Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 02, pp. 1–19.

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

DOI: 10.58997/ejde.2025.02

TRAVELING WAVEFRONTS FOR A DISCRETE DIFFUSIVE

LOTKA-VOLTERRA COMPETITION SYSTEM WITH

NONLOCAL NONLINEARITIES

ZHI-JIAO YANG, GUO-BAO ZHANG, JUAN HE

Abstract. This article concerns the traveling wavefronts of a discrete diffusive
Lotka-Volterra competition system with nonlocal nonlinearities. We first prove

that there exists a c∗ > 0 such that when the wave speed is large than or

equals to c∗, the system admits an increasing traveling wavefront connecting
two boundary equilibria by the upper-lower solutions method. Furthermore,

we prove that (i) all traveling wavefronts with speed c > c∗(> c∗) are globally

stable with exponential convergence rate t−1/2e−ετσt, where σ > 0 and ετ =
ε(τ) ∈ (0, 1) is a decreasing function for the time delay τ > 0; (ii) the traveling

wavefronts with speed c = c∗ are globally algebraically stable in the algebraic

form t−1/2. The approaches are the weighted energy method, the comparison
principle and Fourier transform.

1. Introduction

Consider the discrete diffusive Lotka-Volterra competition system with nonlocal
nonlinearities

ut(x, t) = d1D[u](x, t) + r1u(x, t)
[
1− u(x, t)− b1

∑
i∈Z

g1(i)v(x− i, t− τ)
]
,

vt(x, t) = d2D[v](x, t) + r2v(x, t)
[
1− b2

∑
i∈Z

g2(i)u(x− i, t− τ)− v(x, t)
]
,

(1.1)

where x ∈ R, t > 0, dj , rj , bj (j = 1, 2), τ are positive constants, and

D[w](x, t) = w(x+ 1, t)− 2w(x, t) + w(x− 1, t)

with w = u, v. This model is often used to describe the competing interaction of two
species. The unknown functions u(x, t) and v(x, t) stand for the population densities
of two competitive species at location x and time t, respectively. The parameter dj
is the diffusion coefficient of species j, bj is the inter-specific competition coefficient,
rj is the growth rate, and τ is time delay, j = 1, 2. The kernel functions g1 and g2
are weight functions describing the distribution at past times of the individuals of
the species u or v who are at position x at time t (see Guo and Lin [9]), and satisfy

(H1) gj(i) = gj(−i) ≥ 0, i ∈ Z,
∑

i∈Z gj(i) = 1, j = 1, 2.

2020 Mathematics Subject Classification. 35K55, 35C07, 92D25.
Key words and phrases. Epidemic system; nonlocal dispersal; bistable traveling waves;
stability; time delay.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted July 23, 2024. Published January 4, 2025.

1



2 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

(H2) For every λ > 0,
∑

i∈Z gj(i)e
−λi <∞, j = 1, 2.

It is clear that the spatially homogeneous system of (1.1) is

u′(t) = r1u(t)[1− u(t)− b1v(t− τ)],

v′(t) = r2v(t)[1− b2u(t− τ)− v(t)].
(1.2)

System (1.2) always has three nonnegative equilibria

E0 := (0, 0), E1 := (0, 1), E2 := (1, 0).

When b1, b2 < 1 or b1, b2 > 1, there exists a unique coexistence equilibrium E∗ :=
(u∗, v∗) = ( 1−b1

1−b1b2
, 1−b2
1−b1b2

). In this article, we make the following assumption on

the coefficients b1 and b2 of (1.1):

(H3) 0 < b1 < 1 < b2.

By (H3), we obtain that E1 is unstable and E2 is stable (see [7, 19] for more details).
When gj(0) = 1 and gj(i) = 0 for all i ̸= 0 and τ = 0, system (1.1) is reduced to

the classical Lotka-Volterra competitive system with discrete diffusion

ut(x, t) = d1D[u](x, t) + r1u(x, t)[1− u(x, t)− b1v(x, t)],

vt(x, t) = d2D[v](x, t) + r2v(x, t)[1− b2u(x, t)− v(x, t)],
(1.3)

which has been studied in [7, 32]. More precisely, under assumption (H3), Guo
and Wu [7] obtained the existence of monostable traveling wave solutions of (1.3)
connecting E1 and E2. Later on, Tian and Zhang [32] further studied the stability of
traveling wave solutions of system (1.3) with relatively large speed by the weighted
energy method combining with the comparison principle.

In (1.1), we assume that the migration only happens to the nearest neighbors
and the interaction happens with infinite range. As such, system (1.1) can help
us understand the intricate cumulative effect due to an interaction between time
delay and diffusion through the whole spatial location and the previous time over
[−τ, 0]. It is well known that traveling wave solutions can describe the transitions
between different states of a physical system, propagation of patterns, and domain
invasion of species in population biology (see, e.g., [6]). In mathematics, traveling
wave solutions play an important role in the description of the long-term behaviour
of solutions to initial value problems in reaction-diffusion equations, both in the
spatially continuous cases and in spatially discrete situations[11, 12, 16, 17, 28, 33,
36]. In recent years, many efforts have been made to study the traveling wave
solutions of discrete diffusion equations or systems, see [1, 2, 3, 4, 7, 8, 9, 10, 13,
15, 21] and references therein. Traveling wave solution of (1.1) is a special solution
of the form (u(x, t), v(x, t)) = (ϕ(ξ), φ(ξ)), ξ := x+ ct, where c > 0 is wave speed,
(ϕ, φ) is called wave profile. If ϕ and φ are monotone, then (ϕ, φ) is called a traveling
wavefront. Substituting (ϕ(x+ ct), φ(x+ ct)) into (1.1), we obtain the wave profile
system

cϕ′(ξ) = d1D[ϕ](ξ) + r1ϕ(ξ)
[
1− ϕ(ξ)− b1

∑
i∈Z

g1(i)ϕ(ξ − i− cτ)
]
,

cφ′(ξ) = d2D[φ](ξ) + r2φ(ξ)
[
1− b2

∑
i∈Z

g2(i)φ(ξ − i− cτ)− φ(ξ)
]
,

(1.4)



