Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 55, pp. 1–15.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2023.55

TRAVELING WAVE SOLUTIONS FOR THREE-SPECIES

NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS

HONG-JIE WU, BANG-SHENG HAN, SHAO-YUE MI, LIANG-BIN SHEN

Abstract. By using a two-point boundary-value problem and a Schauder’s

fixed point theorem, we obtain traveling wave solutions connecting (0, 0, 0) to
an unknown positive steady state for speed c ≥ c∗ = max{2, 2

√
d2r2, 2

√
d3r3}.

Then we present some asymptotic behaviors of traveling wave solutions. In
particular we show that the nonlocal effects have a great influence on the final

state of traveling wave solutions at −∞.

1. Introduction

We consider the three-species nonlocal competitive-cooperative system

ut = d1∆u+ r1u[1− a1(φ1 ∗ u)− b1v − c1w],

vt = d2∆v + r2v[1− a2(φ2 ∗ v) + b2w − c2u],

wt = d3∆w + r3w[1− a3(φ3 ∗ w) + b3v − c3u],

(1.1)

where

(φi ∗ u)(x, t) =

∫
R
φi(x− y)u(y, t) dy, x ∈ R, t ∈ R, i = 1, 2, 3.

Here the unknown functions u(x, t), v(x, t) and w(x, t) represent the population
densities of species at position x and time t, and ai, bi, ci, di, ri (i = 1, 2, 3) are
real constants. It is easy to see that the species u competes with the species v
and w which cooperate with each other from (1.1). The positive coefficients di,
ri (i = 1, 2, 3) indicate the diffusion rate and natural growth rate of u, v, w,
respectively. The competition and cooperation coefficients for three species are
denoted by the positive parameters ai, bi and ci (i = 1, 2, 3). To simplify the
notations, we let

t

r1
→ t,

√
d1
r1
x→ x, a1u→ u, a2v → v, a3w → w,

b1
a2
→ b1,

c1
a3
→ c1,

b2
a3
→ b2,

c2
a1
→ c2,

b3
a2
→ b3,

c3
a1
→ c3,

d2
d1
→ d2,

d3
d1
→ d3,

r2
r1
→ r2,

r3
r1
→ r3,

2020 Mathematics Subject Classification. 35A01, 35C07, 35K55, 35K57.
Key words and phrases. Three-species system; competitive-cooperative; nonlocal effect;
traveling wave solution; critical speed.
©2023. This work is licensed under a CC BY 4.0 license.

Submitted April 6, 2023. Published September 4, 2023.

1



2 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

then system (1.1) is converted to

ut = ∆u+ u[1− (φ1 ∗ u)− b1v − c1w],

vt = d2∆v + r2v[1− (φ2 ∗ v) + b2w − c2u],

wt = d3∆w + r3w[1− (φ3 ∗ w) + b3v − c3u],

(1.2)

where the bounded kernel functions φi(x) (i = 1, 2, 3) satisfy the assumptions:

(A1) φi(x) ≥ 0 and
∫
R φi(x)dx = 1, i = 1, 2, 3;

(A2)
∫
R φi(y)eλy dy <∞ for each λ ∈ (0,max{1,

√
r2/d2,

√
r3/d3}).

In addition, as we shown in the paper, it suffices to assume

(A3) 0 < bi, ci < 1 for i = 1, 2, 3.

We point out that if w = 0 in (1.2), then the system of equations is reduced
to a two-species nonlocal Lotka-Volterra competitive system whose traveling waves
have been discussed by Han et al. [6]. When u = 0, (1.2) becomes the two-species
nonlocal cooperative system which have been studied by Huang and Zou [10]. In
summary, two-species Lotka-Volterra systems have been extensively considered [3,
4, 5, 8, 15, 16, 18, 30]. Of course, there are studies on the traveling waves for three-
species competitive systems [17, 28], but little research on cooperative systems
due to the technical treatments of cooperative systems are not as convenient as
competitive systems. Leung and Hou et al. [13, 14] proved that the two-species
Lotka-Volterra cooperative system can be transformed into a competitive system
by using variable transformation which cannot be applied to 3-dimension system
[1, 2, 22]. To mitigate this technical challenge, Hung [12] proposed the following
classical three-species Lotka-Volterra competitive-cooperative system for the first
time

ut = d1uxx + u(λ1 − c11u− c12v + c13w), x ∈ R, t > 0,

vt = d2vxx + v(λ2 − c21u− c22v − c23w), x ∈ R, t > 0,

wt = d3wxx + w(λ3 + c31u− c32v − c33w), x ∈ R, t > 0,

(1.3)

where competition between species u and v (c12, c21 > 0), species v and w (c23, c32 >
0), and cooperation between species u and w (c13, c31 > 0). And by transforming
(1.3) into a monotonic system, they proved the existence of the traveling wave
solutions for (1.3). After that, Meng and Zhang [20] obtained the asymptotic
behavior and uniqueness of the traveling waves for (1.3) by using Ikehare’s theorem.
For more results, we can refer to [9, 21, 23, 27].

To make the model more practical in applications, nonlocal effects and time-
delays have been considered [3, 6, 7, 10, 23]. Subsequently, the traveling wave
solution of this model has been studied and developed, see [11, 18, 19, 25, 29]. But
they are mostly concerned with the quasi-monotone case.

