Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 404696, 26 pages
doi:10.1155/2011/404696
Research Article
Non-Constant Positive Steady States for
a Predator-Prey Cross-Diffusion Model with
Beddington-DeAngelis Functional Response
Lina Zhang and Shengmao Fu
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Shengmao Fu,
Received 13 October 2010; Accepted 30 January 2011
Academic Editor: Dumitru Motreanu
Copyright q 2011 L. Zhang and S. Fu. 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.
This paper deals with a predator-prey model with Beddington-DeAngelis functional response
under homogeneous Neumann boundary conditions. We mainly discuss the following three
problems: 1 stability of the nonnegative constant steady states for the reaction-diffusion system;
2 the existence of Turing patterns; 3 the existence of stationary patterns created by cross-
diffusion.
1. Introduction
Consider the following predator-prey system with diffusion:
u
t
− d
1
Δu r
1
u
x
> 0,v
x, 0
v
0
x
≥ 0,x∈ Ω,
1.1
where Ω ⊂
N
is a bounded domain with smooth boundary ∂Ω and ν is the outward
unit normal vector of the boundary ∂Ω. In the system 1.1, ux, t and vx, t represent
the densities of the species prey and predator, respectively, u
0
x and v
0
x a re given
smooth functions on
Ω which satisfy compatibility conditions. The constants d
1
, d
2
, called
2 Boundary Value Problems
diffusion coefficients, are positive, r
the global stability of the unique positive constant steady state and gained several results for
the non-existence of non-constant positive solutions.
It is known that the prey-dependent functional response means that the p redation
behavior of the predator is only determined by the prey, which contrasts with some realistic
observations, such as the paradox of enrichment 11, 12. The ratio-dependent functional
response reflects the mutual interference between predator and prey, but it usually raises
controversy because of the low-density problem 13. In 1975, Beddington and DeAngelis
14, 15 proposed a function f βu/1 mu nv, commonly known a s Beddington-
DeAngelis functional response. It has an extra term in the denominator which models mutual
interference between predator and prey. In addition, it avoids the low-density problem.
In this paper, we study the system 1.1 with f βu/1 mu nv. Using the scaling
r
1
K
u −→ u,
r
1
Kδ
v −→ v, r
1
−→ λ,
Kδ
r
1
β −→ β,
K
r
1
m −→ m,
Kδ
u
g
2
u, v
,x∈ Ω,t>0,
∂
ν
u ∂
ν
v 0,x∈ ∂Ω,t>0,
u
x, 0
u
0
x
> 0,v
x, 0
v
0
x
2
m n
,v
∗
u
∗
.
1.4
In the system 1.3, the Beddington-DeAngelis functional response is used only in the prey
equation, not the predator, and the predator equation contains a Leslie-Gower term v/δu
16. To our knowledge, there are few known results for 1.3 while there has been relatively
good success fo r the predator-prey model with the full Beddington-DeAngelis functional
responses. For example, Cantrell and Cosner 17 derived criteria for permanence and for
predator extinction, and Chen and Wang 18 proved the nonexistence and existence of
nonconstant positive steady states.
Taking into account the population fluxes of one species due to the presence of the
other species, we consider the following cross-diffusion system:
u
t
− d
1
Δu λu −u
2
−
βuv
1 mu nv
,x∈ Ω,t>0,
> 0,v
x, 0
v
0
x
≥ 0.x∈ Ω,
1.5
where Δd
2
d
3
uv is a cross-diffusion term. If d
3
> 0, the movement of the predator is
directed towards the lower concentration of the prey, which represents that the prey species
congregate and form a huge group to protect themselves from the attack of the predator.
It is clear that such an environment of prey-predator interaction often occurs in reality. For
example, in 19–21, and so forth, with the similar biological interpretation, the authors also
introduced the same cross-diffusion term as in 1.5 to the prey of various prey-predator
models.
