# Existence of Traveling Waves for a Delayed SIRS Epidemic Diffusion Model with Saturation Incidence Rate

Authors: Kai Zhou; Qi-Ru Wang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

This paper is concerned with the existence of traveling waves for a delayed SIRS epidemic diffusion
model with saturation incidence rate. By using the cross-iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling waves to the existence of a pair of upper-lower solutions. By careful analyzsis, we derive the existence of traveling waves connecting the disease-free steady state and the endemic steady state through the establishment of the suitable upper-lower solutions.


---

## Body

## 1. Introduction

Since Kermack and Mckendrick [1] proposed an ordinary differential system to study epidemiology in 1927, various models have been used to describe various kinds of epidemics, and the dynamics of these systems have been investigated. Let S
(
t
) represent the number of individuals who are susceptible to the disease, let I
(
t
) represent the number of infected individuals who are infectious and are able to spread the disease by contact with susceptible individuals, and let R
(
t
) represent the number of individuals who have been infected and then removed from the possibility of being infected again. Mena-Lorca and Hethcote [2] considered the following SIRS epidemic model:
(1)
S
˙
(
t
)
=
A
-
d
S
(
t
)
-
β
S
(
t
)
I
(
t
)
+
δ
R
(
t
)
,
I
˙
(
t
)
=
β
S
(
t
)
I
(
t
)
-
(
γ
+
μ
+
d
)
I
(
t
)
,
R
˙
(
t
)
=
γ
I
(
t
)
-
(
δ
+
d
)
R
(
t
)
,
where the parameters A, d, β, δ, γ, μ are positive constants and A is the recruitment rate of the population, d is the natural death rate of the population, β is the transmission rate, δ is the rate at which recovered individuals lose immunity and return to the susceptible class, γ is the recovery rate of the infective individuals, and μ is the death rate of the infective individuals due to disease. The SIRS model assumes that the recovered individuals have only temporary immunity, which is reasonable in the study of some communicable diseases.However, due to the diseases latency or immunity, the presence of time delays in such models makes them more realistic. On the other hand, the environment in which an individual lives is actually heterogeneous and the mobility of people within a country or even worldwide is large; introducing the spatial diffusion in these epidemic models is unavoidable. In recent years, the dynamics of the delayed epidemic diffusion model have been widely studied by many researchers (see, e.g., [3–6]), and these studies are mainly focused on the global attractivity, basic reproductive number, and especially the epidemic waves. For example, Gan et al. [7] considered the following delayed SIRS epidemic model with spatial diffusion:
(2)
∂
S
∂
t
=
D
S
∂
2
S
∂
x
2
+
A
-
d
S
(
x
,
t
)
mmm
-
β
S
(
x
,
t
)
I
(
x
,
t
-
τ
)
+
δ
R
(
x
,
t
)
,
∂
I
∂
t
=
D
I
∂
2
I
∂
x
2
+
β
S
(
x
,
t
)
I
(
x
,
t
-
τ
)
mmm
-
(
γ
+
α
+
d
)
I
(
x
,
t
)
,
∂
R
∂
t
=
D
R
∂
2
R
∂
x
2
+
γ
I
(
x
,
t
)
-
(
δ
+
d
)
R
(
x
,
t
)
,
and obtained the existence of traveling wave solutions.In systems (1) and (2), the terms β
S
(
t
)
I
(
t
) and β
S
(
x
,
t
)
I
(
x
,
t
-
τ
) are called incidence rate and both of them are bilinear. However, as the number of susceptible individuals is large, it is reasonable to consider the saturation incidence rate (see [8]) instead of the bilinear incidence rate. Motivated by the works mentioned above, we will consider the following delayed SIRS epidemic diffusion model with nonlinear saturation rate
(3)
∂
S
∂
t
=
D
S
∂
2
S
∂
x
2
+
A
-
d
S
(
x
,
t
)
mmm
-
β
S
(
x
,
t
)
I
(
x
,
t
-
τ
)
1
+
α
I
(
x
,
t
-
τ
)
+
δ
R
(
x
,
t
)
,
∂
I
∂
t
=
D
I
∂
2
I
∂
x
2
+
β
S
(
x
,
t
)
I
(
x
,
t
-
τ
)
1
+
α
I
(
x
,
t
-
τ
)
mmm
-
(
γ
+
μ
+
d
)
I
(
x
,
t
)
,
∂
R
∂
t
=
D
R
∂
2
R
∂
x
2
+
γ
I
(
x
,
t
)
-
(
δ
+
d
)
R
(
x
,
t
)
and study its traveling wave solutions. The main tool is the upper-lower solutions coupled with cross-iteration method established by Ma [9]. We point out that the nonlinear terms in (3)do not satisfy the common various (exponential) monotonicity conditions such as in [10–12]; thus the main difficulty is the construction and verification of the upper-lower solutions.