It is worth noting that, compared with three-species delayed Lotka-Volterra
competitive-cooperative systems, there are relatively few studies on nonlocal sys-
tems. Recently, Zhang and Bao [26] introduced nonlocal effect to the diffusion term
which deduced the system

∂u

∂t
= d1(J1 ∗ u− u) + r1u(1− a1u− b1v − c1w),

∂v

∂t
= d2(J2 ∗ v − v) + r2v(1− a2v + b2w − c2u),

∂w

∂t
= d3(J3 ∗ w − w) + r3w(1− a3w + b3v − c3u),

(1.4)



EJDE-2023/55 THREE-SPECIES NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS 3

where

Ji ∗ z − z =

∫
R
Ji(y)[z(x− y, t)− z(x, t)] dy, i = 1, 2, 3,

and gave the existence, uniqueness and asymptotic behavior of the traveling wave
solutions of (1.4) by the comparative lemma, the bilateral Laplace transform and
the sliding method. We should point out that (1.4) does not destroy the com-
parison principle of the classical system which will not hold if nonlocal effects are
introduced to the reaction term, and the method based on the comparison principle
is not applicable. Therefore, inspired by [6, 24, 26], we try to (partially) solve the
existence and asymptotic behavior of the traveling wave solutions of (1.2) by using
the Schauder’s fixed point theorem and a two-point boundary value problem.

Substituting (u, v, w)(x, t) = (U, V,W )(ξ) into (1.2) and denoting ξ = x − ct
where c ≥ {2, 2

√
d2r2, 2

√
d3r3} represents the wave speed, we obtain

−U ′′(ξ)− cU ′(ξ) = U(ξ)[1− (φ1 ∗ U)(ξ)− b1V (ξ)− c1W (ξ)],

−d2V ′′(ξ)− cV ′(ξ) = r2V (ξ)[1− (φ2 ∗ V )(ξ) + b2W (ξ)− c2U(ξ)],

−d3W ′′(ξ)− cW ′(ξ) = r3W (ξ)[1− (φ3 ∗W )(ξ) + b3V (ξ)− c3U(ξ)].

(1.5)

Then, we have the following result.

Theorem 1.1. Assume that (A1)–(A3) hold. Then, for each

c > c∗ = max{2, 2
√
d2r2, 2

√
d3r3},

there exists a traveling wave solution (U, V,W )(ξ) satisfying (1.5) with the boundary
conditions

lim inf
ξ→−∞

(U(ξ) + V (ξ) +W (ξ)) > 0,

lim
ξ→+∞

U(ξ) = lim
ξ→+∞

V (ξ) = lim
ξ→+∞

W (ξ) = 0.
(1.6)

In particular, U , V and W are monotone decreasing on [Z0,+∞) for some Z0 > 0
(which may depend on c). Moreover, such traveling wave solution does not exist for
c < c∗.

This article is organized as follows. In Section 2, we present the super- and
sub-solutions of (1.5). The existence of traveling wave solutions connecting (0, 0)
to an unknown positive stea dy state of (1.1) is obtained in Section 3. In Section
4, we show the proof of Theorem 1.1 .

2. Preliminaries

In this section, we first use super- and sub-solution to construct the range of
traveling waves which will be used for the proof of the existence of the solution in
Section 3. In the following, we construct the super- and sub-solutions of (1.5).

Supersolution. Let

pc(x) = e−λcx, qc(x) = e−ζcx, lc(x) = e−ηcx, for all x ∈ R,

where λc > 0, ζc > 0, ηc > 0 are the smaller roots of the equations

λ2c − cλc + 1 = 0, d2ζ
2
c − cζc + r2 = 0, d3η

2
c − cηc + r3 = 0,

respectively. Then, it holds that

− p′′c − cp′c = pc, −d2q′′c − cq′c = r2qc, −d3l
′′
c − cl

′
c = r3lc. (2.1)



4 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

Subsolution. Let

p
c
(x) = e−λcx −Ae−(λc+ε)x, q

c
(x) = e−ζcx −Be−(ζc+ε)x,

lc(x) = e−ηcx −De−(ηc+ε)x, for all x ∈ R,

where small enough ε ∈ (0,min(λc, ζc, ηc)) satisfies

κc = −(λc + ε)2 + c(λc + ε)− 1 > 0, ιc = −d2(ζc + ε)2 + c(ζc + ε)− r2 > 0,

ϑc = −d3(ηc + ε)2 + c(ηc + ε)− r3 > 0.

Moreover, A,B,D > 1 are large enough such that

lnA

ε
> max

{ 1

λc − ε
ln

3Zc1
Aκc

,
1

ζc − ε
ln

3b1
Aκc

,
1

ηc − ε
ln

3c1
Aκc

}
,

lnB

ε
> max

{ 1

ζc − ε
ln

3r2Z
c
2

Bιc
,

1

ηc
ln

3r2b2D

Bιc
,

1

λc − ε
ln

3r2c2
Bιc

}
,

lnD

ε
> max

{ 1

ηc − ε
ln

3r3Z
c
3

Dϑc
,

1

ζc
ln

3r3b3B

Dϑc
,

1

λc − ε
ln

3r3c3
Dϑc

}
,

where

Zc1 =

∫
R
φ1(y)eλcydy, Zc2 =

∫
R
φ2(y)eζcydy, Zc3 =

∫
R
φ3(y)eηcydy.