The main aim of this paper is to study the effects of the diffusion and cross-
diffusion pressures on the existence of stationary patterns. We will demonstrate that
the unique positive constant steady state u
∗
,v
1.3.InSection3, we investigate the existence of Turing patterns of 1.3 by using the Leray-
Schauder degree theory. In Section 4, we prove the existence of stationary patterns of 1.5.
We end with a brief section on conclusions.
2. The Long Time Behavior of Time-Dependent Solutions
In this section, we discuss the global behavior of solutions for the system 1.3.Bythe
standard theory of parabolic equations 26, 27, we can prove that the problem 1.3 has
a unique classical global solution u, v, which satisfies 0 <ux, t ≤ max{λ, sup
Ω
u
0
} and
0 <vx, t ≤ max{λ, sup
Ω
u
0
, sup
Ω
v
0
} on Ω ×0, ∞.
2.1. Global Attractor and Permanence
First, we show that
0
0,λ ×0,λ is a global attractor for 1.3.
Theorem 2.1. Let ux, t,vx, t be any non-negative solution of 1.3.Then,
lim
t →∞
sup
Ω
u
∈ Ω ×
T, ∞
. 2.2
Let vt be the unique positive solution of
dw
dt
w
1 −
w
λ ε
,t∈
T, ∞
,
w
T
max
Ω
v
x, T
t →∞
inf
Ω
v
x, t
≥ K,
2.5
Boundary Value Problems 5
where
K
1
2m
m −n
λ − 1
m −n
λ − 1
2
4mλ
1 nλ − β
1
1 mu n
λ ε
1
u
2.7
for x, t ∈
Ω ×T, ∞.Letut be the unique positive solution of
dw
dt
−mw
2
mλ −nλ −nε
1
− 1
w λ
nλ − β
λ ε
1
t →∞
u
t
1
2m
m −n
λ − 1 −nε
1
m −n
λ − nε
1
− 1
2
4m
λ
nλ − β
⎞
⎠
. 2.10
It is easy to see that 1 is an eigenvalue of J
1
,thusλ, 0 is unconditionally unstable.
6 Boundary Value Problems
Now, we discuss the Turing instability of u
∗
,v
∗
. Recall that a constant solution is
Turing unstable if it is stable in the absence of diffusion, and it becomes unstable when
diffusion is present 28. More precisely, this requires the following two conditions.
i It is stable as an equilibrium of the system of ordinary differential equations
du
dt
g
1
u, v
,
dv
dt
g
2
u, v
Proof. The linearization matrix of 2.11 at u
∗
,v
∗
is
J
2
a
11
a
12
a
21
a
22
, 2.12
where
a
11
1
β
m
λ − u
∗
−a
11
− a
12
m n
u
2
∗
λ
1
m n
u
∗
, trace J
2
a
11
− 1.
2.14
Clearly, det J
2
> 0. If a
11
< 1, then trace J
2
be the
eigenspace corresponding to μ
i
in H
1
Ω.Let{φ
ij
: j 1, 2, ,dim Eμ
i
}be the orthonormal
basis of Eμ
i
, X H
1
Ω
2
, X
ij
{cφ
ij
: c ∈
2
}. Then,
X
∞
i1
X
i
, X
Boundary Value Problems 7
then 2 ≤ i
0
< ∞. In this case, denote
d
2
min
2≤i≤i
0
d
i
2
,d
i
2
d
1
μ
i
det J
2
μ
i
a
11
− d
1
μ
∗
is locally asymptotically stable if a
11
>d
1
μ
2
and d
2
<
d
2
; u
∗
,v
∗
is unstable if a
11
>d
1
μ
2
and
d
2
>
d
22
are given in 2.13. Suppose φx,ψx
T
is an eigenfunction of L
corresponding to an eigenvalue μ,then
d
1
Δφ
a
11
− μ
φ a
12
ψ, d
2
Δψ a
21
φ
a
22
− μ
ψ
T
μ
i
L
i
a
ij
b
ij
φ
ij
0, where L
i
a
11
− d
1
μ
i
− μa
12
a
21
a
22
i
− trace J
2
,Q
i
−d
2
μ
i
a
11
− d
1
μ
i
d
1
μ
i
det J
2
. 2.23
Clearly, Q
1
> 0sinceμ
1
0. If a
11
Assume that a
11
>d
1
μ
2
.Ifd
2
<
d
2
,thend
1
μ
i
<a
11
and d
2
<d
i
2
for i ∈ 2,i
0
. It follows
that Q
i
> 0foralli ∈ 2,i
0
and d
2
>d
k
2
,sowe
have Q
k
< 0. This implies that u
∗
,v
∗
is unstable.