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 3

where ′ = d
dξ and D[ψ](ξ) = ψ(ξ + 1) − 2ψ(ξ) + ψ(ξ − 1) with ψ = ϕ or φ. With

(1.4) we associate the boundary conditions

(ϕ, φ)(−∞) = E1 and (ϕ, φ)(+∞) = E2. (1.5)

In this article, we shall study the existence and stability of traveling wavefronts
of (1.1) connecting E1 to E2, i.e., solutions of (1.4) and(1.5). We first transform
the competition system (1.1) into a cooperation system. Then, by using upper and
lower solutions, monotone iteration method and a limiting argument, we can prove
the existence of monostable traveling wavefronts for c ≥ c∗, where c∗ is some positive
constant (see Lemma 2.1). The stability of traveling wave solutions for reaction-
diffusion equations with and without time delays has been extensively investigated,
see e.g., [20, 23, 26, 29, 30, 31, 34, 35]. Compared to the rich results for the classical
reaction-diffusion equations, limited results exist for the spatial discrete diffusion
equations, especially for discrete diffusion systems. Chen and Guo [4] employed the
squeezing technique to prove the asymptotic stability of traveling waves for discrete
quasilinear monostable equations without time delay. Guo and Zimmer [10] proved
the global stability of traveling wavefronts for spatially discrete equations with
nonlocal delay effects by using a combination of the weighted energy method and the
Green function technique. Tian and Zhang [32] investigated the global stability of
traveling wavefronts for a discrete diffusive Lotka-Volterra competition system with
two species by the weighted energy method together with the comparison principle.
Later on, Chen, Wu and Hsu [3] employed the similar method to show the global
stability of traveling wavefronts for a discrete diffusive Lotka-Volterra competition
system with three species. For other results on the stability of traveling wavefronts
for a discrete diffusive equations, we refer the reader to [13, 14]. Note that the
comparison principle also works for the transformed system of (1.1). Thus, the
weighted energy method together with the comparison principle can still be used to
prove the stability of traveling wavefronts of (1.1). However, since (1.1) is a system
of two equations and contains nonlocal interaction terms, we can only obtain the
stability of traveling wavefronts with large speed. Hence, in order to establish the
stability of traveling wavefronts with relatively lower speed, in this paper, we shall
employ the method of weighted energy combining with the comparison principle
and Fourier’s transform, which is different from that in [32], to study the stability
of the traveling wavefronts connecting E1 and E2 for (1.1). Our result shows that
all traveling wavefronts with speed c > c∗ are exponentially stable, where c∗ is a
positive constant larger than c∗, defined in Section 2, while the traveling wavefront
with speed c = c∗ is algebraically stable with decay rate t−1/2. In addition, the
time delay τ will slow down the convergence of the solution (u(x, t), v(x, t)) to the
traveling wavefronts (ϕ(x+ ct), φ(x+ ct)) with speed c > c∗.

The rest of this article is organized as follows. In Section 2, we first give some
preliminaries and then present the main results. In Section 3, we show the existence
of traveling wavefronts of (1.1). Section 4 is devoted to proving the stability of the
traveling wavefronts of (1.1).

2. Preliminaries and main results

In this section, we first show the existence of traveling wavefronts of (1.1), then
give the comparison principle of the corresponding initial value problem of (1.1),



4 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

and finally state the main result on the global stability of traveling wavefronts of
(1.1).

Letting u = u and v = 1− v, and dropping the bar for the sake of convenience,
then the competitive system (1.1) becomes the following cooperative system

ut(x, t) = d1D[u](x, t) + r1u(x, t)
[
1− u(x, t)− b1 + b1

∑
i∈Z

g1(i)v(x− i, t− τ)
]
,

vt(x, t) = d2D[u](x, t) + r2(v(x, t)− 1)
[
v(x, t)− b2

∑
i∈Z

g2(i)u(x− i, t− τ)
]
.

(2.1)
The equilibria E1 and E2 of system (1.1) are corresponding to the equilibria E− =
(u−, v−) := (0, 0) and E+ = (u+, v+) := (1, 1) of system (2.1).

Substituting (u(x, t), v(x, t)) = (ϕ(ξ), φ(ξ)), ξ := x + ct into (2.1) leads to the
following wave profile system with the asymptotic boundary conditions

cϕ′(ξ) = d1D[ϕ](ξ) + r1ϕ(ξ)
[
1− ϕ(ξ)− b1 + b1

∑
i∈Z

g1(i)φ(ξ − i− cτ)
]
,

cφ′(ξ) = d2D[φ](ξ) + r2(φ(ξ)− 1)
[
φ(ξ)− b2

∑
i∈Z

g2(i)ϕ(ξ − i− cτ)
]
,

(ϕ, φ)(−∞) = E−, (ϕ, φ)(+∞) = E+.

(2.2)

We define the function

∆(λ, c) = d1(e
λ + e−λ − 2)− cλ+ r1(1− b1).

One can easily show that the following result holds.

Lemma 2.1. Assume that (H1)–(H3) hold. Then there exist λ∗ > 0 and c∗ > 0
such that

∆(λ∗, c∗) = 0 and
∂

∂λ
∆(λ, c∗)

∣∣∣∣∣∣
λ=λ∗

= 0.