Then, for all x > max
{

lnA
ε , lnBε , lnDε

}
, it holds that p

c
> 0, q

c
> 0, lc > 0. So we

have

− p′′
c
− cp′

c
− p

c
+ p

c
(φ1 ∗ pc) + b1pcqc + c1pclc

= (−λ2c + cλc − 1)e−λcx +Ae−(λc+ε)x[(λc + ε)2 − c(λc + ε) + 1]

+ [e−λcx −Ae−(λc+ε)x](Zc1e
−λcx + b1e

−ζcx + c1e
−ηcx)

< −Aκce−(λc+ε)x + e−λcx(Zc1e
−λcx + b1e

−ζcx + c1e
−ηcx)

= e−(λc+ε)x[−Aκc + Zc1e
−(λc−ε)x + b1e

−(ζc−ε)x + c1e
−(ηc−ε)x] < 0,

− d2q′′c − cq
′
c
− r2qc + r2qc(φ2 ∗ qc)− b2r2qclc + c2r2qcpc

= (−d2ζ2c + cζc − r2)e−ζcx +Be−(ζc+ε)x[d2(ζc + ε)2 − c(ζc + ε) + r2]

+ r2[e−ζcx −Be−(ζc+ε)x]
(
Zc2e

−ζcx − b2e−ηcx + b2De
−(ηc+ε)x + c2e

−λcx
)

< e−(ζc+ε)x[−Bιc + r2Z
c
2e
−(ζc−ε)x + r2b2De

−ηcx + r2c2e
−(λc−ε)x] < 0,

and

− d3l′′c − cl
′
c − r3lc + r3lc(φ3 ∗ lc)− b3r3lcqc + c3r3lcpc

= (−d3η2c + cηc − r3)e−ηcx +De−(ηc+ε)x[d3(ηc + ε)2 − c(ηc + ε) + r3]

+ r3[e−ηcx −De−(ηc+ε)x]
(
Zc3e

−ηcx − b3e−ζcx + b3Be
−(ζc+ε)x + c3e

−λcx
)

< e−(ηc+ε)x[−Dϑc + r3Z
c
3e
−(ηc−ε)x + r3b3Be

−ζcx + r3c3e
−(λc−ε)x] < 0,

for all x > max{ lnAε , lnBε , lnCε }. Let

p̃c(x) = max{0, p
c
}, q̃c(x) = max{0, q

c
}, l̃c(x) = max{0, lc}, x ∈ R,



EJDE-2023/55 THREE-SPECIES NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS 5

then combining this with (2.1), we deduce that

−p′′c − cp′c + pc
(
φ1 ∗ p̃c + b1q̃c + c1 l̃c

)
≥ pc,

−d2q′′c − cq′c + r2
(
φ2 ∗ q̃c + c2p̃c

)
qc ≥ r2qc + r2b2lcqc,

−d3l
′′
c − cl

′
c + r3

(
φ3 ∗ l̃c + c3p̃c

)
lc ≥ r3lc + r3b3qclc,

(2.2)

where bi, ci (i = 2, 3) satisfy

φ2 ∗ q̃c − b2lc + c2p̃c ≥ 0 and φ3 ∗ l̃c − b3qc + c3p̃c ≥ 0.

In addition, we can also obtain

− p̃′′c − cp̃′c + p̃c(φ1 ∗ pc − b1qc − c1lc) ≤ p̃c, (2.3)

for each x 6= ln(A)/x,

− d2q̃′′c − cq̃′c + r2(φ2 ∗ qc + c2pc)q̃c ≤ r2q̃c + r2b2q̃c l̃c, (2.4)

for each x 6= ln(B)/x, and

− d3 l̃′′c − cl̃′c + r3(φ3 ∗ lc + c3pc)l̃c ≤ r3 l̃c + r3b3 l̃cq̃c, (2.5)

for each x 6= ln(D)/x.
Based on the above setting, we give the existence of the traveling wave solutions

of (1.5).

3. Existence of traveling wave solutions of (1.5)

The aim of this section is two-fold. Firstly, we provide a specific result assuring
the existence of the solutions for the equation (1.5) in a finite interval by applying
the super- and sub-solution constructed in Section 2 and a well know argument
(Schauder’s fixed point theorem). Secondly, by taking the limit, we derive a exis-
tence criterion of solution to (1.5) on the entire interval.

A three-point boundary value problem. For c > max
{

2, 2
√
d2r2, 2

√
d3r3

}
,

we study the following system in a finite interval (−a, a):

−u′′ − cu′ = u(1− φ1 ∗ u− b1v − c1w),

−d2v′′ − cv′ = r2v(1− φ2 ∗ v + b2w − c2u),

−d3w′′ − cw′ = r3w(1− φ3 ∗ w + b3v − c3u),

u(±a) = p̃c(±a), v(±a) = q̃c(±a), w(±a) = l̃c(±a),

(3.1)

where a > max{ lnAε , lnBε , lnDε } and

u(x) =


u(a), x > a,

u(x), x ∈ [−a, a],

u(−a), x < −a,
v(x) =


v(a), x > a,

v(x), x ∈ [−a, a],

v(−a), x < −a.

w(x) =


w(a), x > a,

w(x), x ∈ [−a, a],

w(−a), x < −a.

