# Existence of Traveling Wave Solutions for Cholera Model

Authors: Tianran Zhang; Qingming Gou; Xiaoli Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

To investigate the spreading speed of cholera, Codeço’s cholera model (2001)
is developed by a reaction-diffusion model that incorporates both indirect
environment-to-human and direct human-to-human transmissions and the
pathogen diffusion. The two transmission incidences are supposed to be
saturated with infective density and pathogen density. The basic reproduction
numberR0 is defined and the formula for minimal wave speed c* is given. It
is proved by shooting method that there exists a traveling wave solution with
speed c for cholera model if and only if c≥c*.


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## Body

## 1. Introduction

Cholera has been a serious threat to human health in the past and at present, which is an acute, diarrheal illness caused by infection of the intestine with the bacteriumVibrio cholera. An estimated 3–5 million cases and over 100,000 deaths occur each year around the world [1]. The cholera bacterium is usually found in water or food sources that have been contaminated by feces from a person infected with cholera. Cholera is most likely to be found and to spread in places with inadequate water treatment, poor sanitation, and inadequate hygiene. Therefore, cholera outbreaks have occurred in developing countries, for example, Iraq (2007-2008), Guinea Bissau (2008), Zimbabwe (2008-2009), Haiti (2010), Democratic Republic of Congo (2011-2012), and Sierra Leone (2012) [2].To understand the propagation mechanism of cholera, many mathematical models were proposed, whose earlier one was established by Capasso and Paveri-Fontana [3] to study the 1973 cholera epidemic in the Mediterranean region as follows:
(1)dIdt=g(B)-a22I,dBdt=-a11B+a12I,
where B(t) and I(t) denote the concentrations of the pathogen and the infective populations, respectively. In addition, Codeço [4] investigated the role of the aquatic pathogen in dynamics of cholera through the following susceptible-infective-pathogen model:
(2)dSdt=n(H-S)-aSBK+B,dIdt=aSBK+B-rI,dBdt=eI-(mb-nb)B,
where S(t) is the susceptible individuals. In this model, human is divided into two groups: the susceptible group and the infective group. As pointed out in [4–8], bacteriumVibrio cholera can spread by direct human-to-human and indirect environment-to-human modes. To understand the complex dynamics of cholera, model (2) is extended by [8–15] and so forth.In all previous models the influences of space distribution of human on the transmission of cholera are omitted. Cholera usually spreads in spatial wave [16]. Cholera bacteria live in rivers and interact with the plankton on the surface of the water [17]. When individuals drink contaminated water and are infected, they will release cholera bacteria through excretion [18]. Capasso et al. [19–23] developed model (1) by incorporating the bacterium diffusion in a bounded area and studied the existence and stability of solutions. To deeply investigate the interaction of transmission modes and bacterium diffusion, Bertuzzo et al. [24, 25] incorporated patchy structure into model (2) and supposed that pathogen in water could diffuse among these patches. Furthermore, Mari et al. [26] studied the influence of diffusion of both human and pathogen on cholera dynamics through a patchy model.Infectious case is usually found firstly at some location and then spreads to other areas. Consequently, the most important question for cholera is what the spreading speed of cholera is. However, the above spatial models mainly focus on the stability of solutions not the spreading speed. Traveling wave solution is an important tool used to study the spreading speed of infectious diseases [27–29]. Based on Capasso’s model (1), Zhao and Wang [30], Xu and Zhao [31], Jin and Zhao [32], and Hsu and Yang [33] studied the influences of pathogen diffusion on the spread speed of cholera.The studies of traveling wave solutions of Capasso’s model (1) incorporating pathogen diffusion provide insight into the spreading speed of cholera. However, some pieces of information are omitted, such as the interaction of direct human-to-human and indirect environment-to-human transmissions. In this paper, a reaction-diffusion model with pathogen diffusion and both transmission paths is proposed by developing Codeço’s model (2). Based on model (2) and ignoring the disease-related death, a general diffusive cholera model can be formulated as the following reaction-diffusion system:
(3)∂S∂t=b(N-S)-f(I)S-g(B)S,∂I∂t=f(I)S+g(B)S-bI,∂B∂t=d∂2B∂x2+eI-mB,
where S=S(x,t) and I=I(x,t) denote the concentrations of susceptible and infected individuals, respectively, and B=B(x,t) is the concentration of the infectious agents. N is the total human population, b stands for the natural birth and death rate, e denotes the contribution of each infected person to the concentration of cholera, and m is the net death rate ofvibrio cholera. f(I) and g(B) are the human-to-human and environment-to-human transmission incidences, respectively. Similar to [10], we assume that f(I) and g(B) satisfy (A1)
f(0)=0, f′(I)≥0, f′′(I)≤0;(A2)
g(0)=0, g′(0)>0, g′(B)≥0, g′′(B)≤0, and g(B) is strictly monotonously increasing in [0,+∞).It is easy to conclude that f(I)≤f′(0)I, g(B)≤g′(0)B, and f(I)/I and g(B)/B are nonincreasing. Obviously, hypotheses (A1) and (A2) imply that the two transmission paths are saturated. In Tian and Wang [10], f(I) and g(B) have the following expressions:
(4)f(I)=β1I,g(B)=β2BK+B.
Obviously, as a special case, such selections satisfy (A1) and (A2).Shooting method is very important in proving the existence of traveling wave solutions, which was proposed by Dunbar [34, 35] and was applied to many models (e.g., [36–40]). In this paper, the existence of traveling wave solutions of system (3) will be proved by shooting method and the formula for minimal wave speed will be given.This paper is organized as follows. In next section, the main theorem and the formula for minimal wave speed will be given. In Section3, the nonexistence of the traveling wave solutions for c<c* is proved by geometric method. Section 4 is devoted to shooting arguments and the construction of Wazewski set. In Section 5, we prove the existence of traveling wave solutions for c>c* and then give the existence of traveling wave solution for c=c* by limit arguments. The final section is devoted to the simulations.


## 2. Main Results

For convenience, we introduce dimensionless variables and parameters. By setting(5)u1=SbN,u2=IbN,u3=mebNB,y=xd,
model (3) has the form
(6)u1,t=1-bu1-f1(u2)u1-g1(u3)u1,u2,t=f1(u2)u1+g1(u3)u1-bu2,u3,t=u3,yy+m(u2-u3),
where f1(u2)=f(bNu2) and g1(u3)=g(ebNu3/m).DenoteR0=[f1′(0)+g1′(0)]/b2, which is the basic reproduction number of (6). Then hypotheses (A1) and (A2) imply that system (6) has two nonnegative constant solutions P1(1/b,0,0) and P2(1/b-u*,u*,u*) if and only if R0>1, where u* is the only one positive root of equation
(7)[f1(u*)+g1(u*)](1b-u*)=bu*
and 0<u*<1/b. Biologically, P1 corresponds to disease-free equilibrium and P2 corresponds to endemic equilibrium. To study the spreading wave of cholera, it is assumed that R0>1 holds in this paper; that is
(8)f1′(0)+g1′(0)>b2.A traveling wave solution of system (6) with speed c is a nonnegative solution of the form
(9)u1(y,t)=u1(s),u2(y,t)=u2(s),u3(y,t)=u3(s),s=y+ct.Substituting traveling profile(u1(s),u2(s),u3(s)) into system (6) yields the following equations:
(10)cu1′=1-bu1-f1(u2)u1-g1(u3)u1,cu2′=f1(u2)u1+g1(u3)u1-bu2,cu3′=u3′′+m(u2-u3),
where ′ denotes d/ds. To investigate invasion question by cholera, we will study the positive solutions of (10) such that
(11)(u1(+∞),u2(+∞),u3(+∞))=(1b-u*,u*,u*),(u1(-∞),u2(-∞),u3(-∞))=(1b,0,0).Before giving the main theorem, we introduce the equation for minimal wave speed(12)Δ(c):=b3c6+b2c4+b1c2+b0=0,
where
(13)ϵ=f1′(0)-b2,b3=b2ϵ2+2b3m(f1′(0)+g1′(0)-b2)+2b3mg1′+b4m2,b2=-2bϵ3+2b2mϵ2+(8b3m2-6b2mg1′)ϵ+4m3b4+18b3m2g1′(0),b1=ϵ4-8mbϵ3-(8b2m2+6bmg1′)ϵ2-36b2m2g1′(0)ϵ-27m2b2g1′(0)2,b0=4m(b2-f1′(0))3(b2-f1′(0)-g1′(0)).Theorem 1.
There exists a constantc*>0 which is the greatest positive root of (12). When c≥c*, system (6) has a traveling wave solution satisfying boundary condition (11). When 0<c<c*, system (6) has no traveling wave solutions satisfying boundary condition (11).