Furthermore,

(i) If c > c∗, then ∆(λ, c) = 0 has two distinct positive real roots λ1(c) and
λ2(c) with λ1(c) < λ∗ < λ2(c), and ∆(λ, c) < 0 for λ ∈ (λ1(c), λ2(c)), and
∆(λ, c) > 0 for λ ∈ (0, λ1(c)) ∪ (λ2(c),+∞).

(ii) If 0 < c < c∗, then ∆(λ, c) > 0 for all λ > 0.

By the method of upper-lower solutions, we can obtain the existence of traveling
wavefronts of system (2.1) under additional assumption

(H4)
∑

i∈Z gj(i)e
−λ1i ≤ 1, j = 1, 2, and b1b2 ≤ 1.

Theorem 2.2 (Existence). Assume that (H1)–(H4) hold and d1 ≥ d2. Then for
any c ≥ c∗, system (2.1) admits an increasing traveling wavefront (ϕ(ξ), φ(ξ)) con-
necting E− and E+.

Remark 2.3. Condition (H4) is a technical assumption, which is only used for
verification of the upper solution constructed in Section 3. We should point out
that (H4) can be replaced with

∑
i∈Z g1(i)e

−λ1i ≤ 1 and b1b2
∑

i∈Z g2(i)e
−λ1i ≤ 1,

or τ > 0 is suitable large.

Throughout this paper, we assume that (2.1) has the following initial data

u(x, s) = u0(x, s), v(x, s) = v0(x, s), (x, s) ∈ R× [−τ, 0]. (2.3)



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 5

Then following comparison principle to the initial value problem (2.1)-(2.3) can
be proved by an argument similar to [19, Lemma 3.2]. Thus, we omit the proof
here.

Lemma 2.4 (Comparison principle). Assume (H1)–(H3), and let (u±, v±)(x, t) be
the solution of system (2.1) with the initial data (u±0 , v

±
0 )(x, s), (x, s) ∈ R× [−τ, 0],

respectively. If

E− ≤ (u−0 , v
−
0 )(x, s) ≤ (u+0 , v

+
0 )(x, s) ≤ E+

for (x, s) ∈ R× [−τ, 0], then

E− ≤ (u−, v−)(x, t) ≤ (u+, v+)(x, t) ≤ E+

for (x, t) ∈ R× R+.

Throughout this article, we introduce some necessary notions at first. C > 0
denotes a generic constant, while Ci(i = 1, 2, . . . ) represents a specific constant. Let
∥ · ∥ and ∥ · ∥∞ denote 1-norm and ∞-norm of the matrix (or vector), respectively.
Let I be an interval, typically I = R. Denote by L1(I) the space of integrable
functions defined on I, and W k,1(I)(k ≥ 0) the Sobolev space of the L1-functions

f(x) defined on the interval I whose derivatives dn

dxn f(n = 1, . . . , k) also belong to

L1(I). Let L1
w(I) be the weighted L1-space with a weight function w(x) > 0 and

its norm is defined by

∥f∥L1
w(I) =

∫
I

w(x)|f(x)|dx,

W k,1
w (I) be the weighted Sobolev space with the norm

∥f∥Wk,1
w (I) =

k∑
i=0

∫
I

w(x)

∣∣∣∣dif(x)dxi

∣∣∣∣ dx.
Let T > 0 be a number and B be a Banach space. We denote by C([0, T ];B)
the space of the B-valued continuous functions on [0, T ], and by L1([0, T ];B) the
space of the B-valued L1-functions on [0, T ]. The corresponding spaces of the B-
valued functions on [0,∞) are defined similarly. For any function f(x), its Fourier
transform is

F [f ](η) = f̂(η) =

∫
R
e−ixηf(x)dx

and the inverse Fourier transform is

F−1[f̂ ](x) =
1

2π

∫
R
eixη f̂(η)dη,

where i is the imaginary unit, i2 = −1.
To obtain the stability of traveling wavefronts, we need the following technical

assumption.

(H5) d1 ≥ d2, r1 > r2, d1 − d2 ≤ r1−r2
2 .

We define the function

∆̃(λ, c) = d1(e
λ + e−λ − 2)− cλ+ r1 + r1b1qG1(λ),

where

G1(λ) =
∑
i∈Z

g1(i)e
−λ(i+cτ) <∞, q = max{1, r2b2

r1b1
}.



6 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

By computations, we have

∆̃(0, c) = r1 + r1b1q
∑
i∈Z

g1(i) = r1 + r1b1q > 0 for all c,

∂∆̃(λ, c)

∂c
= −λ− λτr1b1q

∑
i∈Z

g1(i)e
−λ(i+cτ) < 0 for all λ > 0,

∂∆̃(λ, c)

∂λ

∣∣∣
λ=0

= −c− cτr1b1q < 0, c > 0,

∂2∆̃(λ, c)

∂λ2
= d1(e

λ + e−λ) + c2τ2r1b1q
∑
i∈Z

g1(i)e
−λ(i+cτ) > 0.

Similar to Lemma 2.1, there exist λ∗ > 0 and c∗ > 0 such that ∆̃(λ∗, c∗) = 0 and
∂
∂λ∆̃(λ, c∗)

∣∣∣∣∣∣
λ=λ∗ = 0. Furthermore, when c > c∗, ∆̃(λ, c) = 0 has two distinct

positive real roots λ♮1(c) and λ♮2(c) with λ♮1(c) < λ∗ < λ♮2(c), and ∆̃(λ, c) < 0

for λ ∈ (λ♮1(c), λ
♮
2(c)), and ∆̃(λ, c) > 0 for λ ∈ (0, λ♮1(c)) ∪ (λ♮2(c),+∞). When

0 < c < c∗, ∆̃(λ, c) > 0 for all λ > 0. By computation, we see that

∆̃(λ, c∗) = d1(e
λ + e−λ − 2)− c∗λ+ r1 + r1b1qG1(λ)

≥ −r1(1− b1) + r1 + r1b1qG1(λ) > 0 for λ > 0.