Remark 2.6. From Theorems 2.4 and 2.5, we can conclude that u
∗
,v
∗
is Turing unstable if
d
1
μ
2
<a
11
< 1andd
2
>
d
2
−
1 mK nK
1 mλ nλ
<
1 mu
∗
nv
∗
1 mK nK
. 2.24
Then u
∗
,v
∗
attracts all positive solutions of 1.3.
Proof. Define the Lyapunov function
E
1
t
Ω
u −2u
1
β
1 mu
∗
nv
∗
1 mλ nλ
,
2.26
u, v is a positive solution of 1.3.ThenE
1
t ≥ 0forallt ≥ 0. The straightforward
computations give that
dE
1
dt
Ω
u
2
− u
∗
2
u
2
B
1
u − u
∗
v − v
∗
C
1
v − v
∗
2
dx,
2.27
Boundary Value Problems 9
where
D
1
−
d
1
2u
∗
βmv
∗
1 mu
∗
nv
∗
1 mu nv
,
B
1
δ
1
−
β
u u
∗
1 mu
∗
1 mu
∗
nv
∗
1 mu
∗
nv
∗
1 mK nK
−β
u u
∗
1 mu
∗
1 mK nK
K u
∗
1 mu nv
−
1 mK nK
1 mλ nλ
Ω
1
u
B
1
4
C
1
v − v
∗
2
dx
Ω
D
1
dx
Ω
1
u
δ
1
3
4
δ
1
−
β
u u
∗
1 mu
∗
4
1 mu
∗
nv
∗
1 mu nv
v − v
∗
2
dx
≤ 0
1 mu
∗
nv
∗
1 mK nK
, 2.31
β<
λm λn 2
m n
2
. 2.32
Then, u
∗
,v
∗
attracts all positive solutions of 1.3.
Proof. Define the Lyapunov function
E
2
t
∗
nv
∗
1 mλ nλ, u, v is a positive solution of 1.3.Then
dE
2
dt
Ω
D
2
dx
Ω
1
u
A
2
u −u
∗
2
B
2
u − u
∗
δ
2
d
2
v
∗
v
2
|
∇v
|
2
,
A
2
−1
βmv
∗
1 mu
∗
nv
∗
1 mu nv
,
B
2
1
1 mu
∗
nv
∗
1 mK nK
×
1 mu
∗
nv
∗
1 mK nK
−β
1 mu
∗
1 mK nK
1 mu nv
A
2
B
2
u −u
∗
2
dx
Ω
1
u
B
2
4
C
2
v − v
∗
2
dx
Ω
1
u
−
3
4
δ
2
−
β
1 mu
∗
4
1 mu
∗
nv
∗
1 mu nv
v − v
∗
Ω
v − v
∗
− v
∗
ln
v
v
∗
dx,
2.38
where δ
3
K{1 β/1 mu
∗
nv
∗
1 mλ nλ}, we can also derive the global stability of
u
∗
,v
∗
for 1.3 under a stronger condition than 2.24. Thus, the Lyapunov function defined
by 2.25 is better than 2.38 in discussing the global stability of u
∗
,v
∗
−d
2
Δv v
1 −
v
u
g
2
u, v
in Ω,
∂
ν
u ∂
ν
v 0on∂Ω.
3.1
The existence and non-existence of the non-constant positive solutions of 3.1 will be given.