Next we first define a convex set

Ma =
{

(u, v, w) ∈ C([−a, a],R2) : p̃c(x) ≤ u(x) ≤ pc, x ∈ (−a, a),



6 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

q̃c(x) ≤ v(x) ≤ qc, x ∈ (−a, a), l̃c(x) ≤ w(x) ≤ lc, x ∈ (−a, a),

u(±a) = p̃c(±a), v(±a) = q̃c(±a), w(±a) = l̃c(±a)
}
,

to study the existence of the solution for (3.1). Then, we construct the three-point
boundary value problem

−u′′ − cu′ + (φ1 ∗ u0 + b1v0 + c1w0)u = u0,

−d2v′′ − cv′ + r2(φ2 ∗ v0 + c2u0)v = r2v0 + r2b2w0v0,

−d3w′′ − cw′ + r3(φ3 ∗ w0 + c3u0)w = r3w0 + r3b3v0w0,

u(±a) = p̃c(±a), v(±a) = q̃c(±a), w(±a) = l̃c(±a),

(3.2)

where (u0, v0, w0) ∈Ma and

u0(x) =


u0(a), x > a,

u0(x), x ∈ [−a, a],

u0(−a), x < −a,
v0(x) =


v0(a), x > a,

v0(x), x ∈ [−a, a],

v0(−a), x < −a.

w0(x) =


w0(a), x > a,

w0(x), x ∈ [−a, a],

w0(−a), x < −a.

Now, we define a linear operator Ψa which satisfies Ψa(u0, v0, w0) = (u, v, w).
It is clear that the fixed point of (3.2) is a solution for (3.1). Obviously, Ψa is
compact and continuous. Next, we prove that Ma is an invariant for Ψa. From
the definition of Ψa and Ma, we know that Ma ∈ Ψa(Ma). Following, we prove
Ψa(Ma) ∈ Ma. Since (u, v, w) = (0, 0, 0) is a sub-solution of (3.2), we know that
u(x) > 0, v(x) > 0, w(x) > 0 for each x ∈ (−a, a). Given (u0, v0, w0) ∈ Ma and
combining with (2.2), then for x ∈ (−a, a), it holds that

− p′′c − cp′c + (φ1 ∗ u0 + b1v0 + c1w0)pc

≥ −p′′c − cp′c
= pc

≥ u0
= −u′′ − cu′ + (φ1 ∗ u0 + b1v0 + c1w0)u,

− d2q′′c − cq′c + r2(φ2 ∗ v0 + c2u0)qc

≥ −d2q′′c − cq′c + r2(φ2 ∗ q̃c + c2p̃c)qc

≥ r2qc + r2b2lcqc

≥ r2v0 + r2b2w0v0

= −d2v′′ − cv′ + r2(φ2 ∗ v0 + c2u0)v,

and

− d3l
′′
c − cl

′
c + r3(φ3 ∗ w0 + c3u0)lc

≥ −d3l
′′
c − cl

′
c + r3(φ3 ∗ l̃c + c3p̃c)lc

≥ r3lc + r3b3qclc

≥ r3w0 + r3b3v0w0



EJDE-2023/55 THREE-SPECIES NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS 7

= −d3w′′ − cw′ + r3(φ3 ∗ w0 + c3u0)w.

In addition, we also have u(±a) = p̃c(±a) ≤ pc(±a), v(±a) = q̃c(±a) ≤ qc(±a),

w(±a) = l̃c(±a) ≤ lc(±a). Then by using the maximum principle, we obtain
u(x) ≤ pc(x), v(x) ≤ qc(x), w(x) ≤ lc(x) for each x ∈ (−a, a). On the other hand,
combining with (2.3)-(2.5), it is easy to calculate

− p̃′′c − cp̃′c + (φ1 ∗ u0 + b1v0 + c1w0)p̃c

≤ −p̃′′c − cp̃′c + (φ1 ∗ pc + b1qc + c1lc)p̃c

≤ p̃c
≤ u0
= −u′′ − cu′ + (φ1 ∗ u0 + b1v0 + c1w0)u,

for each x ∈ (ln(A)/ε, a),

− d2q̃′′c − cq̃′c + r2(φ2 ∗ v0 + c2u0)q̃c

≤ −d2q̃′′c − cq̃′c + r2(φ2 ∗ qc + c2pc)q̃c

≤ r2q̃c + r2b2 l̃cq̃c

≤ r2v0 + r2b2w0v0

= −d2v′′ − cv′ + r2(φ2 ∗ v0 + c2u0)v,

for each x ∈ (ln(B)/ε, a), and

− d3 l̃′′c − cl̃′c + r3(φ3 ∗ w0 + c3u0)l̃c

≤ −d3 l̃′′c − cl̃′c + r3(φ3 ∗ lc + c3pc)l̃c

≤ r3 l̃c + r3b3 l̃cq̃c

≤ r3w0 + r3b3w0v0

= −d3w′′ − cw′ + r3(φ3 ∗ w0 + c3u0)w,

for each x ∈ ( lnD
ε , a), where u(a) = p̃c(a), u( lnA

ε ) > 0 = p̃c(
lnA
ε ), v(a) = q̃c(a),

v( lnB
ε ) > 0 = q̃c(

lnB
ε ) and w(a) = l̃c(a), w( lnD

ε ) > 0 = l̃c(
lnD
ε ). The maximum

principle implies u(x) ≥ p̃c for each x ∈ (ln(A)/ε, a), v(x) ≥ q̃c for each x ∈
(ln(B)/ε, a) and w(x) ≥ l̃c for each x ∈ (ln(D)/ε, a). Then we conclude u(x) ≥ p̃c,
v(x) ≥ q̃c, w(x) ≥ l̃c for each x ∈ (−a, a), that is (u, v, w) ∈ Ma holds. Thus
Ψa(Ma) ⊂Ma.

Now, by using the Schauder’s fixed point theorem, we can obtain that Ψa has a
fixed point (ua, va, wa) ∈Ma which is the solution of (3.1).