## 3. Nonexistence of Traveling Wave Solutions forc<c*

From (10), we have
(14)[u1(s)+u2(s)]′=[1-b(u1(s)+u2(s))]c.Consequently, ifu1(0)+u2(0)≠1/b, then
(15)|u1(s)+u2(s)|⟶∞whens⟶-∞.Hence, the traveling profile(u1(s),u2(s),u3(s)) with boundary condition (11) must satisfy
(16)u1(s)+u2(s)=1bforanys∈R.Therefore, to study traveling wave solutions we assume (16) satisfies. Setting u3′=z in system (10) and noticing (16), it follows
(17)u2′=[(f1(u2)+g1(u3))((1/b)-u2)-bu2]c,u3′=z,z′=cz+m(u3-u2).Ifu1(s)=0, then u1′(s)=1/c>0 by system (10). Therefore, we suppose u1(s)=1/b-u2(s)>0 for any s; that is, u2(s)<1/b.Obviously, system (17) has two equilibria E1(0,0,0) and E2(u*,u*,0). A profile solution of (10) which satisfies boundary condition (11) corresponds to the positive solution (u2(s),u3(s),z(s)) of system (17) which satisfies
(18)u(+∞)=(u2*,u2*,0),u(-∞)=(0,0,0),
where u(s)=(u2(s),u3(s),z(s)). Therefore, to study the solutions of (10), it is sufficient to study those of system (17) satisfying boundary condition (18).Firstly, we investigate the dynamics nearE1. Simple calculations show that the characteristic equation of the linearization of system (17) at E1 is
(19)H(λ)=λ3+a2λ2+a1λ+a0=0,
where
(20)a0=m(f1′(0)+g1′(0)-b2)bc,a1=f1′(0)-b2-mbb,a2=b2-f1′(0)-bc2bc.Becausea0>0 (19) has a negative real root, which is denoted by λ3. Let λ1 and λ2 be the other two eigenvalues of (19) and suppose that Reλ1≥Reλ2. To investigate the distribution of roots of (19), denote
(21)p=a1-a223,q=2a2327-a1a23+a0,Δ0=q24+p327
and introduce the following lemma [41].Lemma 2.
(
a
)    If Δ0>0, (19) has one real root and two nonreal complex conjugate roots.
(
b
)   If Δ0=0, (19) has a multiple root and all its roots are real.
(
c
)   If Δ0<0, (19) has three distinct real roots.
Direct calculations show thatΔ0=-Δ/(108b4c4), where Δ is defined by (12).Lemma 3.
(
a
)  The real parts of λ1 and λ2 are positive.
(
b
)  Assume f1′(0)≤b2. Then, there exists c*>0 which is the only positive root of Δ(c)=0. When c≥c*, λ1, and λ2 are real. When 0<c<c*, λ1, and λ2 are complex and nonreal.
(
c
)  Assume that f1′(0)>b2. Then, there exist two positive constants c1*<c* which are all positive roots of Δ(c)=0. λ1 and λ2 are complex and nonreal if and only if c1*<c<c*. If c>c*, then λ*<λ2<λ1; if 0<c≤c1*, then λ2≤λ1<λ*, where λ*=(f1′(0)-b2)/(bc).
(
d
)  λ1=λ2 if and only if c=c* or c1*.Proof.
Supposeλ=βi≠0 is the root of (19). Substituting λ=βi into (19) and comparing real and imaginary parts show that a1=β2>0 and a0=a1a2. Since a0>0, then a2>0. However, it is impossible that a1>0 and a2>0 by the expressions of a1 and a2. Therefore, the real parts of λ1 and λ2 are not zero. Furthermore, since it is impossible that a1>0 and a2>0, Routh-Hurwitz theorem implies that it is impossible that the real parts of both λ1 and λ2 are negative. Consequently, there are two cases: (i) λ1 and λ2 are complex conjugate roots with positive real parts; (ii) λ1 and λ2 are real and at least one is positive. However, Descartes’ rule of signs shows that the number of positive roots of (19) is zero or two. Thus, if case (ii) is true, both of λ1 and λ2 are real and positive. Therefore, (a) is proved.
In this paragraph, we consider the casef1′(0)≤b2. Firstly, suppose that f1′(0)<b2. Obviously, b0<0 and b3>0. By the expression of b2, we have
(22)b2=-2bϵ3+2b2mϵ2-6b2mg1′ϵ+8b3m2(f1′(0)+g1′(0)-b2)+4m3b4+10b3m2g1′(0)>0
since ϵ=f1′(0)-b2<0. Now, assume f1′(0)=b2; that is, ϵ=0. Then. b3>0, b2>0, b1<0, and b0=0. Then, if f1′(0)≤b2, Descartes' rule of signs shows that there exists c*>0 which is the only positive root of Δ(c)=0, where Δ(c)<0 for 0<c<c* and Δ(c)>0 for c>c*. Using Lemma 2 completes the proof of (b).
Suppose thatf1′(0)>b2 in this paragraph and, thus, ϵ>0. Calculations show that
(23)H(λ*)=mg1′bc>0,H′(λ*)=ϵ2-bc2ϵ-mb2c2b2c2
and that H′(λ)=0 has two roots λ1* and λ2*, where
(24)λ1*=bc2+ϵ+b2c4+(3mb2-bϵ)c2+ϵ23bc,λ2*=bc2+ϵ-b2c4+(3mb2-bϵ)c2+ε23bc,
and λ1*>λ2*. By letting c0≜ϵ/bϵ+mb2 and using trivial calculations, we get (see Figure 1)
(25)λ*=λ1*⟺c=c0⟺H′(λ*)=0,λ*>λ1*⟺c<c0⟺H′(λ*)>0,λ*<λ1*⟺c>c0⟺H′(λ*)<0.
Therefore, ifc=c0, then H(λ1*)=H(λ*)>0. Since λ1* is the only minimum-value point of H(λ), and then H(λ)>0 for any λ>0 and both of λ1 and λ2 are not real. Lemma 2 shows that Δ(c0)<0. Thus, since b0>0 and b3>0, there exist two positive roots c1*<c* for equation Δ(c)=0 such that c1*<c0<c*. Then, using (25) and Lemma 2 completes the proof of (c) and (d).Distribution of eigenvalues of (19) when f1′(0)>b2, (a) for c>c* and (b) for c<c1*.
(a)
(b)Direct calculations show that corresponding eigenvectors of eigenvalueλi are
(26)ei=(1-λi(λi-c)m,1,λi),
where i=1,2,3. Since
(27)H(λi)=-m[1-λi(λi-c)m][λi-f1′(0)-b2bc]+mg1′bc=0,
and thus
(28)1-λi(λi-c)m=g1′(0)bcλi+b2-f1′(0)=g1′(0)bc(λi-λ*).Then, we have the following lemma.Lemma 4.
If0<c<c*, there exist no traveling wave solutions which satisfy boundary condition (11).Proof.
Assume thatf1′(0)≤b2 and 0<c<c*. Then, (b) of Lemma 3 implies that λ1 and λ2 are complex conjugate eigenvalues and there exits locally unstable manifold 𝒲u and locally stable manifold 𝒲s. If a solution of (17) tends to E1 when s→-∞, then it will be spiral on 𝒲u. By the structures of e1 and e2, u2(s)<0 at some time s<0, which shows that there exist no traveling wave solutions departing from E1.
Suppose thatf1′(0)>b2. If c1*<c<c*, (c) of Lemma 3 shows that λ1 and λ2 are complex conjugate eigenvalues and similar arguments to that of previous paragraph finish the proof. If 0<c≤c1*, (c) of Lemma 3 shows that λ1 and λ2 are real; however, λ2≤λ1<λ*. If a solution of (17) tends to E1 when s→-∞, structures of e1 and e2 indicate that there is an s<0 such that u2(s)<0. The proof is completed.From Section4 to Section 5.2, we suppose that c>c*, which implies λ*<λ2<λ1.