In the following, t he generic constants C
1
, C
2
, C
∗
, C, C, and so forth, will depend on
the domain Ω and the dimension N. However, as Ω and the dimension N are fixed, we will
12 Boundary Value Problems
j
g
x, w
x
≥ 0inΩ,
∂w
∂ν
≤ 0on∂Ω,
3.2
and wx
0
max
Ω
wx,thengx
0
,wx
0
≥ 0.
ii If w ∈ C
2
Ω ∩C
1
Ω satisfies
Δw
x
0
≤ 0.
Lemma 3.2 Harnack, inequality 31. Let w ∈ C
2
Ω ∩C
1
Ω be a positive solution to Δwx
cxwx0,wherec ∈ C
Ω, satisfying the homogeneous Neumann boundary condition. Then
there exists a positive constant C
∗
which depends only on c
∞
such that
max
Ω
w ≤ C
∗
min
Ω
w.
3.4
The results of upper and lower bounds can be stated as follows.
Theorem 3.3. For any positive number d, there exists a positive constant C
Λ,d such that every
positive solutionu, v of 3.1 satisfies C
<ux, vx <λif d
1
≥ d.
Boundary Value Problems 13
x
1
nv
x
1
≥ 0,
λ − u
y
1
−
βv
y
1
1 mu
y
1
nv
y
1
≤ ux
1
<λ. Moreover, we have
v
y
1
≤ v
x
2
≤ u
x
2
≤ u
x
1
, 3.6
v
y
1
≥ v
− λm
u
y
1
β − λn
v
y
1
− λ ≥ 0.
3.8
Noting that uy
1
≤ vy
1
≤ ux
1
from 3.6 and 3.7, 3.8 implies that max
Ω
uxux
1
>
x
.
3.9
Combining 3.9 with max
Ω
ux >C
1
,wefindthatmin
Ω
ux >C
1
for some positive
constant C
CΛ,d. It follows from 3.7 that min
Ω
vxvy
2
≥ uy
1
>C. The proof
is completed.
3.2. Non-Existence of Non-Constant Positive Steady States
In the following theorem we will discuss the non-constant positive solutions to 3.1 when
the diffusion coefficient d
1
varies while the other parameters d
2
, λ, β, m,andn are fixed.
∇u
|
2
d
2
v
v
2
|
∇v
|
2
dx
Ω
g
1
u, v
u −
u
u
dx
Ω
g
2
−
β
1 m
u
1 m
u nv
1 mu nv
v
uu
u −
u
v − v
dx
Ω
−
1
βm
n
C
3
Ω
u −
u
2
dx C
2
Ω
ε −
1
u
v −
v
2
dx
3.12
for some positive constants C
2
4
Ω
u −
u
2
dx C
2
Ω
ε −
1
u
v −
v
2
dx,
3.13
which implies that u
u constant and v v constant if d
1
>D max{C
4
/μ
2
,d}.
v 0on∂Ω
. 3.14
For the sake of convenience, we define a compact operator F : Y → Y by
F
e
a
11
− d
1
Δ
−1
g
1
u, v
a
11
u
−a
22
Δ
−1
are the inverses of the operators
a
11
− d
1
Δ and −a
22
− d
2
Δ in Y with the homogeneous Neumann boundary conditions.
Then the system 3.1 is equivalent to the equation I −Fe 0. To apply the index theory, we
investigate the eigenvalue of the problem
−
I −F
e
e
∗
Ψ μΨ, Ψ
/
0, 3.16
where Ψ ψ
1
,ψ
2
μ 1
d
1
Δψ
1
−μ 1
a
11
ψ
1
a
12
ψ
2
,
−
μ 1
d
2
Δψ
2
a
21
ψ
−
μ 1
d
1
μ
i
a
12
a
21
μ 1
a
22
−
μ 1
d
2
μ
i
. 3.19
The characteristic equation det B
i
0 can be written as
det J
2
a
11
d
1
μ
i
−a
22
d
2
μ
i
0. 3.20
Note that −d
2
μ
i
a
11
− d
1
μ
i
d
1
μ
μ
n
,μ
n1
for some n ≥ 2 and
2≤i≤n, Q
i
<0
dim Eμ
i
is odd, then the problem 3.1 has at least one
non-constant positive solution for any d
2
>
d
2
,whereQ
i
and
d
2
are given in 2.23 and 2.17,
respectively.