## 2. Preliminaries and Lemmas

Throughout this paper, we employ the usual notations for the standard ordering inℝ
3. That is, for u
=
(
u
1
,
u
2
,
u
3
) and v
=
(
v
1
,
v
2
,
v
3
), we denote u
≤
v if u
i
≤
v
i, i
=
1,2
,
3; u
<
v if u
≤
v but u
≠
v; and u
≪
v if u
≤
v but u
i
≠
v
i, i
=
1,2
,
3. Let ∥
·
∥ denote the Euclidean norm in ℝ
3.First, we assume thatD
S
=
D
I
=
D
R
=
D for (3). Denoting N
=
S
+
I
+
R, then (3) reduces to the following system:
(4)
∂
N
∂
t
=
D
∂
2
N
∂
x
2
+
A
-
d
N
(
x
,
t
)
-
μ
I
(
x
,
t
)
,
∂
I
∂
t
=
D
∂
2
I
∂
x
2
+
β
(
N
-
I
-
R
)
I
(
x
,
t
-
τ
)
1
+
α
I
(
x
,
t
-
τ
)
mmm
-
(
γ
+
μ
+
d
)
I
(
x
,
t
)
,
∂
R
∂
t
=
D
∂
2
R
∂
x
2
+
γ
I
(
x
,
t
)
-
(
δ
+
d
)
R
(
x
,
t
)
.By making changes of variablesN
~
=
A
/
d
-
N, I
~
=
I, R
~
=
R and dropping the tildes, (4) is converted to the following system:
(5)
∂
N
∂
t
=
D
∂
2
N
∂
x
2
-
d
N
(
x
,
t
)
+
μ
I
(
x
,
t
)
,
∂
I
∂
t
=
D
∂
2
I
∂
x
2
+
β
(
A
d
-
N
-
I
-
R
)
I
(
x
,
t
-
τ
)
1
+
α
I
(
x
,
t
-
τ
)
mmm
-
(
γ
+
μ
+
d
)
I
(
x
,
t
)
,
∂
R
∂
t
=
D
∂
2
R
∂
x
2
+
γ
I
(
x
,
t
)
-
(
δ
+
d
)
R
(
x
,
t
)
.Consider the equilibrium equation of system (5):
(6)
μ
I
-
d
N
=
0
,
β
(
A
d
-
N
-
I
-
R
)
I
1
+
α
I
-
(
γ
+
μ
+
d
)
I
=
0
,
γ
I
-
(
δ
+
d
)
R
=
0
.
Obviously, system (5) often has a trivial equilibrium E
0
(
0,0
,
0
). From the first and the third equation of (6), we know that N
=
(
μ
/
d
)
I, R
=
(
γ
/
(
δ
+
d
)
)
I. Substituting the expressions into the second equation of (6), if ℜ
0
:
=
A
β
/
d
(
γ
+
μ
+
d
)
>
1, we get a positive equilibrium E
*
(
k
1
,
k
2
,
k
3
) of system (5), where
(7)
k
1
=
μ
(
δ
+
d
)
[
A
β
-
d
(
γ
+
μ
+
d
)
]
m
m
m
×
(
+
d
2
α
(
γ
+
μ
+
d
)
(
δ
+
d
)
d
β
[
δ
(
μ
+
d
)
+
d
(
γ
+
μ
+
d
)
]
m
m
m
m
m
+
d
2
α
(
γ
+
μ
+
d
)
(
δ
+
d
)
)
-
1
,
k
2
=
(
δ
+
d
)
[
A
β
-
d
(
γ
+
μ
+
d
)
]
β
[
δ
(
μ
+
d
)
+
d
(
γ
+
μ
+
d
)
]
+
d
α
(
γ
+
μ
+
d
)
(
δ
+
d
)
,
k
3
=
γ
[
A
β
-
d
(
γ
+
μ
+
d
)
]
β
[
δ
(
μ
+
d
)
+
d
(
γ
+
μ
+
d
)
]
+
d
α
(
γ
+
μ
+
d
)
(
δ
+
d
)
.By calculating, we can obtain thatk
1
+
k
2
+
k
3
<
A
/
d, which is important in the following text. In fact,
(8)
k
1
+
k
2
+
k
3
=
[
A
β
-
d
(
γ
+
μ
+
d
)
]
β
[
δ
(
μ
+
d
)
+
d
(
γ
+
μ
+
d
)
]
+
d
α
(
γ
+
μ
+
d
)
(
δ
+
d
)
mm
×
[
μ
d
(
δ
+
d
)
+
(
δ
+
d
)
+
γ
]
<
[
A
β
-
d
(
γ
+
μ
+
d
)
]
β
(
μ
δ
+
μ
d
+
δ
d
+
γ
d
+
d
2
)
mm
×
1
d
(
μ
δ
+
μ
d
+
δ
d
+
γ
d
+
d
2
)
=
[
A
β
-
d
(
γ
+
μ
+
d
)
]
β
d
<
A
d
.Now, we study the existence of traveling wave solutions for system (5) connecting E
0 and E
*.SubstitutingN
(
x
,
t
)
=
ϕ
(
x
+
c
t
), I
(
x
,
t
)
=
φ
(
x
+
c
t
), R
(
x
,
t
)
=
ψ
(
x
+
c
t
) into (5), and denoting x
+
c
t still by t, we derive the following wave profile system from (5):
(9)
D
ϕ
″
(
t
)
-
c
ϕ
′
(
t
)
-
d
ϕ
(
t
)
+
μ
φ
(
t
)
=
0
,
D
φ
″
(
t
)
-
c
φ
′
(
t
)
+
β
(
A
d
-
ϕ
(
t
)
-
φ
(
t
)
-
ψ
(
t
)
)
φ
(
t
-
c
τ
)
1
+
α
φ
(
t
-
c
τ
)
-
(
γ
+
μ
+
d
)
φ
(
t
)
=
0
,
D
ψ
″
(
t
)
-
c
ψ
′
(
t
)
+
γ
φ
(
t
)
-
(
δ
+
d
)
ψ
(
t
)
=
0
.Note thatℜ
0
>
1 imply A
β
/
d
-
(
γ
+
μ
)
>
d. Moreover, we have
(10)
μ
k
2
k
1
=
d
,
γ
k
2
k
3
-
δ
=
d
.
