for systems 1.1–1.3 is globally asymptotically stable; that is, the
solution v1 x, t, v2 x, t, v3 x, t, x, t ∈ Ω × R satisfies
lim vi x, t − vi∗ t 0,
t→∞
i 1, 2, 3, uniformly for x ∈ Ω.
3.49
4. Conclusion
This paper presents the use of upper and lower solutions method for systems of nonlinear
reaction-diffusion equations. This method is a powerful tool for solving nonlinear differential
equations in mathematical physics, chemistry, and engineering, and so forth. The technique
constructing a pair of upper and lower solutions and Lyapunov function provides a new
efficient method to handle the nonlinear structure.
We have dealt with the problem of almost periodic solution for a three-species
competition system with grazing rates and diffusions. The general sufficient conditions
have been obtained to ensure the existence and stability of the strictly positive space
homogenous almost periodic solution for the nonlinear reaction-diffusion equations. These
criteria generalize and improve some known results. In particular, the sufficient conditions
that we obtained are very simple, which provide flexibility for the application and analysis
of nonlinear three-species competition system.
Acknowledgments
The authors are grateful to the reviewers for their comments which helped in improving
the paper. This work is supported by the Science and Technology Project of Chongqing
14
Discrete Dynamics in Nature and Society
Municipal Education Committee Grant no. kJ110501 of China, the NSFC Grant nos.
51005264, 40801214 of China, and Natural Science Foundation Project of CQ CSTC Grant
no. 2010BB9401 of China.
References
1
D. P. Jiang, “Some periodic ecological models with grazing rates,” Journal of Biomathematics, vol. 5, no.
1, pp. 80–89, 1990 Chinese.
2
Q. C. Lin, “Some almost periodic biological models with grazing rates,” Journal of Biomathematics, vol.
14, no. 3, pp. 257–263, 1999 Chinese.
3
F. D. Chen and X. X. Chen, “The n-competing Lotka-Volterra almost periodic systems with grazing
rates,” Journal of Biomathematics, vol. 18, no. 4, pp. 411–416, 2003 Chinese.
4
Z. Q. Zhang and Z. C. Wang, “Periodic solution for a two-species nonautonomous competition LotkaVolterra patch system with time delay,” Journal of Mathematical Analysis and Applications, vol. 265, no.
1, pp. 38–48, 2002.
5
Z. Q. Zhang and Z. C. Wang, “Multiple positive periodic solutions for a generalized delayed
population model with an exploited term,” Science in China Series A, vol. 50, no. 1, pp. 27–34, 2007.
6
D. Hu and Z. Q. Zhang, “Four positive periodic solutions of a discrete time delayed predatorprey system with nonmonotonic functional response and harvesting,” Computers & Mathematics with
Applications, vol. 56, no. 12, pp. 3015–3022, 2008.
7
C. V. Pao and Y.M. Wang, “Numerical solutions of a three-competition Lotka-Volterra system,” Applied
Mathematics and Computation, vol. 204, no. 1, pp. 423–440, 2008.
8
Y. Q. Liu, S. L. Xie, and Z. D. Xie, “Existence and stability for periodic solution of competition reactiondiffusion models with grazing rates in population dynamics,” Journal of Systems Science and Systems
Engineering, vol. 10, no. 2, pp. 402–410, 1996.
9
C. Y. Wang, S. Q. An, and C. J. Fang, “Almost periodic solutions and periodic solutions of reactiondiffusion systems with time delays,” Mathematica Applicata, vol. 20, no. 2, pp. 281–285, 2007 Chinese.
10
F. D. Chen and C. L. Shi, “Global attractivity in an almost periodic multi-species nonlinear ecological
model,” Applied Mathematics and Computation, vol. 180, no. 1, pp. 376–392, 2006.
11
C. Y. Wang, “Existence and stability of periodic solutions for parabolic systems with time delays,”
Journal of Mathematical Analysis and Applications, vol. 339, no. 2, pp. 1354–1361, 2008.
12
C. Y. Wang, S. Wang, and L. R. Li, “Periodic solution and almost periodic solution for a nonmonotone
reaction-diffusion system with time delay,” Acta Mathematica Scientia Series A, vol. 30, no. 2, pp. 517–
524, 2010 Chinese.
13
X. Q. Liu, “Periodic or almost periodic solutions to a class of systems of reaction-diffusion equations,”
Chinese Journal of Engineering Mathematics, vol. 11, no. 4, pp. 107–111, 1994 Chinese.
14
Y. Li and Y. Kuang, “Periodic solutions of periodic delay Lotka-Volterra equations and systems,”
Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 260–280, 2001.
15
C. Y. Wang and X. H. Hu, “Existence and uniqueness of bounded solution and periodic solution
of reaction-diffusion equation with time delay,” Journal of Chongqing University of Posts and
Telecommunication (Natural Science Edition), vol. 17, no. 5, pp. 644–646, 2005 Chinese.
16
C. Y. Wang, “Periodic solution of prey predator model with diffusion and distributed delay effects,”
Journal of Chongqing University of Posts and Telecommunication (Natural Science Edition), vol. 18, no. 3,
pp. 409–412, 2006.
17
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, NY,USA, 1992.
18
A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer,
Berlin, Germany, 1974.
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