Thus, c∗ < c∗.
We define the weight function

ω(ξ) :=

{
e−λ∗(ξ−ξ0), ξ ≤ ξ0,

1, ξ > ξ0,

where ξ0 > 0 is a large enough constant defined in the proof of Lemma 4.7 and λ∗

is defined above.

Theorem 2.5 (Stability). Assume that (H1)–(H3), (H5) hold and g1(·) = g2(·).
Let (ϕ(x + ct), φ(x + ct)) be the traveling wavefront connecting E− and E+ with
c ≥ c∗. If the initial data satisfy

(0, 0) ≤ (u0(x, s), v0(x, s)) ≤ (1, 1), (x, s) ∈ R× [−τ, 0]

and the initial perturbation

u0(x, s)− ϕ(x+ cs) ∈ C([−τ, 0];W 1,1
ω (R)),

v0(x, s)− φ(x+ cs) ∈ C([−τ, 0];W 1,1
ω (R)),

∂s(u0 − ϕ) ∈ L1([−τ, 0];L1
ω(R)), ∂s(v0 − φ) ∈ L1([−τ, 0];L1

ω(R)),

then the solution (u(x, t), v(x, t)) of the Cauchy problem (2.1) and (2.3) uniquely
exists and satisfies

(0, 0) ≤ (u(x, t), v(x, t)) ≤ (1, 1), (x, t) ∈ R× [0,+∞)

and

u(x, t)− ϕ(x+ ct) ∈ C([0,+∞);W 1,1
ω (R)),

v(x, t)− φ(x+ ct) ∈ C([0,+∞);W 1,1
ω (R)).



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 7

Furthermore, when c > c∗, the solution (u(x, t), v(x, t)) converges to the traveling
wavefront (ϕ(x+ ct), φ(x+ ct)) as follows:

sup
x∈R

|u(x, t)− ϕ(x+ ct)| ≤ Ct−1/2e−ετσt, ∀t > 0,

sup
x∈R

|v(x, t)− φ(x+ ct)| ≤ Ct−1/2e−ετσt, ∀t > 0,

where σ, C are positive numbers and ετ ∈ (0, 1). When c = c∗, the solution
(u(x, t), v(x, t)) converges to the traveling wavefront (ϕ(x+ct), φ(x+ct)) as follows:

sup
x∈R

|u(x, t)− ϕ(x+ ct)| ≤ Ct−1/2, ∀t > 0,

sup
x∈R

|v(x, t)− φ(x+ ct)| ≤ Ct−1/2, ∀t > 0.

3. Existence

In this section, we shall prove the existence of traveling wavefronts of (1.1) by
upper-lower solutions method. We first define the notion of upper-lower solutions.

Definition 3.1. A continuous function (ϕ, φ) from R to [0, 1] is called an upper
solution (or a lower solution) of (2.2), if each ϕ and φ are continuously differentiable
in R except at finite points and satisfy

cϕ′(ξ) ≥ (≤)d1D[ϕ](ξ) + r1ϕ(ξ)
[
1− ϕ(ξ)− b1 + b1

∑
i∈Z

g1(i)φ(ξ − i− cτ)
]
,

cφ′(ξ) ≥ (≤)d2D[φ](ξ) + r2(φ(ξ)− 1)
[
φ(ξ)− b2

∑
i∈Z

g2(i)ϕ(ξ − i− cτ)
]
,

a.e. ξ ∈ R.

We define two continuous functions as follows:

ϕ̄(ξ) =

{
eλ1ξ, ξ ≤ 0,

1, ξ > 0,
and φ̄(ξ) =

{
eλ1ξ/b1, ξ ≤ ξ1,

1, ξ > ξ1.

Lemma 3.2. Assume that c > c∗ and d1 ≥ d2. Then (ϕ̄(ξ), φ̄(ξ)) is an upper
solution of (2.2).

Proof. When ξ > 0, ϕ̄(ξ) = 1. Note that φ̄(ξ) ≤ 1 for all ξ ∈ R. Then we obtain

cϕ̄′(ξ)− d1D[ϕ̄](ξ)− r1ϕ̄(ξ)
[
1− ϕ̄(ξ)− b1 + b1

∑
i∈Z

g1(i)φ̄(ξ − i− cτ)
]

≥ r1

[
b1 − b1

∑
i∈Z

g1(i)φ̄(ξ − i− cτ)
]
≥ 0,

because of (H1). When ξ < 0, ϕ̄(ξ) = eλ1ξ. In view of φ̄(ξ) ≤ eλ1ξ/b1 for all ξ ∈ R,
we have

cϕ̄′(ξ)− d1D[ϕ̄](ξ)− r1ϕ̄(ξ)
[
1− ϕ̄(ξ)− b1 + b1

∑
i∈Z

g1(i)φ̄(ξ − i− cτ)
]

≥ cλ1e
λ1ξ − d1e

λ1ξ(eλ1 + e−λ1 − 2)− r1e
λ1ξ

[
1− eλ1ξ − b1

+ b1
∑
i∈Z

g1(i)φ̄(ξ − i− cτ)
]



8 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

= eλ1ξ
[
cλ1 − d1(e

λ1 + e−λ1 − 2)− r1(1− b1)
]

− r1e
λ1ξ

[
− eλ1ξ + b1

∑
i∈Z

g1(i)φ̄(ξ − i− cτ)
]

= eλ1ξ∆(λ1, c) + r1e
λ1ξ

[
eλ1ξ − b1

∑
i∈Z

g1(i)φ̄(ξ − i− cτ)
]

≥ r1e
2λ1ξ

[
1− e−λ1cτ

∑
i∈Z

g1(i)e
−λ1i

]
≥ 0,

Because ∆(λ1, c) = 0 for c > c∗ and (H4). Analogously, when ξ > ξ1, φ̄(ξ) = 1. It
is clear that

cφ̄′(ξ)− d2D[φ̄](ξ)− r2(φ̄(ξ)− 1)
[
φ̄(ξ)− b2

∑
i∈Z

g2(i)ϕ̄(ξ − i− cτ)
]
≥ 0.