We can select suitable M
1, M
2, M
3 such that M
i
>
k
i, i
=
1,2
,
3, which satisfy
(11)
A
β
d
-
(
γ
+
μ
)
>
μ
M
2
M
1
>
d
,
A
β
d
-
(
γ
+
μ
)
>
γ
M
2
M
3
-
δ
>
d
,
A
d
>
M
1
+
M
2
+
M
3
.DenoteC
[
0
,
M
]
(
ℝ
,
ℝ
3
)
=
{
(
ϕ
,
φ
,
ψ
)
∈
C
(
ℝ
,
ℝ
3
)
:
0
≤
(
ϕ
(
s
)
,
φ
(
s
)
,
ψ
(
s
)
)
≤
M
}, where M
=
(
M
1
,
M
2
,
M
3
).Denotef
=
(
f
1
,
f
2
,
f
3
)
:
C
[
0
,
M
]
(
ℝ
,
ℝ
3
)
→
C
(
ℝ
,
ℝ
3
):
(12)
f
1
(
ϕ
,
φ
,
ψ
)
(
t
)
=
-
d
ϕ
(
t
)
+
μ
φ
(
t
)
,
f
2
(
ϕ
,
φ
,
ψ
)
(
t
)
=
β
(
A
d
-
ϕ
(
t
)
-
φ
(
t
)
-
ψ
(
t
)
)
m
m
m
m
m
m
m
m
n
×
φ
(
t
-
c
τ
)
1
+
α
φ
(
t
-
c
τ
)
-
(
γ
+
μ
+
d
)
φ
(
t
)
,
f
3
(
ϕ
,
φ
,
ψ
)
(
t
)
=
γ
φ
(
t
)
-
(
δ
+
d
)
ψ
(
t
)
.For(
ϕ
,
φ
,
ψ
)
∈
C
[
0
,
M
]
(
ℝ
,
ℝ
3
), by a careful calculation, we have
(13)
|
f
1
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
-
f
1
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
|
≤
(
d
+
μ
)
|
Φ
(
t
)
-
Ψ
(
t
)
|
ℝ
3
,
|
f
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
-
f
2
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
|
≤
L
|
Φ
(
t
)
-
Ψ
(
t
)
|
ℝ
3
,
|
f
3
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
-
f
3
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
|
≤
(
γ
+
δ
+
d
)
|
Φ
(
t
)
-
Ψ
(
t
)
|
ℝ
3
,
where L
:
=
A
β
/
d
+
β
(
1
+
α
M
2
2
+
M
1
+
M
3
+
2
M
2
+
2
M
2
/
(
1
+
α
M
2
)
)
+
(
γ
+
μ
+
d
), Φ
=
(
ϕ
1
,
φ
1
,
ψ
1
), Ψ
=
(
ϕ
2
,
φ
2
,
ψ
2
).For the positive constantsρ
1, ρ
2, ρ
3, we define H
:
C
[
0
,
M
]
(
ℝ
,
ℝ
3
)
→
C
(
ℝ
,
ℝ
3
) by
(14)
H
1
(
ϕ
,
φ
,
ψ
)
(
t
)
=
f
1
(
ϕ
,
φ
,
ψ
)
(
t
)
+
ρ
1
ϕ
(
t
)
,
H
2
(
ϕ
,
φ
,
ψ
)
(
t
)
=
f
2
(
ϕ
,
φ
,
ψ
)
(
t
)
+
ρ
2
φ
(
t
)
,
H
3
(
ϕ
,
φ
,
ψ
)
(
t
)
=
f
3
(
ϕ
,
φ
,
ψ
)
(
t
)
+
ρ
3
ψ
(
t
)
.Then operatorsH
1, H
2, H
3 have the following properties.Lemma 1.
For0
≤
ϕ
2
(
t
)
≤
ϕ
1
(
t
)
≤
M
1, 0
≤
φ
2
(
t
)
≤
φ
1
(
t
)
≤
M
2, 0
≤
ψ
2
(
t
)
≤
ψ
1
(
t
)
≤
M
3, one has(i)
(15)
H
1
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
≥
H
1
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
,
H
3
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
≥
H
3
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
;
(ii)
(16)
H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
≥
H
2
(
ϕ
1
,
φ
2
,
ψ
1
)
(
t
)
,
H
2
(
ϕ
2
,
φ
1
,
ψ
1
)
(
t
)
≥
H
1
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
,
H
2
(
ϕ
1
,
φ
1
,
ψ
2
)
(
t
)
≥
H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
.Proof.
According to the definitions off and H, we have
(17)
H
1
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
-
H
1
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
=
(
ρ
1
-
d
)
(
ϕ
1
(
t
)
-
ϕ
2
(
t
)
)
+
μ
(
φ
1
(
t
)
-
φ
2
(
t
)
)
,
H
3
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
-
H
3
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
=
(
ρ
3
-
d
-
δ
)
(
ψ
1
(
t
)
-
ψ
2
(
t
)
)
+
γ
(
ϕ
1
(
t
)
-
ϕ
2
(
t
)
)
.
Let ρ
1
=
d, ρ
3
=
d
+
δ; we obtain the properties for H
1 and H
3.
For (ii), we have(18)
H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
-
H
2
(
ϕ
1
,
φ
2
,
ψ
1
)
(
t
)
=
β
(
A
d
-
ϕ
1
(
t
)
-
ψ
1
(
t
)
)
×
(
φ
1
(
t
-
c
τ
)
1
+
α
φ
1
(
t
-
c
τ
)
-
φ
1
(
t
-
c
τ
)
1
+
α
φ
1
(
t
-
c
τ
)
)
-
β
(
φ
1
(
t
)
φ
1
(
t
-
c
τ
)
1
+
α
φ
1
(
t
-
c
τ
)
-
φ
2
(
t
)
φ
2
(
t
-
c
τ
)
1
+
α
φ
2
(
t
-
c
τ
)
)
+
[
ρ
2
-
(
γ
+
μ
+
d
)
]
(
φ
1
(
t
)
-
φ
2
(
t
)
)
.