Lemma 3.1. There exists a constant M which is independent of the number a and
c > c∗ (c∗ = max{2, 2

√
d2r2, 2

√
d3r3}) such that each solution of problem (3.1)

satisfies

0 ≤ ua ≤M, 0 ≤ va ≤M, 0 ≤ wa ≤M (3.3)

for all a > max
{

1
ε ln A(λc+ε)

λc
, 1ε ln B(ζc+ε)

ζc
, 1ε ln D(ηc+ε)

ηc

}
and all x ∈ [−a, a].

Proof. Suppose that the maximum points of ua(x), va(x) and wa(x) are xM , xN ,
xH ∈ [−a, a] respectively, that is

Mu = max
x∈[−a,a]

ua(x) = ua(xM ), Mv = max
x∈[−a,a]

va(x) = va(xN ),



8 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

Mw = max
x∈[−a,a]

wa(x) = wa(xH).

Then we have u′a(xM ) = 0, u′′a(xM ) ≤ 0, v′′a(xN ) = 0, v′′a(xN ) ≤ 0, w′a(xH) = 0 and
w′′a(xH) ≤ 0.

Apart from this, we can also prove xM , xN , xH ∈ [−a, a). Since ua(a) = p̃c(a),

va(a) = q̃c(a) and wa(a) = l̃c(a), and p̃c(x), q̃c(x), l̃c(x) are decreasing for x >

max
{

1
ε ln A(λc+ε)

λc
, 1ε ln B(ζc+ε)

ζc
, 1ε ln D(ηc+ε)

ηc

}
, so it holds that xM , xN , xH ∈ [−a, a].

Next we prove the lemma. From the value of

−u′′a − cu′a = ua(1− φ1 ∗ ua − b1va − c1wa)

at xM ,
−d2v′′a − cv′a = r2va(1− φ2 ∗ va + b2wa − c2ua)

at xN and
−d3w′′a − cw′a = r3wa(1− φ3 ∗ wa + b3va − c3ua)

at xH , we obtain

1− (φ1 ∗ ua)(xM )− b1va(xM )− c1wa(xM ) ≥ 0,

1− (φ2 ∗ va)(xN ) + b2wa(xN )− c2ua(xN ) ≥ 0,

1− (φ3 ∗ wa)(xH) + b3va(xH)− c3ua(xH) ≥ 0,

which implies

(φ1 ∗ ua)(xM ) < 1, (φ2 ∗ va)(xN ) < 1 and (φ3 ∗ wa)(xH) < 1, (3.4)

and

−u′′a − cu′a ≤ ua ≤Mu,

−d2v′′a − cv′a ≤ r2va ≤ r2Mv,

−d3w′′a − cw′a ≤ r3wa ≤ r3Mw,

for small enough b2 and b3. Thus, it holds that

(u′ae
cx)′ ≥ −Mue

cx,
(
d2v
′
ae

c
d2
x
)′
≥ −r2Mve

c
d2
x,

(
d3w

′
ae

c
d3
x
)′
≥ −r3Mwe

c
d3
x.

Integrating the above inequalities from xM to x > xM , xN to x > xN and xH to
x > xH , respectively, we obtain

u′a(x) ≥ −Mu

c
(1− e−c(x−xM )), x ∈ [xM , a),

v′a(x) ≥ −r2Mv

c
(1− e−

c
d2

(x−xN )), x ∈ [xN , a),

w′a(x) ≥ −r3Mw

c
(1− e−

c
d3

(x−xH)), x ∈ [xH , a).

Integrating the above inequalities in the same interval again, we obtain

ua(x) ≥Mu −
Mu

c
(x− xM ) +

Mu

c

∫ x

xm

e−c(s−xM )ds

= Mu

[
1− x− xM

c
+

1− e−c(x−xM )

c2
]

= Mu[1− (x− xM )2h(c(x− xM ))]

≥Mu

[
1− 1

2
(x− xM )2

]
,



EJDE-2023/55 THREE-SPECIES NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS 9

va(x) ≥Mv −
r2Mv

c
(x− xN ) +

r2Mv

c
e
c
d2
xN

∫ x

xN

e−
c
d2
sds

= Mv

[
1− r2

d2
(x− xN )2

( 1
c(x−xN )

d2

+
e−

c
d2

(x−xN )

c2(x−xN )2

d22

− 1
c2(x−xN )2

d22

)]
= Mv

[
1− r2

d2
(x− xN )2h

( c
d2

(x− xN )
)]

≥Mv

[
1− r2

2d2
(x− xN )2

]
,

and

wa(x) ≥Mw

[
1− r3

2d3
(x− xH)2

]
,

where h(y) = e−y+y−1
y2 ≤ 1/2 for y > 0. Since ua(x), va(x), wa(x) ∈Ma, we have

ua(a) = p̃c(a) ≤ pc(a) = e−λca ≤ 1,

va(a) = q̃c(a) ≤ qc(a) = e−ζca ≤ 1,

wa(a) = l̃c(a) ≤ lc(a) = e−ηca ≤ 1,

which can further imply

Mu[1− 1

2
(a− xM )2] ≤ 1,

Mv[1−
r2

2d2
(a− xN )2] ≤ 1,

Mw[1− r3
2d3

(a− xH)2] ≤ 1.

(3.5)

Taking x0 = 1/2, if xM ∈ (a− x0, a), it follows from (3.5) that

Mu ≤
[
1− 1

2
(a− xM )2

]−1 ≤ (1− 1

2
x20

)−1
≤ 4

3
.