## 4. Shooting Method and Wazewski Set

To prove the existence of traveling wave, shooting method developed by Dunbar [34] is used. Firstly, we give the shooting arguments.Consider the differential equation(29)dyds=f(y),
where f(y) from Rn to Rn satisfies Lipschitz condition about y. Let y(s;y0) denote the unique solution of (29) with initial value y(0)=y0. It is convenient to give the notations y0·s≜y(s;y0) and y0·S≜{y0·s∣s∈S⊂R}. To describe the shooting method (or Wazewski theorem), some definitions are necessary.Definition 5.
(a) ForW⊆Rn, define immediate exit set W- of W as
(30)W-≜{y0∈W∣∀s>0,y0·[0,s)⊈W}.
(b) ForΣ⊆W, let Σ0≜{y0∈Σ∣∃s0>0suchthaty0·s0∉W}.
(c) Giveny0∈Σ0, define exit time T(y0) of y0 by
(31)T(y0)≜sup⁡{s∣y0·[0,s)⊆W}.
Then, Wazewski theorem is formulated as follows.Lemma 6 (see [34]).
Suppose that(1)
ify0∈Σ and y0·[0,s]⊆cl⁡(W), then y0·[0,s]⊆W.(2)
Ify0∈Σ, y0·s∈W and y0·s∉W-, then there exists an open set Vs about y0·s disjoint from W-.(3)
IfΣ=Σ0, Σ is compact and Σ intersects a trajectory of (29) only once.
Then, the mappingH(y0)=y0·T(y0) is a homeomorphism from Σ to its image on W-.A setW⊆Rn satisfying conditions (1) and (2) of Lemma 6 is called a Wazewski set. In the following, we first construct the Wazewski set W. Fundamental idea to construct a Wazewski set is that the characteristic vectors corresponding eigenvalues with positive real parts should be removed from W and that those characteristic vectors corresponding eigenvalues with negative real parts should be included. Therefore, we set
(32)W=ℝ3∖(P∪Q),
where
(33)P={(u1,u2,u3,z):u3>u2>u*,z>0},Q={(u1,u2,u3,z):0<u3<u2<u*,z<0}.It is obvious that∂W=∂P∪∂Q. Firstly, we give the construction of W-, which is described in Figure 2.Figure 2
The construction ofW and W-.Lemma 7.
The construction ofW- is as follows:
(34)W-=∂W∖(J∪E2),
where J={(u2,u3,z):0≤u2≤u*,u3=0,z≤0}.Proof.
It is enough to analyze the behavior of solution on∂P∪∂Q. We only study ∂Q and omit the proof of ∂P since the analysis of ∂P is similar to that of ∂Q and is simpler. In the process of this proof, we use some notations to simplify the proof. Set
(35)ui′=duids|(u2,u3,z)∈∂Q,z′=dzds|(u2,u3,z)∈∂Q,i=2,3,h(u2)=[f1(u2)u2+g1(u2)u2](1b-u2)-b.
From hypotheses (A1) and (A2), we find thatf1(u2)/u2 and g1(u2)/u2 are monotonously decreasing, h(u2) is strictly monotonously decreasing for u2∈(0,1/b), and u* is the only positive root of h(u2)=0. The set ∂Q is classified into two cases according to variable z. (a) Casez<0. This case is classified as follows.
(1) Case0=u3<u2<u*. Then u3′=z<0 and the solution of (17) will enter int⁡(W).(2) Case0<u3=u2<u*. Then
(36)(u3-u2)′=z-h(u2)u2c<0.The solution of (17) will enter Q.(3) Case0<u3<u2=u*. Then
(37)u2′=[(f1(u*)/u*+g1(u3)/u*)(1/b-u*)-b]u*c<h(u*)u*c=0.The solution of (17) will enter Q.(4) Case0=u3=u2<u*. Then u3′=z<0 and the solution of (17) will enter int⁡(W).(5) Case0=u3 and u2=u*. The solution of (17) will enter int⁡(W).(6) Caseu3=u2=u*. Then u2′=0,
(38)u2′′=[(f1′(0)(u*)u2′+g1′(0)(u*)z)(1b-u*)-(f1(u*)+g1(u*))u2′-b(1b-u*)]×(c)-1=[g1′(0)(u*)z((1/b)-u*)-b]c<0,and(u3-u2)′=z<0. Therefore, the solution of (17) will enter Q.(b)  Casez=0. This case is classified as follows.
(1) Case0<u3<u2<u*. Then z′=m(u3-u2)<0 and the solution of (17) will enter Q.(2) Case0=u3<u2<u*. Then u3′=z=0, u3′′=z′=-mu2<0. The solution of (17) will enter int⁡(W).(3) Case0<u3=u2<u*. Then (u3-u2)′=-h(u2)u2/c<0, z′=0, and z′′=cz′+m(u3-u2)′<0. The solution of (17) will enter Q.(4) Case0<u3<u2=u*. Then u2′<0 and z′=m(u3-u2)<0. The solution of (17) will enter Q.(5) Case0=u3=u2<u*. In this case, (0,0,0) is equilibrium and is constant.(6) Case0=u3 and u2=u*. Then u3′=z=0 and u3′′=z′=-mu2<0. The solution of (17) will enter int⁡(W).(7) Caseu3=u2=u*. Then (u*,u*,0) is equilibrium and is constant.
The proof is completed.


## 5. Existence of Traveling Wave Solution forc≥c*

In this section, we prove the existence of traveling wave solution forc≥c*. Firstly, we study the behaviors of solutions near E1.