Note that M
1
+
M
3
<
A
/
d, and x
/
(
1
+
α
x
) is nondecreasing; we have that the first term of the last formula is nonnegative, and the second term is bigger than -
(
β
M
2
/
(
1
+
α
M
2
)
)
(
φ
1
(
t
)
-
φ
2
(
t
)
). Let ρ
2
=
β
M
2
/
(
1
+
α
M
2
)
+
γ
+
μ
+
d; we have H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
≥
H
2
(
ϕ
1
,
φ
2
,
ψ
1
)
(
t
). Since
(19)
H
2
(
ϕ
2
,
φ
1
,
ψ
1
)
(
t
)
-
H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
=
[
β
φ
1
(
t
-
c
τ
)
1
+
α
φ
1
(
t
-
c
τ
)
+
ρ
2
]
(
φ
1
(
t
)
-
φ
2
(
t
)
)
,
H
2
(
ϕ
1
,
φ
1
,
ψ
2
)
(
t
)
-
H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
=
[
β
φ
1
(
t
-
c
τ
)
1
+
α
φ
1
(
t
-
c
τ
)
+
ρ
2
]
(
ψ
1
(
t
)
-
ψ
2
(
t
)
)
,
then, for any positive constant ρ
2, we have H
2
(
ϕ
2
,
φ
1
,
ψ
1
)
(
t
)
≥
H
1
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
), H
2
(
ϕ
1
,
φ
1
,
ψ
2
)
(
t
)
≥
H
2
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
).Remark 2.
ForH
2, we can further conclude that H
2
(
ϕ
2
,
φ
1
,
ψ
2
)
(
t
)
≥
H
2
(
ϕ
1
,
φ
2
,
ψ
1
)
(
t
) from Lemma 1(ii).According to the definition ofH, system (9) can be written as
(20)
D
ϕ
″
(
t
)
-
c
ϕ
′
(
t
)
-
ρ
1
ϕ
(
t
)
+
H
1
(
ϕ
,
φ
,
ψ
)
(
t
)
=
0
,
D
φ
″
(
t
)
-
c
φ
′
(
t
)
-
ρ
2
φ
(
t
)
+
H
2
(
ϕ
,
φ
,
ψ
)
(
t
)
=
0
,
D
ψ
″
(
t
)
-
c
ψ
′
(
t
)
-
ρ
3
ψ
(
t
)
+
H
3
(
ϕ
,
φ
,
ψ
)
(
t
)
=
0
.Define(21)
λ
1
=
c
-
c
2
+
4
ρ
1
D
2
D
,
λ
2
=
c
+
c
2
+
4
ρ
1
D
2
D
,
λ
3
=
c
-
c
2
+
4
ρ
2
D
2
D
,
λ
4
=
c
+
c
2
+
4
ρ
2
D
2
D
,
λ
5
=
c
-
c
2
+
4
ρ
3
D
2
D
,
λ
6
=
c
+
c
2
+
4
ρ
3
D
2
D
,
and operator F
=
(
F
1
,
F
2
,
F
3
)
:
C
[
0
,
M
]
(
ℝ
,
ℝ
3
)
→
C
(
ℝ
,
ℝ
3
) by
(22)
F
1
(
ϕ
,
φ
,
ψ
)
(
t
)
=
1
D
(
λ
2
-
λ
1
)
[
∫
-
∞
t
‍
e
λ
1
(
t
-
s
)
H
1
(
ϕ
,
φ
,
ψ
)
(
s
)
d
s
mmmmmmmmm
+
∫
t
+
∞
‍
e
λ
2
(
t
-
s
)
H
1
(
ϕ
,
φ
,
ψ
)
(
s
)
d
s
]
,
F
2
(
ϕ
,
φ
,
ψ
)
(
t
)
=
1
D
(
λ
4
-
λ
3
)
[
∫
-
∞
t
‍
e
λ
3
(
t
-
s
)
H
2
(
ϕ
,
φ
,
ψ
)
(
s
)
d
s
mmmmmmmmm
+
∫
t
+
∞
‍
e
λ
4
(
t
-
s
)
H
2
(
ϕ
,
φ
,
ψ
)
(
s
)
d
s
]
,
F
3
(
ϕ
,
φ
,
ψ
)
(
t
)
=
1
D
(
λ
6
-
λ
5
)
[
∫
-
∞
t
‍
e
λ
5
(
t
-
s
)
H
3
(
ϕ
,
φ
,
ψ
)
(
s
)
d
s
mmmmmmmmm
+
∫
t
+
∞
‍
e
λ
6
(
t
-
s
)
H
3
(
ϕ
,
φ
,
ψ
)
(
s
)
d
s
]
.
It is easy to see that F
i
(
ϕ
,
φ
,
ψ
)
(
t
)
(
i
=
1,2
,
3
) satisfy system (20); thus the fixed point of operator F satisfies (9), which is a traveling wave solution of system (5). Therefore, we will use Schauder's fixed point theorem to find the fixed point of F, where the continuity of F is required. For this purpose, let ν
>
0; we define a norm for Φ
(
t
)
=
(
ϕ
,
φ
,
ψ
)
(
t
)
∈
C
(
ℝ
,
ℝ
3
) by
(23)
|
Φ
|
ν
=
sup
⁡
t
∈
ℝ
e
-
ν
|
t
|
∥
Φ
(
t
)
∥
.Define(24)
B
ν
(
ℝ
,
ℝ
3
)
=
{
Φ
∈
C
(
ℝ
,
ℝ
3
)
:
|
Φ
|
ν
<
∞
}
.
Then it is obvious that (
B
ν
(
ℝ
,
ℝ
3
)
,
|
·
|
ν
) is a Banach space. We also need the following definition of upper and lower solutions for system (9).Definition 3.