16 Boundary Value Problems
Proof. The proof, which is by contradiction, is based on the homotopy invariance of the
topological degree. Suppose, on the contrary, that the assertion is not true for some d
2
θ; e
⎛
⎜
⎝
a
11
−
θd
1
1 −θ
d
1
Δ
−1
g
1
u, v
a
⎞
⎟
⎠
. 3.22
Then, e is a positive solution of 3.1 if and only if it is a positive solution of F1; ee.
It is obvious that e
∗
is the unique constant positive solution of 3.22 for any 0 ≤ θ ≤ 1.
By Theorem 3.3, there exists a positive constant C such that, for all 0 ≤ θ ≤ 1, the positive
solutions of the problem Fθ; ee are contained in BC
{e ∈ Y | C
−1
<u,v<C}.
Since Fθ; e
/
e for all e ∈ ∂BC and Fθ; · : BC × 0, 1 → Y is compact, we can see that
the degree degI −Fθ; ·,BC, 0 is well defined. Moreover, by the homotopy invariance
property of the topological degree, we have
deg
I −F
0; ·
,B
C
, 0
d
i
2
in 2.17.Since
d
2
˘
d
2
>
d
2
,thenQ
k
< 0forsomek,2≤ k ≤ n.Leti k. Then, 3.20 has one positive root
and a negative root. Furthermore, we have Q
i
> 0fori 1andalli ≥ n 1. Therefore, when
i 1andi ≥ n 1, the characteristic equation 3.20 has no roots with non-negative real parts.
In addition, if
2≤i≤n, Q
i
<0
dim Eμ
i
is odd, we have
index
,B
C
, 0
index
I −F
1; ·
, e
∗
−1. 3.25
Similar argument shows μ is an eigenvalue of −I −F
e
0; e
∗
if and only if it is a root
of the characteristic equation
μ
2
2
d
1
1
μ
i
det J
2
a
11
d
1
μ
i
−a
22
d
2
μ
i
0. 3.26
It is easy to check that all eigenvalues of 3.26 have negative real parts for all i ≥ 1, which
implies
index
I −F
I −F
0; ·
, e
∗
1. 3.28
This contradicts 3.23, and the proof is complete.
Example 3.6. Let Ω0, 1. Then, the p arameters λ 2, β 6, m 3, n 0.1, d
1
0.0152,
and d
2
4.1309 satisfy all the conditions of Theorem 3.5. This means that u
∗
,v
∗
2
√
159−
4/31, 2
√
159 − 4/31 is a locally asymptotically stable equilibrium point for the system
du
dt
2u − u
2
−
− 4.1309v
xx
v
1 −
v
u
,x∈
0, 1
,t>0 ,
u
x
v
x
0,x 0, 1,t>0,
u
x, 0
u
0
x
> 0,v
x, 0
2
,thereexistsad
2
∈ d
2
− δ, d
2
δ such that 3.1 has a non-constant
positive solution close to e
∗
. Otherwise, we say that d
2
; e
∗
is a regular point 27.
We will consider the bifurcation of 3.1 at the equilibrium points
d
2
; e
∗
, while all
other parameters are fixed. From 2.23,wedefine
Q
d
2
; μ
d
1
d
2
a
11
d
1
2
4d
1
d
2
a
12
> 0,
3.32
18 Boundary Value Problems
then Qd
2
,μ0hastwodifferent real roots with same symbols. Let
S
p
μ
1
,μ
2
,μ
3
| d
2
d
i
2
d
1
μ
i
− det J
2
μ
i
a
11
− d
1
μ
i
,μ
i
> 0,d
1
μ
i
<a
11
2
dim Eμ
i
is odd, then d
2
; e
∗
is a
bifurcation point of 3.1 with respect to the curve d
2
; e
∗
,d
2
> 0. In this case, there exists
an interval σ
1
,σ
2
⊂ R
,where
i d
2
σ
1
<σ
2
< ∞ and σ
2
, 3.1 admits a non-constant positive solution.