When ξ < ξ1, φ̄(ξ) = eλ1ξ/b1. Thus, we can derive that

cφ̄′(ξ)− d2D[φ̄](ξ)− r2(φ̄(ξ)− 1)
[
φ̄(ξ)− b2

∑
i∈Z

g2(i)ϕ̄(ξ − i− cτ)
]

≥ cλ1
1

b1
eλ1ξ − d1

1

b1
eλ1ξ(eλ1 + e−λ1 − 2)

− r2(
1

b1
eλ1ξ − 1)

[ 1

b1
eλ1ξ − b2

∑
i∈Z

g2(i)ϕ̄(ξ − i− cτ)
]

≥ 1

b1
eλ1ξ[cλ1 − d1(e

λ1 + e−λ1 − 2)]

+ r2(1−
1

b1
eλ1ξ)

[ 1

b1
eλ1ξ − b2e

λ1ξ
∑
i∈Z

g2(i)e
−λ1(i+cτ)

]
=

1

b1
eλ1ξ

[
r1(1− b1) + r2(1−

1

b1
eλ1ξ)

(
1− b1b2e

−λ1cτ
∑
i∈Z

g2(i)e
−λ1i

)]
≥ 0,

since d1 ≥ d2 and (H4) holds. Thus, we prove (ϕ̄(ξ), φ̄(ξ)) is an upper solution of
(2.2). The proof is complete. □

Let φ(ξ) = 0 for all ξ ∈ R, and ϕ(ξ) be positive and satisfy

cϕ′(ξ) = d1D[ϕ](ξ) + r1ϕ(ξ)[1− b1 − ϕ(ξ)]

and
lim

ξ→−∞
ϕ(ξ)e−λ1ξ = 1.

For the existence of ϕ(ξ), we refer readers to Chen and Guo [4, 5]. We can prove
that (φ(ξ), ϕ(ξ)) is a lower solution of (2.2). Since the proof is easy, we omit it
here.

Lemma 3.3. (ϕ(ξ), φ(ξ)) is a lower solution of (2.2).

Based on Lemmas 3.2 and 3.3, Theorem 2.2 can be easily obtained. More pre-
cisely, the existence of traveling wavefronts with speed c > c∗ can be proved by the
monotone iteration method [5, 14], Schauder’s fixed point theorem [22], or trun-
cated method [7]. By applying the limiting arguments, we can obtain the existence
of traveling wavefronts with speed c = c∗. We refer the reader to [14, 27], and here
we skip the details.



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 9

4. Global stability

In this section, we are going to prove the stability of the traveling wavefronts by
the weighted energy method together with the comparison principle and Fourier’s
transform. The global existence and uniqueness of the solution for Cauchy problem
(2.1)-(2.3) can be proved by the standard energy method and continuity extension
method [25] or the theory of abstract functional differential equations[24]. There-
fore, we present the following proposition and omit the proof.

Proposition 4.1. Assume that (H1)–(H3), (H5) hold. Let (ϕ(x+ ct), φ(x+ ct)) =
(ϕ(ξ), φ(ξ)) be the traveling wavefront of (2.1) connecting E− and E+ with c ≥ c∗.
If the initial data satisfy

(0, 0) ≤ (u0(x, s), v0(x, s)) ≤ (1, 1), (x, s) ∈ R× [−τ, 0].

then the Cauchy problem (2.1) and (2.3) admits a unique solution (u(x, t), v(x, t))
satisfying

(0, 0) ≤ (u(x, t), v(x, t)) ≤ (1, 1), (x, t) ∈ R× [0,+∞).

If, in addition, the initial perturbation satisfies

u0(x, s)−ϕ(x+cs) ∈ C([−τ, 0];W 1,1
ω (R)), v0(x, s)−φ(x+cs) ∈ C([−τ, 0];W 1,1

ω (R)),

then

u(x, t)−ϕ(x+ct) ∈ C([0,+∞);W 1,1
ω (R)), v(x, t)−φ(x+ct) ∈ C([0,+∞);W 1,1

ω (R)).

Before proving the stability of the traveling wavefronts, we recall some properties
of the solutions to linear delayed differential system.

Lemma 4.2 ([18, Theorem 1]). Let z(t) be the solution to the linear delayed dif-
ferential system

d

dt
z(t) = Az(t) +Bz(t− τ), t ≥ 0, τ > 0,

z(s) = z0(s), s ∈ [−τ, 0].
(4.1)

where A,B ∈ CN×N , N ≥ 2, and z0(s) ∈ C1([−τ, 0],CN ). Then

z(t) = eA(t+τ)eB̄t
τ z0(−τ) +

∫ 0

−τ

eA(t−s)eB̄(t−τ−s)
τ [z′0(s) +Az0(s)]ds,

where B̄ = BeAτ and eB̄t
τ is the so-called delayed exponential function in the form

eB̄t
τ =



0, −∞ < t < −τ,
I, −τ ≤ t < 0,

I + B̄ t
1! , 0 ≤ t < τ,

I + B̄ t
1! + B̄2 (t−τ)2

2! , τ ≤ t < 2τ,
...

...

I + B̄ t
1! + B̄2 (t−τ)2

2! + · · ·+ B̄m [t−(m−1)τ ]m

m! , (m− 1)τ ≤ t < mτ,
...

...

where 0, I ∈ CN×N , and 0 is zero matrix and I is the unit matrix.