### 5.1. Behaviors of Solutions NearE1

Lemma 8.
Suppose(u2(s),u3(s),z(s)) is a solution of (17) satisfying initial conditions
(39)z(0)>ku3(0),u3(0)>bck+b2-f1′(0)g1′(0)u2(0)>0,
where k=(λ1+λ2)/2. Then, for every s>0, we have
(40)z(s)>ku3(s),u3(s)>bck+b2-f1′(0)g1′(0)u2(s)>0.Proof.
From Lemma3, we have (bck+b2-f1′(0))/g1′(0)>0. To finish the proof, it is sufficient to prove that the set
(41)Ψ={(u2,u3,z):z>ku3,u3>bck+b2-f1′(0)g1′(0)u2>0}
is positively invariant. It is obvious that
(42)∂Ψ=∂Ψ1∪∂Ψ2∪∂Ψ3∪E1,
where
(43)∂Ψ1={(u2,u3,z):z=ku3,u3≥bck+b2-f1′(0)g1′(0)u2>0},∂Ψ2={(u2,u3,z):z>ku3,u3=bck+b2-f1′(0)g1′(0)u2≥0},∂Ψ3={(u2,u3,z):z≥ku3,u3>u2=0}.
Suppose that(u2(s0),u3(s0),z(s0))∈∂Ψ1. Then, z(s0)=ku3(s0) and
(44)dds[z(s)-ku3(s)]s=s0=cz(s0)+m[u3(s0)-u2(s0)]-kz(s0)=(c-k)z(s0)+m[u3(s0)-u2(s0)]=(c-k)ku3(s0)+m[u3(s0)-u2(s0)]=[(c-k)k+m]u3(s0)-mu2(s0)≥{[(c-k)k+m]bck+b2-f1′(0)g1′(0)-m}u2(s0)=-bcg1′(0)H(k)u2(s0)>0.
The last inequality is given sinceλ2<k<λ1. Suppose that (u2(s0),u3(s0),z(s0))∈∂Ψ2. If u2(s0)>0, then
(45)dds[u3(s)-bck+b2-f1′(0)g1′(0)u2(s)]s=s0={z-bck+b2-f1′(0)g1′(0)·1c×[(f1(u2)+g1(u3))(1b-u2)-bu2]f1′b}s=s0>{ku3-bck+b2-f1′(0)cg1′×[f1′(0)u2+g1′(0)u3b-bu2]}s=s0={b2-f1′(0)bcg1′[(bck+b2-f1′(0))u2-g1′(0)u3]}s=s0=0.
Ifu2(s0)=0, we have
(46)dds[u3(s)-bck+b2-f1′(0)g1′(0)u2(s)]s=s0=z(s0)>0.
Consequently, the solution of system (17) departing from Ψ cannot intersect ∂Ψ1∪∂Ψ2. If (u2(s0),u3(s0),z(s0))∈∂Ψ3, then u2′(s0)=g1(u3(s0))/(bc)>0. Since E1 is equilibrium, in summary, Ψ is positive invariant.Sinceλ1>λ2>0, stable manifold theorem implies that there exists a one-dimensional strong unstable manifold 𝒲1 tangent to e1 at E1 such that the point on 𝒲1 near E1 can be expressed by
(47)G1(ε)=εe1+o(ε).
Furthermore, there is a two-dimensional unstable manifold 𝒲2 tangent to span {e1,e2} at E1 such that 𝒲2 near E1 can be expressed by
(48)G2(ε1,ε2)=ε1e1+ε2e2+o(ε12+ε22).Lemma 9.
Suppose thatu(s)≜(u2(s),u3(s),z(s)) is a solution of (17) such that u(0)∈𝒲1 for small ε>0. Then, u(s) will leave W and enter P.Proof.
Obviously,u(s) satisfies initial condition (39) by the structure of e1, and Lemma 8 implies u(s)>0 (u(s)>0 means that ui(s)>0 and z(s)>0, i=2,3) for every s>0.
Furthermore, Lemma8 shows that u3′(s)=z(s)>ku3(s), implying lim⁡s→+∞u3(s)=+∞. Since u2(s)<1/b, it follows lim⁡s→+∞z(s)=+∞. Suppose that u2(s)<u* for every s>0. Then
(49)u2′>[((f1(u*)/u*)+(g1(u3)/u*))((1/b)-u*)-b]u2c>[((f1(u*)/u*)+(g1(2u*)/u*))((1/b)-u*)-b]u2c=Mu2c>0
for large s since u3(s) and g1(u3) are strictly monotonous increasing with respect to s and u3, respectively. Thus, we have that lim⁡s→+∞u2(s)=+∞, contradicting u2(s)<1/b for any s∈R. Therefore, there exists s1>0 such that u2(s1)=u*. Without losing generality, let s1=inf⁡{s>0:u2(s)=u*}. Obviously, we have u2′(s1)≥0. If u3(s1)<u*, then(50)u2′(s1)=[((f1(u*)/u*)+(g1(u3(s1))/u*))((1/b)-u*)-b]u*c<0,which is a contradiction. Therefore, u3(s1)≥u* and u(s1)∈∂P. Then, the construction of W- shows that u(s) will leave W and enter P.LetC be a small circle on 𝒲2 centered at E1. Then, points on C can be expressed in terms of local coordinate by
(51)F(θ)≜G2(εcos⁡⁡θ,εsin⁡θ)=ε[e1cos⁡⁡θ+e2sin⁡θ+O(ε)],
where θ∈[θ1,2π+θ1), ε>0, and θ1 is chosen such that F(θ1) lies on 𝒲1 with z>0. Then, stable manifold theorem shows that θ1→0 when ε→0. Denote F(θ)≜(u-2(θ),u-3(θ),z-(θ)).Lemma 10.
There exists aθ2∈(π/2,3π/4) such that
(52)z-(θ2)=0,0<u-3(θ2)<u-2(θ2)<u*,
and that
(53)z-(θ)>0,0<u-2(θ2)<u*,0<u-3(θ)<u*
for θ∈[θ1,θ2).Proof.
From (51), we have
(54)z-(θ)=ε[λ1cos⁡⁡θ+λ2sin⁡θ+O(ε)]=ελ12+λ22×[λ1λ12+λ22cos⁡⁡θ+λ2λ12+λ22sin⁡θ+O(ε)]=ελ12+λ22[sin⁡(φ0+θ)+O(ε)],
where sin(φ0)=λ1/λ12+λ22, cos⁡(φ0)=λ2/λ12+λ22, and φ0∈(π/4,π/2) since λ1>λ2. Therefore, z-(θ2)=0 and θ2∈[0,π] imply that θ2=π-φ0+O(ε)∈(π/2,3π/4). Obviously, z-(θ)>0 for any θ∈[θ1,θ2). However,
(55)u-2(θ)=εg1′(0)bc[cos⁡⁡θλ1-λ*+sin⁡θλ2-λ*+O(ε)]=εm[(-λ12+λ1c+m)cos⁡⁡θhhhh+(-λ22+λ2c+m)sin⁡θ+O(ε)],u-3(θ)=ε[cos⁡⁡θ+sin⁡θ+O(ε)],u-2(θ)-u-3(θ)=εm[λ1(c-λ1)cos⁡⁡θ+λ2(c-λ2)sin⁡θ+O(ε)].
Then, equalityz-(θ2)=ε[λ1cos⁡θ2+λ2sinθ2+O(ε)]=0, together with the last of (55), reveals u-2(θ2)-u-3(θ2)=ε[(λ1-λ2)λ2sinθ2+O(ε)]/m>0; that is, u-2(θ2)>u-3(θ2). For θ∈[θ1,θ2], the first and second equalities of (55) imply that 0<u-i(θ)<u* where i=2,3 since λ1>λ2 and 0<ε≪1.Let(56)Σ={F(θ)∣θ∈[θ1,θ2],εissmallenough}.By Lemma10, Σ is an arc of circle, Σ⊆W, and the solution of (17) with initial value being the endpoint F(θ2) will enter Q since F(θ2)∈W-∩∂Q. From Lemma 9, the solution of (17) with initial value being the endpoint F(θ1) will enter P.