4. Stationary Patterns for the PDE System with Cross-Diffusion
In this section, we discuss the corresponding steady-state problem of the system 1.5:
−d
1
Δu λu − u
2
−
βuv
1 mu nv
in Ω,
−d
2
Δ
1 d
3
u
v v
1 −
v
u
in Ω,
∂
ν
u ∂
, ∀x ∈ Ω.
4.2
Boundary Value Problems 19
Proof. We first prove that there exists a positive constant C CΛ,d,D such that
max
Ω
u ≤ Cmin
Ω
u, max
Ω
v ≤ Cmin
Ω
v.
4.3
A direct application of Lemma 3.1 to the first equation of 4.1 gives u<λon
Ω.From
Lemma 3.2, we have max
Ω
u ≤ Cmin
Ω
u for some positive constant CΛ,d,D.Define
ϕxd
2
1 d
3
uv and ϕx
0
max
Ω
ϕ. Applying Lemma 3.1 again to the second equation
u
ϕ in Ω,
∂ϕ
∂ν
0on∂
Ω.
4.5
Denote cxu − v/d
2
1 d
3
uu.wehave
c
x
∞
≤
1
d
2
max
Ω
v
d
2
min
Ω
1 d
3
λ
u
x
0
d
2
min
Ω
u
≤
1
d
2
1 d
3
λ
d
2
·
max
Ω
u
Ω
ϕ
·
max
Ω
1 d
3
u
min
Ω
1 d
3
u
≤ C
·
max
Ω
u
min
Ω
u
≤ C. 4.7
Thus, 4.3 is proved.
Note that min
Ω
of 4.1 satisfy
min
Ω
u
i
−→ 0ormin
Ω
v
i
−→ 0, as i −→ ∞,
4.8
20 Boundary Value Problems
and u
i
,v
i
satisfies
−d
1,i
Δu
i
λu
i
− u
2
i
−
βu
i
v
i
∂
ν
v
i
0on∂Ω.
4.9
Integrating by parts, we obtain that
Ω
u
i
λ − u
i
−
βv
i
1 mu
i
nv
i
dx 0,
Ω
v
i
1 −
max
Ω
u
i
−→ 0, max
Ω
v
i
−→ 0asi −→ ∞.
4.12
So we have
λ −u
i
−
βv
i
1 mu
i
nv
i
> 0onΩ, ∀i 1.
4.13
Integrating the first equation of 4.9 over Ω by parts, we have
Ω
u
i
λ − u
i
d
1
|
∇u
|
2
d
2
1 d
3
u
|
∇v
|
2
d
2
d
3
v∇u ·∇v
dx
Ω
λ −
u
1 m
u nv
1 mu nv
v
2
uu
u −
u
v − v
dx
Ω
1 −
v
v
u
v −
v
C
ε
u −
u
2
1 ε
v − v
2
d
2
2
d
2
3
v
2
4ε
|
∇u
|
2
ε
dx ≤
Ω
C
ε
1 d
2
2
d
2
3
|
∇u
|
2
1
μ
2
ε
|
∇v
3
u
∗
v
∗
,then4.1 can be rewritten
as
−d
1
Δu λu −u
2
−
βuw
1 d
3
u
1 mu
nw
g
1
u, w
in Ω,
−d
2
Δw
∗
,w
∗
. The linearization matrix of
Gu, wg
1
u, w, g
2
u, w
T
at u
∗
,v
∗
is
J
3
m
11
m
12
m
21
m
22
, 4.19
where
∗
1 d
3
u
∗
,m
22
−
1
1 d
3
u
∗
.