If xM ∈ [−a, a− x0), then combining this with (3.4), we have

1 ≥ (φ1 ∗ ua)(xM )

=

∫
R
φ1(y)ua(xM − y) dy

≥
∫ 0

−x0

φ1(y)ua(xM − y) dy

≥Mu

∫ 0

−x0

φ1(y)
(
1− y2

2

)
dy

From the definition of x0, we obtain

Mu ≤
[ ∫ 0

−x0

φ1(y)(1− y2

2
) dy

]−1
≤ 4

3

[ ∫ 0

−
√

1
2

φ1(y) dy
]−1

.

From (A1), we know that

4

3

(∫ 0

−
√

1
2

φ1(y) dy
)−1
≥ 4

3
.



10 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

Thus, for each x ∈ [−a, a), we have

Mu ≤
4

3

(∫ 0

−
√

1
2

φ1(y) dy
)−1

.

Similarly, take y0 =
√
d2/(2r2) and z0 =

√
d3/(2r3), then for each x ∈ [−a, a), it

holds

Mv ≤
4

3

(∫ 0√
− d2

2r2

φ2(y) dy
)−1

and Mw ≤
4

3

(∫ 0√
− d3

2r3

φ3(y) dy
)−1

.

Let

M = max
{4

3

(∫ 0

−
√

1
2

φ1(y) dy
)−1

,
4

3

(∫ 0√
− d2

2r2

φ2(y) dy
)−1

,

4

3

(∫ 0√
− d3

2r3

φ3(y) dy
)−1}

,

(3.6)

then the inequality (3.3) follows. The proof is complete �

Limit of (ua, va, wa) as a→ +∞. From Lemma 3.1 and the standard elliptic esti-
mates, we know that there existsM0 > 0 such that for each a > max{ lnAε , lnBε , lnDε }
and a constant α ∈ (0, 1), it holds that

‖ua‖C2,α(− a2 ,
a
2 )
≤M0, ‖va‖C2,α(− a2 ,

a
2 )
≤M0, ‖wa‖C2,α(− a2 ,

a
2 )
≤M0.

Letting a → +∞ (possibly along a subsequence), we have ua → u, va → v and
wa → w in C2

loc(R), and (u(x), v(x), w(x)) satisfies

−u′′ − cu′ = u(1− φ1 ∗ u− b1v − c1w), x ∈ R,
−d2v′′ − cv′ = r2v(1− φ2 ∗ v + b2w − c2u), x ∈ R,
−d3w′′ − cw′ = r3w(1− φ3 ∗ w + b3v − c3u), x ∈ R,

and

p̃c ≤ u(x) ≤ min{M,pc}, q̃c ≤ v(x) ≤ min{M, qc}. l̃c ≤ w(x) ≤ min{M, lc},

which implies

lim
x→+∞

u(x) = lim
x→+∞

v(x) = lim
x→+∞

w(x) = 0. (3.7)

4. Proof of Theorem 1.1

To prove Theorem 1.1 we use the following lemmas.

Lemma 4.1. There exists a Z0 > 0 such that u(x), v(x) and w(x) are monotoni-
cally decreasing for x > Z0.

Proof. On the contrary, suppose that u(x) is not always monotonic as x → +∞.
From (3.7), then there exists a sequence xn → +∞ (n→ +∞) such that u(xn)→ 0,
v(xn) → 0, w(xn) → 0 (n → +∞) and u(x) achieves a local minimum at xn, that
is u′(xn) = 0, u′′(xn) ≥ 0. Since

−u′′(xn)− cu′(xn) = u(xn)(1− (φ1 ∗ u)(xn)− b1v(xn)− c1w(xn)),

for each n ∈ N, it holds that

(φ1 ∗ u)(xn) + b1v(xn) + c1w(xn) ≥ 1. (4.1)



EJDE-2023/55 THREE-SPECIES NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS 11

On the other hand, from limx→+∞ u(x) = 0 and the boundedness of u(x) on C2(R),
it is easy to find that limn→+∞(φ1 ∗u)(xn) = 0. Then combining this with the fact
that limn→+∞ v(xn) = 0, limn→+∞ w(xn) = 0, one can obtain a contradiction of
(4.1). Therefore, u(x) is always monotonically decreasing on [Z0,+∞). In the same
way, v(x) and w(x) are also monotonically decreasing on [Z0,+∞). This completes
the proof. �

Lemma 4.2. There exists no traveling wave solutions of (1.5) for speed c < c∗.

Proof. Using a contradiction argument, we suppose that there exists a traveling
wave solution satisfying (1.5) and (1.6) for c < c∗. Take a sequence {zn} satisfying
zn → +∞ as n→ +∞. Denote un(x) = u(x+ zn)/u(zn), vn(x) = v(x+ zn)/u(zn),
wn(x) = w(x+ zn)/w(zn), then we have

−u′′n(x)− cu′n(x) = un(x)(1− (φ1 ∗ ũn)(x)− b1ṽn(x)− c1w̃n(x)), x ∈ R,
−d2v′′n(x)− cv′n(x) = r2vn(x)(1− (φ2 ∗ ṽn)(x) + b2w̃n(x)− c2ũn(x)), x ∈ R,
−d3w′′n(x)− cw′n(x) = r3wn(x)(1− (φ3 ∗ w̃n)(x) + b3ṽn(x)− c3ũn(x)), x ∈ R,

where ũn(x) = u(x + zn), ṽn(x) = v(x + zn), w̃n(x) = w(x + zn). Notice that
un(0) = vn(0) = wn(0) = 1 and un(x), vn(x), wn(x) are monotonic decreasing on
[Z0 − zn,+∞) for n ∈ N (where Z0 is defined by Lemma 4.1. Since u(x) → 0,
v(x)→ 0 and w(x)→ 0 as x→ +∞, it follows that (ũn, ṽn, w̃n)→ (0, 0, 0) locally
uniformly at x as n → +∞. Let (un, vn, wn) → (û(x), v̂(x), ŵ(x)) in C2

loc(R) as
n→ +∞, so we have

−û′′ − cû′ = û, x ∈ R,
−d2v̂′′ − cv̂′ = r2v̂, x ∈ R,
−d3ŵ′′ − cŵ′ = r3ŵ, x ∈ R.