A pair of continuous functionsΦ
¯
=
(
ϕ
¯
,
φ
¯
,
ψ
¯
) and Φ
_
=
(
ϕ
_
,
φ
_
,
ψ
_
) are called an upper solution and a lower solution of (9), respectively, if there exist finite points T
1
,
T
2
,
…
,
T
m such that Φ
¯, Φ
_ are twice differentiable and bounded on ℝ
∖
{
T
i
}, i
=
1,2
,
…
,
m, and satisfy
(25)
D
ϕ
¯
″
(
t
)
-
c
ϕ
¯
′
(
t
)
+
f
1
(
ϕ
¯
,
φ
¯
,
ψ
_
)
(
t
)
≤
0
,
D
φ
¯
″
(
t
)
-
c
φ
¯
′
(
t
)
+
f
2
(
ϕ
_
,
φ
¯
,
ψ
_
)
(
t
)
≤
0
,
D
ψ
-
″
(
t
)
-
c
ψ
¯
′
(
t
)
+
f
3
(
ϕ
¯
,
φ
¯
,
ψ
¯
)
(
t
)
≤
0
,
D
ϕ
_
″
(
t
)
-
c
ϕ
_
′
(
t
)
+
f
1
(
ϕ
_
,
φ
_
,
ψ
_
)
(
t
)
≥
0
,
D
φ
_
″
(
t
)
-
c
φ
_
′
(
t
)
+
f
2
(
ϕ
¯
,
φ
_
,
ψ
¯
)
(
t
)
≥
0
,
D
ψ
_
″
(
t
)
-
c
ψ
_
′
(
t
)
+
f
3
(
ϕ
_
,
φ
_
,
ψ
_
)
(
t
)
≥
0
,
for t
∈
ℝ
∖
{
T
i
}, respectively.
We assume that a pair of upper-lower solutionsΦ
¯
=
(
ϕ
¯
,
φ
¯
,
ψ
¯
) and Φ
_
=
(
ϕ
_
,
φ
_
,
ψ
_
) are given such that(P1)
(26)
(
0,0
,
0
)
≤
(
ϕ
_
(
t
)
,
φ
_
(
t
)
,
ψ
_
(
t
)
)
≤
(
ϕ
¯
(
t
)
,
φ
¯
(
t
)
,
ψ
¯
(
t
)
)
≤
(
M
1
,
M
2
,
M
3
)
;
(P2)
(27)
lim
⁡
t
→
-
∞
(
ϕ
_
(
t
)
,
φ
_
(
t
)
,
ψ
_
(
t
)
)
=
lim
⁡
t
→
-
∞
(
ϕ
¯
(
t
)
,
φ
¯
(
t
)
,
ψ
¯
(
t
)
)
=
(
0,0
,
0
)
,
lim
⁡
t
→
+
∞
(
ϕ
_
(
t
)
,
φ
_
(
t
)
,
ψ
_
(
t
)
)
=
lim
⁡
t
→
+
∞
(
ϕ
¯
(
t
)
,
φ
¯
(
t
)
,
ψ
¯
(
t
)
)
=
(
k
1
,
k
2
,
k
3
)
;
(P3)
(28)
Φ
¯
′
(
t
+
)
≤
Φ
¯
′
(
t
-
)
,
Φ
_
′
(
t
+
)
≥
Φ
_
′
(
t
-
)
for
t
∈
ℝ
.
Define the set(29)
Γ
=
{
(
ϕ
,
φ
,
ψ
)
(
t
)
∈
C
[
0
,
M
]
(
ℝ
,
ℝ
3
)
:
ϕ
_
(
t
)
≤
ϕ
(
t
)
≤
ϕ
¯
(
t
)
,
m
φ
_
(
t
)
≤
φ
(
t
)
≤
φ
¯
(
t
)
,
ψ
_
(
t
)
≤
ψ
(
t
)
≤
ψ
¯
(
t
)
}
.
Then by the property of H, we have the property of F.Lemma 4.
For0
≤
ϕ
2
(
t
)
≤
ϕ
1
(
t
)
≤
M
1, 0
≤
φ
2
(
t
)
≤
φ
1
(
t
)
≤
M
2, 0
≤
ψ
2
(
t
)
≤
ψ
1
(
t
)
≤
M
3, one has
(30)
F
1
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
≥
F
1
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
,
F
2
(
ϕ
2
,
φ
1
,
ψ
2
)
(
t
)
≥
F
2
(
ϕ
1
,
φ
2
,
ψ
1
)
(
t
)
,
F
3
(
ϕ
1
,
φ
1
,
ψ
1
)
(
t
)
≥
F
3
(
ϕ
2
,
φ
2
,
ψ
2
)
(
t
)
.Similar to Lemmas 4.4–4.6 in [7], we have the following lemmas and omit their proofs.Lemma 5.
F
=
(
F
1
,
F
2
,
F
3
) is continuous with respect to the norm |
·
|
ν in (
B
ν
(
ℝ
,
ℝ
3
)
).Lemma 6.
ConsiderF
:
Γ
→
Γ.Lemma 7.
F
:
Γ
→
Γ is compact with respect to the norm |
·
|
ν.Theorem 8.
Assume thatA
β
/
d
(
γ
+
μ
+
d
)
>
1. If (9) has a pair of upper-lower solutions Φ
¯
=
(
ϕ
¯
,
φ
¯
,
ψ
¯
) and Φ
_
=
(
ϕ
_
,
φ
_
,
ψ
_
) and satisfies (P1), (P2), and (P3), then system (5) has a traveling wave solution.Proof.