10 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

Lemma 4.3 ([23, Theorem 3.1]). Suppose µ(A) := µ1(A)+µ∞(A)
2 < 0, where µ1(A)

and µ∞(A) denote the matrix measure of A induced by matrix 1-norm ∥ · ∥1 and

∞-norm ∥ · ∥∞, respectively. If ν(B) := µ1(B)+µ∞(B)
2 ≤ −µ(A), then there exists a

decreasing function ετ = ε(τ) ∈ (0, 1) for τ > 0 such that any solution of system
(4.1) satisfies

∥z(t)∥ ≤ C0e
−ετσt, t > 0,

where C0 is a positive constant depending on initial data z0(s), s ∈ [−τ, 0] and
σ = |µ(A)| − ν(B). In particular,

∥eAteB̄t
τ ∥ ≤ C0e

−ετσt, t > 0,

where eB̄t
τ is defined in Lemma 4.2.

Remark 4.4. It can be seen from the proof of [23, Theome 3.1] that

µ1(A) = lim
θ→0+

∥I + θA∥ − 1

θ
= max

1≤j≤N

[
Re(ajj) +

N∑
j ̸=i

|aij |
]

and

µ∞(A) = lim
θ→0+

∥I + θA∥∞ − 1

θ
= max

1≤i≤N

[
Re(aii) +

N∑
i ̸=j

|aij |
]
.

The following lemma can be found in [10, Lemma 3.2].

Lemma 4.5. For t > 0,

1

2π

∫
R
exp{−2tdϵτ cosh(λ

∗)(1− cos η)}dη ≤
√

π

dtϵτ
,

where cosh(λ∗) = (eλ
∗
+ e−λ∗

)/2.

Now we are ready to prove the stability of traveling wavefronts to (2.1) with a
specific convergence rate. For any c ≥ c∗, we define the functions

Ũ+
0 (x, s) = max{u0(x, s), ϕ(x+ cs)}, Ṽ +

0 (x, s) = max{v0(x, s), φ(x+ cs)},

Ũ−
0 (x, s) = min{u0(x, s), ϕ(x+ cs)}, Ṽ −

0 (x, s) = min{v0(x, s), φ(x+ cs)}

for (x, s) ∈ R× [−τ, 0]. It is easy to see that

0 = u− ≤ Ũ−
0 (x, s) ≤ u0(x, s) ≤ Ũ+

0 (x, s) ≤ u+ = 1,

0 = u− ≤ Ũ−
0 (x, s) ≤ ϕ(x+ cs) ≤ Ũ+

0 (x, s) ≤ u+ = 1,

0 = v− ≤ Ṽ −
0 (x, s) ≤ v0(x, s) ≤ Ṽ +

0 (x, s) ≤ v+ = 1,

0 = v− ≤ Ṽ −
0 (x, s) ≤ φ(x+ cs) ≤ Ṽ +

0 (x, s) ≤ v+ = 1

for (x, s) ∈ R× [−τ, 0].
It is clear that the initial data (Ũ±

0 (x, s), Ṽ ±
0 (x, s)) are piecewise continuous

and have a poor regularity, which may also cause the absence of regularity for
the corresponding solutions. To overcome such a shortcoming, we choose smooth
functions (U±

0 (x, s), V ±
0 (x, s)) instead of these initial data as the new initial data

such that

(0, 0) ≤ (U−
0 (x, s), V −

0 (x, s)) ≤ (Ũ−
0 (x, s), Ṽ −

0 (x, s)) ≤ (u0(x, s), v0(x, s))

≤ (Ũ+
0 (x, s), Ṽ +

0 (x, s)) ≤ (U+
0 (x, s), V +

0 (x, s)) ≤ (1, 1).



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 11

We denote (U+(x, t), V +(x, t)) and (U−(x, t), V −(x, t)) as the corresponding so-
lutions of (2.1) with respect to the above mentioned initial data

(U+
0 (x, s), V +

0 (x, s)), (U−
0 (x, s), V −

0 (x, s)),

respectively. It then follows from Lemma 2.4 that

0 ≤ U−(x, t) ≤ u(x, t) ≤ U+(x, t) ≤ 1,

0 ≤ U−(x, t) ≤ ϕ(x+ ct) ≤ U+(x, t) ≤ 1,

0 ≤ V −(x, t) ≤ v(x, t) ≤ V +(x, t) ≤ 1,

0 ≤ V −(x, t) ≤ φ(x+ ct) ≤ V +(x, t) ≤ 1

for (x, t) ∈ R× [0,+∞).
We shall complete the proof of Theorem 2.5 in the following three steps:

(1) (U+(x, t), V +(x, t)) converges to (ϕ(x+ ct), φ(x+ ct)).
(2) (U−(x, t), V −(x, t)) converges to (ϕ(x+ ct), φ(x+ ct)).
(3) (u(x, t), v(x, t)) converges to (ϕ(x+ ct), φ(x+ ct)).

We do only Step 1, since Step 2 can be done by a similar way. Using a squeezing
technique, we can easily get the conclusion of Step 3, which implies Theorem 2.5.

Let ξ = x+ ct and

U1(ξ, t) := U+(x, t)− ϕ(x+ ct), V1(ξ, t) := V +(x, t)− φ(x+ ct)

with the initial data

U10(ξ, s) := U+(x, s)− ϕ(x+ cs), V10(ξ, s) := V +(x, s)− φ(x+ cs) (4.2)

for t > 0, s ∈ [−τ, 0], ξ ∈ R. It is easy to see that

U1(ξ, t) ≥ 0, V1(ξ, t) ≥ 0, ∀(ξ, t) ∈ R× [0,+∞).