(4.2)

Evidently, v̂, v̂, ŵ are monotonically decreasing and û(0) = v̂(0) = ŵ(0) = 1. In
addition, it is easy to get that û, v̂, ŵ are positive. Take û to say, if there exists a
point x0 ∈ R such that û(x0) = 0, then from the monotonicity of the nonnegative
function û, we know that for each x ≥ x0, û(x0) = 0. By the uniqueness of the
solutions of ordinary differential equations (4.2), we can obtain û(x) = 0 in R, which
contradicts with the fact û(0) = 1. Therefore, (4.2) admits such a solution (û, v̂, ŵ)
if and only if c ≥ max

{
2, 2
√
d2r2, 2

√
d3r3

}
, that is, there exists no traveling wave

solutions for speed c < max
{

2, 2
√
d2r2, 2

√
d3r3

}
. This completes the proof. �

Lemma 4.3. Under assumptions (A1)–(A3), the traveling wave (u(x), v(x), w(x))
of system (1.5) satisfies

lim inf
x→−∞

(u(x) + v(x) + w(x)) > 0.

Proof. Since u(x), v(x), w(x) are non-negative, lim infx→−∞(u(x)+v(x)+w(x)) ≥
0. Using a contradiction argument, we suppose that lim infx→−∞(u(x) + v(x) +
w(x)) = 0 which will lead to a sequence yn satisfying u(yn) → 0, v(yn) → 0,
w(yn) → 0 as yn → −∞(n → +∞). Taking ũ(x) = u(−x), ṽ(x) = v(−x),
w̃(x) = w(−x) and c̃ = −c, then (ũ(−yn), ṽ(−yn), w̃(−yn)) → (0, 0, 0), and
(ũ(x), ṽ(x), w̃(x)) satisfies

−ũ′′ − c̃ũ′ = ũ(1− φ1 ⊗ ũ− b1ṽ − c1w̃),

−d2ṽ′′ − c̃ṽ′ = r2ṽ(1− φ2 ⊗ ṽ + b2w̃ − c2ũ),



12 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

−d3w̃′′ − c̃w̃′ = r3w̃(1− φ3 ⊗ w̃ + b3ṽ − c3ũ),

where (φi⊗ z)(x) =
∫
R φi(y)z(x+ y) dy (i = 1, 2, 3). As in the proof of Lemma 4.2,

we can obtain c̃ ≥ max{2, 2
√
d2r2, 2

√
d3r3}; that is

c ≤ min{−2,−2
√
d2r2,−2

√
d3r3}

which contradicts Lemma 4.2. The proof is complete. �

Lemma 4.4. Assume (A1)–(A3) hold If b1 <
1

2M , ci <
1

2M (i = 1, 2, 3), then the
traveling wave (u(x), v(x), w(x)) of the system (1.5) satisfies

lim inf
x→−∞

u(x) > 0, lim inf
x→−∞

v(x) > 0, lim inf
x→−∞

w(x) > 0.

Proof. Since u(x), v(x), and w(x) are nonnegative functions, we know that

lim inf
x→−∞

u(x) ≥ 0, lim inf
x→−∞

v(x) ≥ 0, lim inf
x→−∞

w(x) ≥ 0.

We will prove this lemma in two steps.

Step 1. We prove that lim infx→−∞ u(x) > 0. By a contradiction argument assume
that lim infx→−∞ u(x) = 0. Then there must hold one of the following two cases:
u(x)→ 0 in an oscillating or a monotonous manner as x→ −∞.

Case 1.1. There exists a sequence xn → −∞ as n→ +∞, such that u(x) attains
local minimum at xn and u(xn) → 0 as n → +∞. A well know argument (the

Harnack inequality) shows that for each Z > 0 and δ ∈ (0, 1−(b1+c1)M2 ), there exists
a constant N > 0 such that u(x) ≤ δ for each n > N and x ∈ (xn − Z, xn + Z)
which can conclude that limn→+∞(φ1 ∗ u)(xn) = 0 and

−u′′(xn)− cu′(xn) = u(xn)[1− (φ1 ∗ u)(xn)− b1v(xn)− c1w(xn)]

≥ u(xn)[1− (b1 + c1)M − (φ1 ∗ u)(xn)] > 0,

for large enough n. On the other hand, it is easy to obtain that

−u′′(xn)− cu′(xn) ≤ 0,

because u(x) attains local minimum at xn which implies u′(xn) = 0 and u′′(xn) ≥ 0.
At this point, we reach a contradiction.

Case 1.2. limx→−∞ u(x) = 0 and there exists a large enough constant Z > 0 such
that u′(x) ≥ 0 for all x < −Z. From Lemma 4.3, we know that

lim inf
x→−∞

(v(x) + w(x)) > 0

which can be divided into the following two situations.