### 5.2. Traveling Wave Solution forc>c*

Lemma 11.
Letu(s)=(u2(s),u3(s),z(s)) be a solution of (17) such that u(0)∈Λ. If u(s)∈W for any s≥0, then u(s)∈Λ for any s>0, where
(57)Λ={(u2,u3,z):0<u2<u*,u3>0,0<z<ku3}
and k=c+c2+4m.Proof.
Sets0=inf⁡{s:u(s)∉Λ,s≥0}. Suppose the conclusion is false; that is, s0<+∞. Obviously, s0>0 and u(s0)∈∂Λ where
(58)∂Λ=(∪i=17∂Λi)∪E2,∂Λ1={(u2,u3,z):u2=u*,u3≥u*,0≤z≤ku3}∖E2,∂Λ2={(u2,u3,z):0<u3≤u2≤u*,z=0}∖E2,∂Λ3={(u2,u3,z):u2=0,u3>0,0≤z≤ku3},∂Λ4={(u2,u3,z):u2=u*,0<u3<u*,0<z≤ku3},∂Λ5={(u2,u3,z):0≤u2≤u*,u3>0,z=ku3},∂Λ6={(u2,u3,z):u3>u2,0≤u2<u*,u3>0,z=0},∂Λ7={(u2,u3,z):0≤u2≤u*,u3=z=0}.
In Figure3, we find ∂Λ1={unboundedareaBE2CD}, ∂Λ2={triangleOAE2O}, ∂Λ3={unboundedcone⁡u3OE}, ∂Λ4={triangleAE2CA}, ∂Λ5={unboundedareaDCAOE}, ∂Λ6={unboundedareaBE2Ou3}, and ∂Λ7={segmentOA}.
Since∂Λ1∪∂Λ2⊂W-, thus u(s0)∉∂Λ1∪∂Λ2. If u(s0)∈∂Λ3, we have u2′(s0)≤0 because u2(s)>0 for 0<s<s0 and u2(s0)=0. However, u2′(s0)=g1(u3(s0))/(bc)>0 which is a contradiction. Therefore, u(s0)∉∂Λ3. If u(s0)∈∂Λ4, then(59)u2′(s0)=[((f1(u*)/u*)+(g1(u3(s0))/u*))((1/b)-u*)-b]u*c<0,contradicting u2′(s0)≥0. If u(s0)∈∂Λ5, then
(60)[z(s)-ku3(s)]s=s0′=(c-k)z(s0)+m[u3(s0)-u2(s0)]=[(c-k)k+m]u3(s0)-mu2(s0)<0
since (c-k)k+m<0, contradicting [z(s)-ku3(s)]s=s0′≥0. If u(s0)∈∂Λ6, then z′(s0)=m[u3(s0)-u2(s0)]>0 which is a contradiction. In conclusion, u(s0)∉∂Λ4∪∂Λ5∪∂Λ6. If u(s0)∈∂Λ7, then u3(s)>0 and z(s)>0 for any 0<s<s0. Hence, u3′(s)=z(s)>0 for any 0<s<s0, which implies that u3(s0)>u3(0)>0. From this contradiction we find u(s0)∉∂Λ7. Because E2 is a constant solution, we get u(s0)≠E2. In summary, u(s0)∉∂Λ and s0=+∞. The proof is completed.Figure 3
The construction of∂Λ.Lemma 12.
There exists a pointu0=(u20,u30,z0)∈Σ such that the solution u(s;u0)=(u2(s),u3(s),z(s)) of (17) with initial value being u0 will stay in W for any s>0.Proof.
It is sufficient to proveΣ≠Σ0. Suppose that Σ=Σ0. Firstly, we verify Conditions (1) and (2) of Lemma 6. Condition (1) of Lemma 6 is valid since W is closed.
Supposeu0=(u20,u30,z0)∈Σ, s<T(u0) and u(s;u0)∈W∖W-. Then, u(s;u0)∈int⁡W∪J and u0≠F(θ2) since F(θ2)∈W-. The structure of Σ implies that u20>0, u30>0, and z0>0. By the proof of Lemma 11, we have that u(s;u0)>0 for s<T(u0). Therefore, u(s;u0)∉J and u(s;u0)∈int⁡W. Condition (2) of Lemma 6 holds.
Lemma6 shows that Σ is homeomorphic to H(Σ). Since
(61)H(F(θ1))∈∂P∩W-,H(F(θ2))∈∂Q∩W-,
and W- is disconnected, we have that H(Σ) is disconnected, contradicting the connection of Σ. Thus, Σ≠Σ0 and the proof is completed.Lemma 13.
Letc>c*. Then, there exists a positive solution u(s)=(u2(s),u3(s),z(s)) of (17) such that
(62)u(+∞)=E2,u(-∞)=E1.Proof.
By Lemma12 there exists a point u0=(u20,u30,z0)∈Σ such that the solution u(s;u0)=(u2(s),u3(s),z(s)) of (17) with initial value being u0 will stay in W for any s>0. Furthermore, Lemma 11 shows u(s;u0)>0 for any s≥0. Stable manifold theorem implies that u(s;u0)>0 for any s≤0 and lim⁡s→-∞u(s;u0)=E1. Therefore, u(s;u0) is a positive solution.
To complete the proof, it is sufficient to show thatlim⁡s→+∞u(s;u0)=E2. By Lemma 11, we know that u2(s)<u* for any s>0 since u(s;u0) remains in W for all s. Because u3′(s)=z(s)>0, then the limit of u3(s) exists; that is, lim⁡s→+∞u3(s)=u3* and 0<u3*≤+∞. Suppose that u*<u3*≤+∞. The first equation of (17) shows that
(63)u2′>[((f1(u*)/u*)+(g1(u3)/u*))((1/b)-u*)-b]u2c>([(f1(u*)u*+g1((u3*+u*)/2)u*)(1b-u*)-b]u2)×(c)-1=Mu2c>0
for large s, which implies that there is an s*>0 such that u2(s*)>u*. This is a contradiction, and thus 0<u3*≤u*. From the first equation of (17), we have lim⁡s→+∞u2(s)=u2* where u2* is the only positive root of algebra equation
(64)[f1(u2)+g1(u3*)](1b-u2)-bu2=0.
At the same time, the third equation of (17) implies lim⁡s→+∞z(s)=z* and z*=m(u2*-u3*)/c or ±∞. It is impossible that z*=±∞ due to the boundedness of u3(s). In conclusion, the limit lim⁡s→+∞u(s)=(u2*,u3*,z*) exists and is finite. By [42], (u2*,u3*,z*) must be equilibrium. Since u3*>0, then (u2*,u3*,z*)=E2.Noticing the relation of systems (17) and (10) completes the proof of Theorem 1 for case c>c*.