According to Lemmas5–7 and applying Schauder’s fixed point theorem, operator F has a fixed point Φ
*
=
(
ϕ
*
,
φ
*
,
ψ
*
)
∈
Γ, which is traveling wave for system (5). Furthermore, by (P2) we have
(31)
lim
⁡
t
→
-
∞
(
ϕ
*
,
φ
*
,
ψ
*
)
=
(
0,0
,
0
)
,
lim
⁡
t
→
+
∞
(
ϕ
*
,
φ
*
,
ψ
*
)
=
(
k
1
,
k
2
,
k
3
)
.
Therefore, the fixed point is a traveling wave solution for (5) connecting E
0 and E
*. The proof is complete.


## 3. Existence of Traveling Waves

To prove the existence of traveling wave solutions for (5), we only need to construct a pair of upper-lower solutions.Consider the following functions:(32)
Δ
1
(
η
,
c
)
=
D
η
2
-
c
η
-
d
+
μ
M
2
M
1
,
Δ
2
(
η
,
c
)
=
D
η
2
-
c
η
+
A
β
d
-
(
γ
+
μ
+
d
)
,
Δ
3
(
η
,
c
)
=
D
η
2
-
c
η
-
(
δ
+
d
)
+
γ
M
2
M
3
.
Note that (11) and ℜ
0
>
1; we know that there exist positive numbers c
1
*, c
2
*, c
3
* such that
(33)
Δ
1
(
η
,
c
)
=
0
has
two
zeros
0
<
η
1
<
η
2
,
for
c
>
c
1
*
,
Δ
2
(
η
,
c
)
=
0
has
two
zeros
0
<
η
3
<
η
4
,
for
c
>
c
2
*
,
Δ
3
(
η
,
c
)
=
0
has
two
zeros
0
<
η
5
<
η
6
,
for
c
>
c
3
*
.Denotec
*
=
max
⁡
{
c
1
*
,
c
2
*
,
c
3
*
}. According to [5, Lemma 3.8], we have η
1
<
η
3 and η
5
<
η
3.Assume thatμ
/
d
+
γ
/
(
δ
+
d
)
<
1; we can select ε
i
>
0
(
i
=
1,2
,
…
,
6
), ε
1
,
ε
2
∈
(
0
,
k
1
), ε
3
,
ε
4
∈
(
0
,
k
2
), ε
5
,
ε
6
∈
(
0
,
k
3
) satisfying the following inequalities:
(34)
μ
(
k
2
+
ε
3
)
-
d
(
k
1
+
ε
1
)
<
0
,
β
(
A
d
-
k
1
+
ε
2
-
k
2
-
ε
3
-
k
3
+
ε
6
)
-
(
γ
+
μ
+
d
)
<
0
,
γ
(
k
2
+
ε
3
)
-
(
δ
+
d
)
(
k
3
+
ε
5
)
<
0
,
d
(
k
1
-
ε
2
)
-
μ
(
k
2
-
ε
4
)
<
0
,
(
γ
+
μ
+
d
)
-
β
1
+
α
(
k
2
-
ε
4
)
m
m
m
h
m
×
(
A
d
-
k
1
-
ε
1
-
k
2
+
ε
4
-
k
3
-
ε
5
)
<
0
,
(
δ
+
d
)
(
k
3
-
ε
6
)
-
γ
(
k
2
-
ε
4
)
<
0
.In fact, we first chooseε
3
,
ε
4
∈
(
0
,
k
2
) such that
(35)
(
μ
d
+
γ
δ
+
d
)
ε
3
<
ε
4
,
(
μ
d
+
γ
δ
+
d
)
ε
4
<
ε
3
.Forε
3
∈
(
0
,
k
2
), noting that k
1
=
(
μ
/
d
)
k
2 and k
3
=
(
γ
/
(
δ
+
d
)
)
k
2, we can find ε
1
∈
(
0
,
k
1
), ε
5
∈
(
0
,
k
3
) such that
(36)
k
1
>
ε
1
>
μ
d
ε
3
=
μ
d
(
k
2
+
ε
3
)
-
k
1
mmmmmmm
⟹
μ
(
k
2
+
ε
3
)
-
d
(
k
1
+
ε
1
)
<
0
,
k
3
>
ε
5
>
γ
δ
+
d
ε
3
=
γ
δ
+
d
(
k
2
+
ε
3
)
-
k
3
mmmmmmmnm
⟹
γ
(
k
2
+
ε
3
)
-
(
δ
+
d
)
(
k
3
+
ε
5
)
<
0
.Forε
4
∈
(
0
,
k
2
), we can find ε
2
∈
(
0
,
k
1
), ε
6
∈
(
0
,
k
3
) such that
(37)
k
1
>
ε
2
>
μ
d
ε
4
=
k
1
-
μ
d
(
k
2
-
ε
4
)
mmmmmmm
⟹
d
(
k
1
-
ε
2
)
-
μ
(
k
2
-
ε
4
)
<
0
,
k
3
>
ε
6
>
γ
δ
+
d
ε
4
=
k
3
-
γ
δ
+
d
(
k
2
-
ε
4
)
mmmmmmmmn
⟹
(
δ
+
d
)
(
k
3
-
ε
6
)
-
γ
(
k
2
-
ε
4
)
<
0
.Furthermore, forε
1
>
(
μ
/
d
)
ε
3, ε
5
>
(
γ
/
(
δ
+
d
)
)
ε
3 and (35), we can find suitable ε
1, ε
5 satisfying ε
1
+
ε
5
<
ε
4. Similarly, we can find suitable ε