(a) Without loss of generality, we suppose that

lim inf
x→−∞

v(x) > 0, lim inf
x→−∞

w(x) = 0

which admits a sequence xn → −∞(n→ +∞) such that

lim
n→+∞

u(xn) = 0, lim
n→+∞

v(xn) = lim inf
x→−∞

v(x) = A > 0, lim
n→+∞

w(xn) = 0,

where A is a constant. Let un(x) = u(x+xn)/u(xn), vn(x) = v(x+xn)/v(xn) and
wn(x) = w(x+ xn)/w(xn), then we have

−u′′n(x)− cu′n(x) = un(x)[1− (φ1 ∗ ûn)(x)− b1v̂n(x)− c1ŵn(x)] for all x ∈ R,
where u′n(x), v′n(x) and w′n(x) are defined by Lemma 4.2. Suppose that un(x) →
ũ(x), vn(x) → ṽ(x), wn(x) → w̃(x) in C2

loc(R) as n → +∞. Then by the Harnack



EJDE-2023/55 THREE-SPECIES NONLOCAL COMPETITIVE-COOPERATIVE SYSTEMS 13

inequality, we can conclude that ûn(x)→ 0, v̂n(x)→ v̂(x), ŵn(x)→ 0 (n→ +∞).
Since u′(x) ≥ 0 for x < −Z, it follows that ũ′(x) ≥ 0 for each x ∈ R. Also we that
ũ(x) satisfies

−ũ′′(x)− cũ′(x) = ũ(x)(1− b1v̂(x)) for all x ∈ R.

After integrating from 0 to x > 0, we obtain that

−ũ′(x) + ũ′(0)− cũ(x) + cũ(0) =

∫ x

0

ũ(y)(1− b1v̂(y)) dy

> (1− b1M)ũ(0)x.

(4.3)

Since ũ(x) > 0, ũ′(x) ≥ 0, ũ(0) = 1, it follows that (4.3) does hold for large enough
x.

(b) lim infx→−∞ v(x) > 0, lim infx→−∞ w(x) > 0. Processing w(x) as we did for
v(x) in case (a), we obtain

−ũ′(x) + ũ′(0)− cũ(x) + cũ(0) > [1− (b1 + c1)M ]ũ(0)x,

which also does not hold for large enough x. Hence,

lim inf
x→−∞

u(x) > 0 for all x ∈ R.

Step 2. We prove that lim infx→−∞ v(x) > 0. As in the case above, we assume
that lim infx→−∞ v(x) = 0 which implies that there exists a sequence yn → −∞ as
n→ +∞ such that u(yn)→ 0(n→ +∞). We also analyze the following two cases.

Case 2.1. v(x) attains local minimum at yn. The proof is same as the Case 1.1
in Step 1, except that we take ε ∈ (0, 1−c2M2 ) here such that v(x) < ε for each
x ∈ (yn − Z, yn + Z) and n > N . Then we have

−d2v′′(yn)− cv′(yn) ≥ r2v(yn)[1− c2M − (φ2 ∗ v)(yn)] > 0,

which contradicts −d2v′′(yn)− cv′(yn) ≤ 0, because v(x) attains local minimum at
yn.

Case 2.2. There exists large enough Z > 0 such that v′(x) ≥ 0 for each x < −Z.
As in Case 1.2 in Step 1, one can obtain

− d2ṽ′′(x)− cṽ′(x) = r2ṽ(x)(1 + b2ŵ(x)− c2û(x)) for all x ∈ R. (4.4)

From Lemma 4.3, we know that lim infx→−∞(u(x) +w(x)) > 0 which implies that
lim infx→−∞ u(x) > 0, and either lim infx→−∞ w(x) > 0 or lim infx→−∞ w(x) = 0.
Then, after integrating (4.4) from 0 to x > 0, we obtain

−d2ṽ′′(x) + d2ṽ
′(0)− cṽ(x) + cṽ(0) = r2

∫ x

0

ṽ(y)(1 + b2ŵ(y)− c2û(y)) dy

≥ r2
∫ x

0

ṽ(y)(1− c2û(y)) dy

≥ r2(1− c2M)ṽ(0)x.

which does not hold for large enough x because ṽ(x) > 0, ṽ′(x) ≥ 0 and ṽ(0) = 1.
Therefore,

lim inf
x→−∞

v(x) > 0.



14 H.-J. WU, B.-S. HAN, S.-Y. MI, L.-B. SHEN EJDE-2023/55

Note that the proof of lim infx→−∞ v(x) > 0 is similar to the the proof of
lim infx→−∞ w(x) > 0; so we omit it. This completes the proof. �

Now the proof of Theorem 1.1 follows from Lemmas 4.1, 4.2, 4.2, 4.3, and 4.4.

Acknowledgment. We are very grateful to anonymous referees for their careful
reading and valuable comments. This work was supported by the Natural Sci-
ence Foundation of China (12161052, 11801470, U22A20231), by the Natural Sci-
ence Foundation of Sichuan Province(2022NSFSC1819), by the Central Government
Funds for Guiding Local Scientific and Technological Development (2021ZYD0010),
and by the Fundamental Research Funds for the Central Universities (2682022
ZTPY080).

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Hong-Jie Wu

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China
Email address: m17882275716@163.com

Bang-Sheng Han (corresponding author)

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China
Email address: hanbangsheng@swjtu.edu.cn

Shao-Yue Mi
School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Email address: mishaoyue@my.swjtu.edu.cn

Liang-Bin Shen

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

Email address: m18871027873@163.com


	1. Introduction
	2. Preliminaries
	Supersolution
	Subsolution

	3. Existence of traveling wave solutions of (??)
	A three-point boundary value problem
	Limit of (ua,va,wa) as a+

	4. Proof of Theorem ??
	Acknowledgment

	References