By (2.1) and (2.2), we can verify that (U1(ξ, t), V1(ξ, t)) satisfies

U1t + cU1ξ = d1D[U1] + r1M(ϕ, φ)U1 − r1U
2
1 + r1b1U1

∑
i∈Z

g1(i)V
τ
1

+ r1b1ϕ
∑
i∈Z

g1(i)V
τ
1 ,

V1t + cV1ξ = d2D[V1]− r2N(ϕ, φ)V1 + r2V
2
1 − r2b2V1

∑
i∈Z

g2(i)U
τ
1

+ r2b2(1− φ)
∑
i∈Z

g2(i)U
τ
1 ,

(4.3)

where

M(ϕ, φ) = 1− 2ϕ− b1 + b1
∑
i∈Z

g1(i)φ
τ , N(ϕ, φ) = 1− 2φ+ b2

∑
i∈Z

g2(i)ϕ
τ ,

ϕτ = ϕ(ξ − i− cτ), φτ = φ(ξ − i− cτ),

Uτ
1 = U1(ξ − i− cτ, t− τ), V τ

1 = V1(ξ − i− cτ, t− τ).



12 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

Furthermore, system (4.3) can be rewritten as

U1t + cU1ξ = d1D[U1] + r1U1

[
1− b1 − (U1 + ϕ) + b1

∑
i∈Z

g1(i)(V
τ
1 + φτ )

]
+ r1ϕ

[
b1

∑
i∈Z

g1(i)V
τ
1 − U1

]
,

V1t + cV1ξ = d2D[V1] + r2V1

[
(V1 + φ)− b2

∑
i∈Z

g2(i)(U
τ
1 + ϕτ )

]
+ r2(1− φ)

[
b2

∑
i∈Z

g2(i)U
τ
1 − V1

]
.

(4.4)

Note that U1 ≥ 0, V1 ≥ 0, ϕ, φ ∈ (0, 1), U1+ϕ ∈ (0, 1) and V1+φ ∈ (0, 1). Then
we obtain

r1U1

[
1− b1 − (U1 + ϕ) + b1

∑
i∈Z

g1(i)(V
τ
1 + φτ )

]
+ r1ϕ

[
b1

∑
i∈Z

g1(i)V
τ
1 − U1

]
≤ r1U1 + r1b1

∑
i∈Z

g1(i)V
τ
1

(4.5)

and

r2V1

[
(V1 + φ)− b2

∑
i∈Z

g2(i)(U
τ
1 + ϕτ )

]
+ r2(1− φ)

[
b2

∑
i∈Z

g2(i)U
τ
1 − V1

]
≤ r2V1 + r2b2

∑
i∈Z

g2(i)U
τ
1 .

(4.6)

Using (4.5) and (4.6) in (4.4), we have

U1t + cU1ξ ≤ d1D[U1] + r1U1 + r1b1
∑
i∈Z

g1(i)V
τ
1 ,

V1t + cV1ξ ≤ d2D[V1] + r2V1 + r2b2
∑
i∈Z

g2(i)U
τ
1

for all (ξ, t) ∈ R× R+.
Let (U+

1 (ξ, t), V +
1 (ξ, t)) be the solution of the initial value problem

U+
1t(ξ, t) + cU+

1ξ(ξ, t) = d1D[U+
1 ](ξ, t) + r1U

+
1 (ξ, t)

+ r1b1
∑
i∈Z

g1(i)V
+
1 (ξ − i− cτ, t− τ),

V +
1t (ξ, t) + cV +

1ξ (ξ, t) = d2D[V +
1 ](ξ, t) + r2V

+
1 (ξ, t)

+ r2b2
∑
i∈Z

g2(i)U
+
1 (ξ − i− cτ, t− τ),

(U+
1 (ξ, s), V +

1 (ξ, s)) = (U10(ξ, s), V10(ξ, s))

(4.7)

for (ξ, t) ∈ R × R+, where (U10(ξ, s), V10(ξ, s)) is defined in (4.2). Then, by the
comparison principle, we have

(U1(ξ, t), V1(ξ, t)) ≤ (U+
1 (ξ, t), V +

1 (ξ, t)), ∀(ξ, t) ∈ R× R+. (4.8)

We introduce the transformation

U(ξ, t) = e−λ∗(ξ−ξ0)U+
1 (ξ, t), V (ξ, t) = e−λ∗(ξ−ξ0)V +

1 (ξ, t),



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 13

where ξ0 is a large enough positive constant. Then by (4.7), (U(ξ, t), V (ξ, t)) satis-
fies

Ut(ξ, t) + cUξ(ξ, t) = d1[e
λ∗
U(ξ + 1, t) + e−λ∗

U(ξ − 1, t)] + k1U(ξ, t)

+ r1b1
∑
i∈Z

g1(i)e
−λ∗(i+cτ)V (ξ − i− cτ, t− τ),

Vt(ξ, t) + cVξ(ξ, t) = d2[e
λ∗
V (ξ + 1, t) + e−λ∗

V (ξ − 1, t)] + k2V (ξ, t)

+ r2b2
∑
i∈Z

g2(i)e
−λ∗(i+cτ)U(ξ − i− cτ, t− τ),

(4.9)

where

k1 := r1 − cλ∗ − 2d1, k2 := r2 − cλ∗ − 2d2.

Taking the Fourier transform to system (4.9) and denoting the Fourier transform

of Z(ξ, t) := (U(ξ, t), V (ξ, t))T by Ẑ(η, t) := (Û(η, t), V̂ (η, t))T , we obtain

Ût(η, t) = [d1(e
λ∗+iη + e−(λ∗+iη))− icη + k1]Û(η, t)

+ r1b1
∑
i∈Z

g1(i)e
−λ∗(i+cτ)e−(i+cτ)iηV̂ (η, t− τ),

V̂t(η, t) = [d2(e
λ∗+iη + e−(λ∗+iη))− icη + k2]V̂ (η, t)

+ r2b2
∑
i∈Z

g2(i)e
−λ∗(i+cτ)e−(i+cτ)iηÛ(η, t− τ).