2, ε
6 satisfying ε
2
+
ε
6
<
ε
3. Thus we have
(38)
β
(
A
d
-
k
1
+
ε
2
-
k
2
-
ε
3
-
k
3
+
ε
6
)
-
(
γ
+
μ
+
d
)
<
0
,
(
γ
+
μ
+
d
)
-
β
1
+
α
(
k
2
-
ε
4
)
×
(
A
d
-
k
1
-
ε
1
-
k
2
+
ε
4
-
k
3
-
ε
5
)
<
0
.We define continuous functionsΦ
(
t
)
=
(
ϕ
1
(
t
)
,
φ
1
(
t
)
,
ψ
1
(
t
)
) and Ψ
(
t
)
=
(
ϕ
2
(
t
)
,
φ
2
(
t
)
,
ψ
2
(
t
)
) as follows:
(39)
ϕ
1
(
t
)
=
{
k
1
e
η
1
t
,
t
≤
t
1
,
k
1
+
ε
1
e
-
η
t
,
t
>
t
1
,
ϕ
2
(
t
)
=
{
0
,
t
≤
t
2
,
k
1
-
ε
2
e
-
η
t
,
t
>
t
2
,
φ
1
(
t
)
=
{
k
2
e
η
3
t
,
t
≤
t
3
,
k
2
+
ε
3
e
-
η
t
,
t
>
t
3
,
φ
2
(
t
)
=
{
0
,
t
≤
t
4
,
k
2
-
ε
4
e
-
η
t
,
t
>
t
4
,
ψ
1
(
t
)
=
{
k
3
e
η
5
t
,
t
≤
t
5
,
k
3
+
ε
5
e
-
η
t
,
t
>
t
5
,
ψ
2
(
t
)
=
{
0
,
t
≤
t
6
,
k
3
-
ε
6
e
-
η
t
,
t
>
t
6
,
where t
1
,
t
3
,
t
5
>
0, t
2
,
t
4
,
t
6
<
0, and η
>
0 is a proper constant to be chosen later.Furthermore, we can conclude thatt
3
≥
max
⁡
{
t
1
,
t
5
} and t
4
≤
min
⁡
{
t
2
,
t
6
}, which can help us verify the upper-lower solution for system (9). We point out that Φ
(
t
) and Ψ
(
t
) satisfy (P1), (P2), and (P3) for proper parameters.Lemma 9.
Supposeμ
/
d
+
γ
/
(
δ
+
d
)
<
1. Then the functions Φ
(
t
) and Ψ
(
t
) defined above are upper and lower solutions of (9), respectively.Proof.
Ift
≤
t
1, ϕ
1
(
t
)
=
k
1
e
η
1
t, φ
1
(
t
)
=
k
2
e
η
3
t, we have
(40)
p
1
(
t
)
:
=
D
ϕ
1
″
(
t
)
-
c
ϕ
1
′
(
t
)
-
d
ϕ
1
(
t
)
+
μ
φ
1
(
t
)
≤
(
D
η
1
2
-
c
η
1
-
d
+
μ
M
2
M
1
)
k
1
e
η
1
t
=
0
.
If t
>
t
1, ϕ
1
(
t
)
=
k
1
+
ε
1
e
-
η
t, φ
1
(
t
)
≤
k
2
+
ε
3
e
-
η
t, we know
(41)
p
1
(
t
)
≤
I
1
(
η
)
,
where I
1
(
η
)
=
(
D
ε
1
η
2
+
c
ε
1
η
)
e
-
η
t
-
d
(
k
1
+
ε
1
e
-
η
t
)
+
μ
(
k
2
+
ε
3
e
-
η
t
). Then I
1
(
0
)
=
μ
(
k
2
+
ε
3
)
-
d
(
k
1
+
ε
1
). It follows from (34) that I
1
(
0
)
<
0 and there exists η
1
*
>
0 such that p
1
(
t
)
<
0 for all η
∈
(
0
,
η
1
*
).
Ift
≤
t
3, φ
1
(
t
)
=
k
2
e
η
3
t, we obtain that
(42)
p
2
(
t
)
:
=
D
φ
1
′
′
(
t
)
-
c
φ
1
′
(
t
)
mmmm
+
β
(
A
d
-
ϕ
2
(
t
)
-
φ
1
(
t
)
-
ψ
2
(
t
)
)
φ
1
(
t
-
c
τ
)
1
+
α
φ
1
(
t
-
c
τ
)
mmmm
-
(
γ
+
μ
+
d
)
φ
1
(
t
)
mmm
≤
(
D
η
2
3
-
c
η
3
-
γ
-
μ
-
d
)
k
2
e
η
3
t
+
A
β
d
k
2
e
η
3
(
t
-
c
τ
)
mmm
≤
[
D
η
2
3
-
c
η
3
+
A
β
d
-
γ
-
μ
-
d
]
k
2
e
η
3
t
=
0
.
If t
>
t
3, φ
1
(
t
)
=
k
2
+
ε
3
e
-
η
t, ϕ
2
(
t
)
=
k
1
-
ε
2
e
-
η
t, ψ
2
(
t
)
=
k
3
-
ε
6
e
-
η
t, we have
(43)
p
2
(
t
)
≤
I
2
(
η
)
,
where
(44)
I
2
(
η
)
=
(
D
ε
3
η
2
+
c
ε
3
η
)
e
-
η
t
mmmm
-
(
γ
+
μ
+
d
)
(
k
2
+
ε
3
e
-
η
t
)
mmmm
+
β
[
A
d
-
k
1
+
ε
2
e
-
η
t
-
k
2
-
ε
3
e
-
η
t
-
k
3
+
ε
6
e
-
η
t
]
mmmm
×
(
k
2
+
ε
3
e
-
η
(
t
-
c
τ
)
)
.
It follows from (34) that I
2
(
0
)
<
0 and there exists η
2
*
>
0 such that p
2
(
t
)
<
0 for all η
∈
(
0
,
η
2
*
).