### 5.3. Traveling Wave Solution forc=c*

Firstly, supposec>c* and let u(s;u0)=(u2(s),u3(s),z(s)) be the traveling wave solution of (17). Then, Lemma 11 implies that u(s;u0)∈Λ for all s. From the proof of Lemma 13, we find u3(s)≤u*. Therefore, for all s, we have u(s;u0)∈Π where
(65)Π={(u2,u3,z):0<u2<u*,0<u3≤u*,0<z<ku3}.Let{cn} be a sequence such that c*<cn<cn+1 for any n and lim⁡n→∞cn=c*. Set kn=cn+cn2+4m and
(66)Πn={(u2,u3,z):0<u2≤u*,0<u3≤u*,0<z≤knu3}.
Then, Πn⊆Π1 for any n.Lemma13 shows that there is a positive solution wn(s)=(u2,n(s),u3,n(s),zn(s)) for system
(67)u2′=[(f1(u2)+g1(u3))((1/b)-u2)-bu2]cn,u3′=z,z′=cnz+m(u3-u2),
satisfying boundary condition (62) such that wn(s)∈Πn⊆Π1 for any s.Lemma 14.
Letc=c*. Then, there exists a traveling wave solution for system (6) satisfying boundary condition (11).Proof.
It is sufficient to prove that there exists a positive solutionu(s)=(u2(s),u3(s),z(s)) of (17) satisfying boundary condition (62).
Firstly, we show that sequences{u2,n}, {u3,n}, {zn}, {u2,n′}, {u3,n′}, and {zn′} are uniformly bounded and equicontinuous. The idea of Lemma 11 in [34] is used. Obviously, {u2,n}, {u3,n}, and {zn} are uniformly bounded since wn(s)⊆Π1 for any s. Because wn(s)=(u2,n(s),u3,n(s),zn(s)) is the solution of (67), {u2,n′}, {u3,n′}, and {zn′} are also uniformly bounded. Since |zn(s1)-zn(s2)|=zn′(s3)|s1-s2| where s1<s3<s2, then {zn} is equicontinuous. Similarly, {u2,n} and {u3,n} are also equicontinuous. By differentiating the equations of (67) and using the previous bounds, we can get that {u2,n′′}, {u3,n′′}, and {zn′′} are uniformly bounded, and hence {u2,n′}, {u3,n′}, and {zn′} are equicontinuous.
The previous paragraph and Arzelà-Ascoli theorem imply that there exist subsequences, again denoted by{u2,n}, {u3,n}, and {zn} and functions u2, u3, and z such that
(68)u2,n⟶u2,u3,n⟶u3,zn⟶z
uniformly on compact subsets of ℝ, thus pointwise on ℝ. Same arguments imply that {u2,n′}, {u3,n′}, and {zn′} are also uniformly convergent on compact subsets of ℝ and pointwise convergent on ℝ. Consequently, we get
(69)u2,n′⟶u2′,u3,n′⟶u3′,zn′⟶z′.
Since (u2,n,u3,n,zn) is the solution of (67), then u(s)=(u2(s),u3(s),z(s)) is the solution of (17) for c=c* and u(s)∈cl⁡(Π1), where cl⁡(Π1) is the closer of Π1. Because system (67) is autonomous and (u2,n,u3,n,zn) satisfies boundary condition (62), we can assume that u3,n(0)=u*/2 for all n; thus, u3(0)>0. Then, similar to the proof of Lemma 13, we have that the solution u(s) satisfies boundary condition (62).