4.20
If
m
11
a
11
− a
12
d
3
u
∗
1 d
3
u
∗
Δ
−1
g
2
u, w
− m
22
w
, 4.22
where h ux,wx
T
, m
11
− d
1
Δ
−1
,and−m
22
− d
2
Δ
−1
are the inverses of the operators
index
I −Φ, h
∗
−1
γ
, 4.24
where γ is the sum of the algebraic multiplicities of the positive eigenvalues of −I −Φ
h
h
∗
.
Notice that 4.23 can be rewritten as
−
μ 1
d
1
Δψ
1
−μ 1
m
Boundary Value Problems 23
As the proof of Theorem 2.5, we can conclude that μ is an eigenvalue of −I − Φ
h
h
∗
on X
ij
if and only if it is a root of the characteristic equation det B
i
0, where
B
i
−μ 1
m
11
−
μ 1
d
1
μ
i
m
12
m
d
3
; μ
i
μ M
2
d
3
; μ
i
0, 4.27
where
M
1
d
3
; μ
i
2d
1
μ
i
m
11
μ
i
det J
3
m
11
d
1
μ
i
−m
22
d
2
μ
i
. 4.28
When i 1,
P
1
μ
μ
2
−
m
1
d
3
; μ
i
> 0. Consider the following limit:
lim
d
3
→∞
M
2
d
3
; μ
d
1
μ −
a
11
− a
12
d
1
μ a
n1
for some n ≥ 2 and the
sum
n
i2
dim Eμ
i
is odd. Then, there exists a positive constant D such that for d
3
> D,
indexΦ·,
h−1.
Theorem 4.4. Under the same assumption of Lemma 4.3, there exists a positive constant
D such that
for d
3
> D, the problem 4.1 has at least one non-constant positive solution.
Proof. From Lemma 4.3, there exists a positive constant
D such that, when d
3
> D,
indexF·, u−1. We shall prove that for any d
3
> D, 4.1 has at least one non-constant
positive solution. The proof, which is by contradiction, is based on the homotopy invariance
of the topological degree. Suppose, on the contrary, that the assertion is not true for some
24 Boundary Value Problems
d
⎜
⎜
⎜
⎝
m
11
−
θd
1
1 −θ
d
1
Δ
−1
λu − u
2
−
βuw
1 θd
3
3
u
1 −
w
1 θd
3
u
u
− m
22
w
⎞
⎟
⎟
⎟
⎠
.
4.32
It is obvious that
h is the unique constant positive solution of 4.32 for any 0 ≤ θ ≤ 1. By
Theorem 4.1 and w 1 d
3
uv, there exists a positive constant C such that, for all 0 ≤ θ ≤ 1,
the positive solutions of the problem Φθ; h0 are contained in BC
, 0
. 4.33
By our supposition and Lemma 4.3,theequationΦ1; h0 has only the p ositive solution
h in BC, and hence degΦ1; ·,BC, 0indexΦ1; ·,
h−1. Similar argument shows
degΦ0; ·,BC, 0indexΦ0; ·,
h1. This contradicts with 4.33, and then the proof is
completed.
Example 4.5. Let Ω0, 1. Then, the p arameters λ 2, β 6, m 3, n 0.1, d
1
0.0743,
d
2
2, and d
3
100 satisfy all the conditions of Theorem 4.4.Inthiscase,u
∗
,v
∗
2
√
159 −
4/31, 2
√
159 − 4/31 is a locally asymptotically stable steady state for the system
u
v
x
0,x 0, 1,t>0,
u
x, 0
u
0
x
> 0,v
x, 0
v
0
x
≥ 0,x∈
0, 1
.
4.34
However, it is an unstable steady state for the system
u
t
u
x
v
x
0,x 0, 1,t>0,
u
x, 0
u
0
x
> 0,v
x, 0
v
0
x
≥ 0,x∈
0, 1
.
4.35
Boundary Value Problems 25
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