Ift
≤
t
5, ψ
1
(
t
)
=
k
3
e
η
5
t, φ
1
(
t
)
=
k
2
e
η
3
t, we have
(45)
p
3
(
t
)
:
=
D
ψ
1
′
′
(
t
)
-
c
ψ
1
′
(
t
)
+
γ
φ
1
(
t
)
-
(
δ
+
d
)
ψ
1
(
t
)
≤
[
D
η
5
2
-
c
η
5
-
(
δ
+
d
)
+
γ
M
2
M
3
]
k
3
e
η
5
t
=
0
.
If t
>
t
5, ψ
1
(
t
)
=
k
3
+
ε
5
e
-
η
t, φ
1
(
t
)
≤
k
2
+
ε
3
e
-
η
t,
(46)
p
3
(
t
)
≤
I
3
(
η
)
,
where I
3
(
η
)
=
(
D
ε
5
η
2
+
c
ε
5
η
)
e
-
η
t
+
γ
(
k
2
+
ε
3
e
-
η
t
)
-
(
δ
+
d
)
(
k
3
+
ε
5
e
-
η
t
). We can derive from (34) that there exists η
3
*
>
0 such that p
3
(
t
)
<
0 for all η
∈
(
0
,
η
3
*
).
Ift
≤
t
2, ϕ
2
(
t
)
=
0, we have
(47)
q
1
(
t
)
:
=
D
ϕ
2
′
′
(
t
)
-
c
ϕ
2
′
(
t
)
-
d
ϕ
2
(
t
)
+
μ
φ
2
(
t
)
=
μ
φ
2
(
t
)
≥
0
.
If t
>
t
2, ϕ
2
(
t
)
=
k
1
-
ε
2
e
-
η
t, φ
2
(
t
)
=
k
2
-
ε
4
e
-
η
t, we have
(48)
q
1
(
t
)
≥
I
4
(
η
)
,
where I
4
(
η
)
=
-
(
D
ε
2
η
2
+
c
ε
2
η
)
e
-
η
t
-
d
(
k
1
-
ε
2
e
-
η
t
)
+
μ
(
k
2
-
ε
4
e
-
η
t
). It follows from (34) that I
4
(
0
)
>
0 and there exists η
4
*
>
0 such that q
1
(
t
)
>
0 for all η
∈
(
0
,
η
4
*
).
Ift
≤
t
4, φ
2
(
t
)
=
0, we obtain that
(49)
q
2
(
t
)
:
=
D
φ
2
′
′
(
t
)
-
c
φ
2
′
(
t
)
mmmm
+
β
(
A
d
-
ϕ
1
(
t
)
-
φ
2
(
t
)
-
ψ
1
(
t
)
)
φ
2
(
t
-
c
τ
)
1
+
α
φ
2
(
t
-
c
τ
)
mmmm
-
(
γ
+
μ
+
d
)
φ
2
(
t
)
=
0
.
If t
>
t
4, φ
2
(
t
)
=
k
2
-
ε
4
e
-
η
t, φ
2
(
t
-
c
τ
)
≥
k
2
-
ε
4
e
-
η
(
t
-
c
τ
), ϕ
1
(
t
)
≤
k
1
+
ε
1
e
-
η
t, ψ
1
(
t
)
≤
k
3
+
ε
5
e
-
η
t, we have
(50)
q
2
(
t
)
≥
I
5
(
η
)
,
where
(51)
I
5
(
η
)
=
-
(
D
ε
4
η
2
+
c
ε
4
η
)
e
-
η
t
-
(
γ
+
μ
+
d
)
(
k
2
-
ε
4
e
-
η
t
)
+
β
(
A
d
-
k
1
-
ε
1
e
-
η
t
-
k
2
+
ε
4
e
-
η
t
-
k
3
m
m
m
-
ε
5
e
-
η
t
A
d
)
k
2
-
ε
4
e
-
η
(
t
-
c
τ
)
1
+
α
k
2
-
α
ε
4
e
-
η
(
t
-
c
τ
)
.
It follows from (34) that I
5
(
0
)
>
0 and there exists η
5
*
>
0 such that q
2
(
t
)
>
0 for all η
∈
(
0
,
η
5
*
).
Ift
≤
t
6, ψ
2
(
t
)
=
0, we have
(52)
q
3
(
t
)
:
=
D
ψ
2
′
′
(
t
)
-
c
ψ
2
′
(
t
)
+
γ
φ
2
(
t
)
-
(
δ
+
d
)
ψ
2
(
t
)
≥
0
.
If t
>
t
6, ψ
2
(
t
)
=
k
3
-
ε
6
e
-
η
t, φ
2
(
t
)
=
k
2
-
ε
4
e
-
η
t, we have
(53)
q
3
(
t
)
≥
I
6
(
η
)
,
where I
6
(
η
)
=
-
(
D
ε
6
η
2
+
c
ε
6
η
)
e
-
η
t
+
γ
(
k
2
-
ε
4
e
-
η
t
)
-
(
δ
+
d
)
(
k
3
-
ε
6
e
-
η
t
). It follows from (34) that I
6
(
0
)
>
0 and there exists η
6
*
>
0 such that q
3
(
t
)
>
0 for all η
∈
(
0
,
η
6
*
).
Thus, takingη
∈
(
0
,
min
⁡
1
≤
i
≤
6
{
η
i
*
}
), we prove that Φ
(
t
) and Ψ
(
t
) are upper and lower solutions of (9).Now we obtain and state the main result in this paper.Theorem 10.
Assume thatA
β
/
d
(
γ
+
μ
+
d
)
>
1 and μ
/
d
+
γ
/
(
δ
+
d
)
<
1; then, for any c
>
c
*, (5) has a traveling wave solution connecting two equilibria E
0 and E
*. Furthermore, system (4) has a traveling wave solution with speed c, which connects two states (
A
/
d
,
0,0
) and (
A
/
d
-
k
1
,
k
2
,
k
3
).



---

*Source: 369072-2014-04-30.xml*