## 5.1. Behaviors of Solutions NearE1

Lemma 8.
Suppose(u2(s),u3(s),z(s)) is a solution of (17) satisfying initial conditions
(39)z(0)>ku3(0),u3(0)>bck+b2-f1′(0)g1′(0)u2(0)>0,
where k=(λ1+λ2)/2. Then, for every s>0, we have
(40)z(s)>ku3(s),u3(s)>bck+b2-f1′(0)g1′(0)u2(s)>0.Proof.
From Lemma3, we have (bck+b2-f1′(0))/g1′(0)>0. To finish the proof, it is sufficient to prove that the set
(41)Ψ={(u2,u3,z):z>ku3,u3>bck+b2-f1′(0)g1′(0)u2>0}
is positively invariant. It is obvious that
(42)∂Ψ=∂Ψ1∪∂Ψ2∪∂Ψ3∪E1,
where
(43)∂Ψ1={(u2,u3,z):z=ku3,u3≥bck+b2-f1′(0)g1′(0)u2>0},∂Ψ2={(u2,u3,z):z>ku3,u3=bck+b2-f1′(0)g1′(0)u2≥0},∂Ψ3={(u2,u3,z):z≥ku3,u3>u2=0}.
Suppose that(u2(s0),u3(s0),z(s0))∈∂Ψ1. Then, z(s0)=ku3(s0) and
(44)dds[z(s)-ku3(s)]s=s0=cz(s0)+m[u3(s0)-u2(s0)]-kz(s0)=(c-k)z(s0)+m[u3(s0)-u2(s0)]=(c-k)ku3(s0)+m[u3(s0)-u2(s0)]=[(c-k)k+m]u3(s0)-mu2(s0)≥{[(c-k)k+m]bck+b2-f1′(0)g1′(0)-m}u2(s0)=-bcg1′(0)H(k)u2(s0)>0.
The last inequality is given sinceλ2<k<λ1. Suppose that (u2(s0),u3(s0),z(s0))∈∂Ψ2. If u2(s0)>0, then
(45)dds[u3(s)-bck+b2-f1′(0)g1′(0)u2(s)]s=s0={z-bck+b2-f1′(0)g1′(0)·1c×[(f1(u2)+g1(u3))(1b-u2)-bu2]f1′b}s=s0>{ku3-bck+b2-f1′(0)cg1′×[f1′(0)u2+g1′(0)u3b-bu2]}s=s0={b2-f1′(0)bcg1′[(bck+b2-f1′(0))u2-g1′(0)u3]}s=s0=0.
Ifu2(s0)=0, we have
(46)dds[u3(s)-bck+b2-f1′(0)g1′(0)u2(s)]s=s0=z(s0)>0.
Consequently, the solution of system (17) departing from Ψ cannot intersect ∂Ψ1∪∂Ψ2. If (u2(s0),u3(s0),z(s0))∈∂Ψ3, then u2′(s0)=g1(u3(s0))/(bc)>0. Since E1 is equilibrium, in summary, Ψ is positive invariant.Sinceλ1>λ2>0, stable manifold theorem implies that there exists a one-dimensional strong unstable manifold 𝒲1 tangent to e1 at E1 such that the point on 𝒲1 near E1 can be expressed by
(47)G1(ε)=εe1+o(ε).
Furthermore, there is a two-dimensional unstable manifold 𝒲2 tangent to span {e1,e2} at E1 such that 𝒲2 near E1 can be expressed by
(48)G2(ε1,ε2)=ε1e1+ε2e2+o(ε12+ε22).Lemma 9.
Suppose thatu(s)≜(u2(s),u3(s),z(s)) is a solution of (17) such that u(0)∈𝒲1 for small ε>0. Then, u(s) will leave W and enter P.Proof.
Obviously,u(s) satisfies initial condition (39) by the structure of e1, and Lemma 8 implies u(s)>0 (u(s)>0 means that ui(s)>0 and z(s)>0, i=2,3) for every s>0.
Furthermore, Lemma8 shows that u3′(s)=z(s)>ku3(s), implying lim⁡s→+∞u3(s)=+∞. Since u2(s)<1/b, it follows lim⁡s→+∞z(s)=+∞. Suppose that u2(s)<u* for every s>0. Then
(49)u2′>[((f1(u*)/u*)+(g1(u3)/u*))((1/b)-u*)-b]u2c>[((f1(u*)/u*)+(g1(2u*)/u*))((1/b)-u*)-b]u2c=Mu2c>0
for large s since u3(s) and g1(u3) are strictly monotonous increasing with respect to s and u3, respectively. Thus, we have that lim⁡s→+∞u2(s)=+∞, contradicting u2(s)<1/b for any s∈R. Therefore, there exists s1>0 such that u2(s1)=u*. Without losing generality, let s1=inf⁡{s>0:u2(s)=u*}. Obviously, we have u2′(s1)≥0. If u3(s1)<u*, then(50)u2′(s1)=[((f1(u*)/u*)+(g1(u3(s1))/u*))((1/b)-u*)-b]u*c<0,which is a contradiction. Therefore, u3(s1)≥u* and u(s1)∈∂P. Then, the construction of W- shows that u(s) will leave W and enter P.LetC be a small circle on 𝒲2 centered at E1. Then, points on C can be expressed in terms of local coordinate by
(51)F(θ)≜G2(εcos⁡⁡θ,εsin⁡θ)=ε[e1cos⁡⁡θ+e2sin⁡θ+O(ε)],
where θ∈[θ1,2π+θ1), ε>0, and θ1 is chosen such that F(θ1) lies on 𝒲1 with z>0. Then, stable manifold theorem shows that θ1→0 when ε→0. Denote F(θ)≜(u-2(θ),u-3(θ),z-(θ)).Lemma 10.
There exists aθ2∈(π/2,3π/4) such that
(52)z-(θ2)=0,0<u-3(θ2)<u-2(θ2)<u*,
and that
(53)z-(θ)>0,0<u-2(θ2)<u*,0<u-3(θ)<u*
for θ∈[θ1,θ2).Proof.
From (51), we have
(54)z-(θ)=ε[λ1cos⁡⁡θ+λ2sin⁡θ+O(ε)]=ελ12+λ22×[λ1λ12+λ22cos⁡⁡θ+λ2λ12+λ22sin⁡θ+O(ε)]=ελ12+λ22[sin⁡(φ0+θ)+O(ε)],
where sin(φ0)=λ1/λ12+λ22, cos⁡(φ0)=λ2/λ12+λ22, and φ0∈(π/4,π/2) since λ1>λ2. Therefore, z-(θ2)=0 and θ2∈[0,π] imply that θ2=π-φ0+O(ε)∈(π/2,3π/4). Obviously, z-(θ)>0 for any θ∈[θ1,θ2). However,
(55)u-2(θ)=εg1′(0)bc[cos⁡⁡θλ1-λ*+sin⁡θλ2-λ*+O(ε)]=εm[(-λ12+λ1c+m)cos⁡⁡θhhhh+(-λ22+λ2c+m)sin⁡θ+O(ε)],u-3(θ)=ε[cos⁡⁡θ+sin⁡θ+O(ε)],u-2(θ)-u-3(θ)=εm[λ1(c-λ1)cos⁡⁡θ+λ2(c-λ2)sin⁡θ+O(ε)].
Then, equalityz-(θ2)=ε[λ1cos⁡θ2+λ2sinθ2+O(ε)]=0, together with the last of (55), reveals u-2(θ2)-u-3(θ2)=ε[(λ1-λ2)λ2sinθ2+O(ε)]/m>0; that is, u-2(θ2)>u-3(θ2). For θ∈[θ1,θ2], the first and second equalities of (55) imply that 0<u-i(θ)<u* where i=2,3 since λ1>λ2 and 0<ε≪1.Let(56)Σ={F(θ)∣θ∈[θ1,θ2],εissmallenough}.By Lemma10, Σ is an arc of circle, Σ⊆W, and the solution of (17) with initial value being the endpoint F(θ2) will enter Q since F(θ2)∈W-∩∂Q. From Lemma 9, the solution of (17) with initial value being the endpoint F(θ1) will enter P.


## 5.2. Traveling Wave Solution forc>c*

Lemma 11.
Letu(s)=(u2(s),u3(s),z(s)) be a solution of (17) such that u(0)∈Λ. If u(s)∈W for any s≥0, then u(s)∈Λ for any s>0, where
(57)Λ={(u2,u3,z):0<u2<u*,u3>0,0<z<ku3}
and k=c+c2+4m.Proof.
Sets0=inf⁡{s:u(s)∉Λ,s≥0}. Suppose the conclusion is false; that is, s0<+∞. Obviously, s0>0 and u(s0)∈∂Λ where
(58)∂Λ=(∪i=17∂Λi)∪E2,∂Λ1={(u2,u3,z):u2=u*,u3≥u*,0≤z≤ku3}∖E2,∂Λ2={(u2,u3,z):0<u3≤u2≤u*,z=0}∖E2,∂Λ3={(u2,u3,z):u2=0,u3>0,0≤z≤ku3},∂Λ4={(u2,u3,z):u2=u*,0<u3<u*,0<z≤ku3},∂Λ5={(u2,u3,z):0≤u2≤u*,u3>0,z=ku3},∂Λ6={(u2,u3,z):u3>u2,0≤u2<u*,u3>0,z=0},∂Λ7={(u2,u3,z):0≤u2≤u*,u3=z=0}.
In Figure3, we find ∂Λ1={unboundedareaBE2CD}, ∂Λ2={triangleOAE2O}, ∂Λ3={unboundedcone⁡u3OE}, ∂Λ4={triangleAE2CA}, ∂Λ5={unboundedareaDCAOE}, ∂Λ6={unboundedareaBE2Ou3}, and ∂Λ7={segmentOA}.
Since∂Λ1∪∂Λ2⊂W-, thus u(s0)∉∂Λ1∪∂Λ2. If u(s0)∈∂Λ3, we have u2′(s0)≤0 because u2(s)>0 for 0<s<s0 and u2(s0)=0. However, u2′(s0)=g1(u3(s0))/(bc)>0 which is a contradiction. Therefore, u(s0)∉∂Λ3. If u(s0)∈∂Λ4, then(59)u2′(s0)=[((f1(u*)/u*)+(g1(u3(s0))/u*))((1/b)-u*)-b]u*c<0,contradicting u2′(s0)≥0. If u(s0)∈∂Λ5, then
(60)[z(s)-ku3(s)]s=s0′=(c-k)z(s0)+m[u3(s0)-u2(s0)]=[(c-k)k+m]u3(s0)-mu2(s0)<0
since (c-k)k+m<0, contradicting [z(s)-ku3(s)]s=s0′≥0. If u(s0)∈∂Λ6, then z′(s0)=m[u3(s0)-u2(s0)]>0 which is a contradiction. In conclusion, u(s0)∉∂Λ4∪∂Λ5∪∂Λ6. If u(s0)∈∂Λ7, then u3(s)>0 and z(s)>0 for any 0<s<s0. Hence, u3′(s)=z(s)>0 for any 0<s<s0, which implies that u3(s0)>u3(0)>0. From this contradiction we find u(s0)∉∂Λ7. Because E2 is a constant solution, we get u(s0)≠E2. In summary, u(s0)∉∂Λ and s0=+∞. The proof is completed.Figure 3
The construction of∂Λ.Lemma 12.
There exists a pointu0=(u20,u30,z0)∈Σ such that the solution u(s;u0)=(u2(s),u3(s),z(s)) of (17) with initial value being u0 will stay in W for any s>0.Proof.
It is sufficient to proveΣ≠Σ0. Suppose that Σ=Σ0. Firstly, we verify Conditions (1) and (2) of Lemma 6. Condition (1) of Lemma 6 is valid since W is closed.
Supposeu0=(u20,u30,z0)∈Σ, s<T(u0) and u(s;u0)∈W∖W-. Then, u(s;u0)∈int⁡W∪J and u0≠F(θ2) since F(θ2)∈W-. The structure of Σ implies that u20>0, u30>0, and z0>0. By the proof of Lemma 11, we have that u(s;u0)>0 for s<T(u0). Therefore, u(s;u0)∉J and u(s;u0)∈int⁡W. Condition (2) of Lemma 6 holds.
Lemma6 shows that Σ is homeomorphic to H(Σ). Since
(61)H(F(θ1))∈∂P∩W-,H(F(θ2))∈∂Q∩W-,
and W- is disconnected, we have that H(Σ) is disconnected, contradicting the connection of Σ. Thus, Σ≠Σ0 and the proof is completed.Lemma 13.
Letc>c*. Then, there exists a positive solution u(s)=(u2(s),u3(s),z(s)) of (17) such that
(62)u(+∞)=E2,u(-∞)=E1.Proof.
By Lemma12 there exists a point u0=(u20,u30,z0)∈Σ such that the solution u(s;u0)=(u2(s),u3(s),z(s)) of (17) with initial value being u0 will stay in W for any s>0. Furthermore, Lemma 11 shows u(s;u0)>0 for any s≥0. Stable manifold theorem implies that u(s;u0)>0 for any s≤0 and lim⁡s→-∞u(s;u0)=E1. Therefore, u(s;u0) is a positive solution.
To complete the proof, it is sufficient to show thatlim⁡s→+∞u(s;u0)=E2. By Lemma 11, we know that u2(s)<u* for any s>0 since u(s;u0) remains in W for all s. Because u3′(s)=z(s)>0, then the limit of u3(s) exists; that is, lim⁡s→+∞u3(s)=u3* and 0<u3*≤+∞. Suppose that u*<u3*≤+∞. The first equation of (17) shows that
(63)u2′>[((f1(u*)/u*)+(g1(u3)/u*))((1/b)-u*)-b]u2c>([(f1(u*)u*+g1((u3*+u*)/2)u*)(1b-u*)-b]u2)×(c)-1=Mu2c>0
for large s, which implies that there is an s*>0 such that u2(s*)>u*. This is a contradiction, and thus 0<u3*≤u*. From the first equation of (17), we have lim⁡s→+∞u2(s)=u2* where u2* is the only positive root of algebra equation
(64)[f1(u2)+g1(u3*)](1b-u2)-bu2=0.
At the same time, the third equation of (17) implies lim⁡s→+∞z(s)=z* and z*=m(u2*-u3*)/c or ±∞. It is impossible that z*=±∞ due to the boundedness of u3(s). In conclusion, the limit lim⁡s→+∞u(s)=(u2*,u3*,z*) exists and is finite. By [42], (u2*,u3*,z*) must be equilibrium. Since u3*>0, then (u2*,u3*,z*)=E2.Noticing the relation of systems (17) and (10) completes the proof of Theorem 1 for case c>c*.


## 5.3. Traveling Wave Solution forc=c*

Firstly, supposec>c* and let u(s;u0)=(u2(s),u3(s),z(s)) be the traveling wave solution of (17). Then, Lemma 11 implies that u(s;u0)∈Λ for all s. From the proof of Lemma 13, we find u3(s)≤u*. Therefore, for all s, we have u(s;u0)∈Π where
(65)Π={(u2,u3,z):0<u2<u*,0<u3≤u*,0<z<ku3}.Let{cn} be a sequence such that c*<cn<cn+1 for any n and lim⁡n→∞cn=c*. Set kn=cn+cn2+4m and
(66)Πn={(u2,u3,z):0<u2≤u*,0<u3≤u*,0<z≤knu3}.
Then, Πn⊆Π1 for any n.Lemma13 shows that there is a positive solution wn(s)=(u2,n(s),u3,n(s),zn(s)) for system
(67)u2′=[(f1(u2)+g1(u3))((1/b)-u2)-bu2]cn,u3′=z,z′=cnz+m(u3-u2),
satisfying boundary condition (62) such that wn(s)∈Πn⊆Π1 for any s.Lemma 14.
Letc=c*. Then, there exists a traveling wave solution for system (6) satisfying boundary condition (11).Proof.
It is sufficient to prove that there exists a positive solutionu(s)=(u2(s),u3(s),z(s)) of (17) satisfying boundary condition (62).
Firstly, we show that sequences{u2,n}, {u3,n}, {zn}, {u2,n′}, {u3,n′}, and {zn′} are uniformly bounded and equicontinuous. The idea of Lemma 11 in [34] is used. Obviously, {u2,n}, {u3,n}, and {zn} are uniformly bounded since wn(s)⊆Π1 for any s. Because wn(s)=(u2,n(s),u3,n(s),zn(s)) is the solution of (67), {u2,n′}, {u3,n′}, and {zn′} are also uniformly bounded. Since |zn(s1)-zn(s2)|=zn′(s3)|s1-s2| where s1<s3<s2, then {zn} is equicontinuous. Similarly, {u2,n} and {u3,n} are also equicontinuous. By differentiating the equations of (67) and using the previous bounds, we can get that {u2,n′′}, {u3,n′′}, and {zn′′} are uniformly bounded, and hence {u2,n′}, {u3,n′}, and {zn′} are equicontinuous.
The previous paragraph and Arzelà-Ascoli theorem imply that there exist subsequences, again denoted by{u2,n}, {u3,n}, and {zn} and functions u2, u3, and z such that
(68)u2,n⟶u2,u3,n⟶u3,zn⟶z
uniformly on compact subsets of ℝ, thus pointwise on ℝ. Same arguments imply that {u2,n′}, {u3,n′}, and {zn′} are also uniformly convergent on compact subsets of ℝ and pointwise convergent on ℝ. Consequently, we get
(69)u2,n′⟶u2′,u3,n′⟶u3′,zn′⟶z′.
Since (u2,n,u3,n,zn) is the solution of (67), then u(s)=(u2(s),u3(s),z(s)) is the solution of (17) for c=c* and u(s)∈cl⁡(Π1), where cl⁡(Π1) is the closer of Π1. Because system (67) is autonomous and (u2,n,u3,n,zn) satisfies boundary condition (62), we can assume that u3,n(0)=u*/2 for all n; thus, u3(0)>0. Then, similar to the proof of Lemma 13, we have that the solution u(s) satisfies boundary condition (62).


## 6. Simulations

In this section, we present some simulations to confirm the theoretical results. Set(70)f(I)=β1IKh+I,g(B)=β2BKe+B,
and assign numerical values to parameters as follows:
(71)b=0.01,e=1,m=0.5,Ke=6,Kh=2,β1=0.62,β2=0.001,N=200,d=2.Obviously, such selection forf(I) and g(B) satisfies (A1) and (A2). Then, the traveling wave solution is described in Figure 4.Figure 4
The wave profiles forS andI and their movements.



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*Source: 201094-2014-03-26.xml*