(4.10)

Let

A(η) =

(
d1(e

λ∗+iη + e−(λ∗+iη))− icη + k1 0

0 d2(e
λ∗+iη + e−(λ∗+iη))− icη + k2

)
and

B(η) =

(
0 r1b1

∑
i∈Z e

−i(i+cτ)ηg1(i)e
−λ∗(i+cτ)

r2b2
∑

i∈Z e
−i(i+cτ)ηg2(i)e

−λ∗(i+cτ) 0

)
.

Then system (4.10) can be rewritten as

d

dt
Ẑ(η, t) = A(η)Ẑ(η, t) +B(η)Ẑ(η, t− τ). (4.11)

By using the solution formula of (4.1) in Lemma 4.2, one can solve the linear
time-delayed ordinary differential system (4.11) as follows

Ẑ(η, t) = eA(η)(t+τ)eB(η)t
τ Ẑ0(η,−τ)

+

∫ 0

−τ

eA(η)(t−s)eB(η)(t−s−τ)
τ

[
∂sẐ0(η, s)−A(η)Ẑ0(η, s)

]
ds

=: J1(η, t) +

∫ 0

−τ

J2(η, t− s)ds,

(4.12)



14 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

where B(η) = B(η)eA(η)τ . Taking the inverse Fourier transform to (4.12), one has

Z(ξ, t)

= F−1[J1](ξ, t) +

∫ 0

−τ

F−1[J2](ξ, t− s)ds

=
1

2π

∫ +∞

−∞
eiξηeA(η)(t+τ)eB(η)t

τ Ẑ0(η,−τ)dη

+
1

2π

∫ 0

−τ

∫ +∞

−∞
eiξηeA(η)(t−s)eB(η)(t−s−τ)

τ

[
∂sẐ0(η, s)−A(η)Ẑ0(η, s)

]
dηds.

(4.13)
Now we present the lemma about the decay rate of U1(ξ, t) and V1(ξ, t) for

ξ ∈ I := (−∞, ξ0].

Lemma 4.6. Let assumptions in Theorem 2.5 hold. Then there exists a decreasing
function ετ = ε(τ) ∈ (0, 1) such that

∥U1(·, t)∥L∞(I) + ∥V1(·, t)∥L∞(I) ≤ Ct−1/2e−ετσ1t, ∀t > 0, (4.14)

where C is a positive constant and

σ1 := −∆̃(λ∗, c)

{
> 0 for c > c∗,

= 0 for c = c∗.
(4.15)

Proof. By (4.13), it suffices to estimate F−1[J1](ξ, t) and
∫ 0

−τ
F−1[J2](ξ, t − s)ds,

respectively. By the definition of µ(A) and ν(B), we have

µ(A) =
µ1(A) + µ∞(A)

2

= max
{
d1(e

λ∗
cos η + e−λ∗

cos η) + k1, d2(e
λ∗

cos η + e−λ∗
cos η) + k2

}
= d1(e

λ∗
+ e−λ∗

) cos η + k1

= d1(e
λ∗

+ e−λ∗
− 2)− cλ∗ + r1 −m(η),

where
m(η) = d1(e

λ∗
+ e−λ∗

)(1− cos η) ≥ 0,

since d1 ≥ d2, r1 > r2, d1 − d2 ≤ r1−r2
2 by (H5), and

ν(B) ≤ max{r1b1G1(λ
∗), r2b2G2(λ

∗)},
since ∣∣rjbj ∑

i∈Z
e−i(i+cτ)ηgj(i)e

−λ∗(i+cτ)
∣∣ ≤ rjbj

∑
i∈Z

|e−i(i+cτ)ηgj(i)e
−λ∗(i+cτ)|

= rjbj
∑
i∈Z

gj(i)e
−λ∗(i+cτ), j = 1, 2.

Note that ∆̃(λ∗, c) = d1(e
λ∗

+ e−λ∗ − 2) − cλ∗ + r1 + r1b1qG1(λ
∗) ≤ 0 for c ≥ c∗.

It then follows that d1(e
λ∗

+ e−λ∗ − 2)− cλ∗ + r1 ≤ −r1b1qG1(λ
∗) < 0. Thus, we

can derive that µ(A) < 0 and

µ(A) + ν(B) ≤ d1(e
λ∗

+ e−λ∗
− 2)− cλ∗ + r1 −m(η)

+ max{r1b1G1(λ
∗), r2b2G2(λ

∗)}

= d1(e
λ∗

+ e−λ∗
− 2)− cλ∗ + r1 + r1b1qG1(λ

∗)−m(η)



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 15

= ∆̃(λ∗, c)−m(η) ≤ 0,

because g1(·) = g2(·), where q = max{1, r2b2r1b1
}. Furthermore,

|µ(A)| − ν(B) = −µ(A)− ν(B) = −∆̃(λ∗, c) +m(η) ≥ 0,

i.e. |µ(A)| ≥ ν(B). Then by Lemma 4.3, there exists a decreasing function ετ =
ε(τ) ∈ (0, 1) such that∣∣eA(η)(t+τ)eB̄(η)t

τ

∣∣ ≤ Ce−ετ (|µ(A)|−ν(B))t ≤ Ce−ετσ1te−ετm(η)t, (4.16)

where C is a positive constant and σ1 is defined in (4.15).
By the definition of Fourier’s transform, we obtain

sup
η∈R

∥Ẑ0(η,−τ)∥ ≤
∫ ∞

−∞
∥Z0(ξ,−τ)∥dξ =

2∑
i=1

∥Zi0(·,−τ)∥L1(R).

Applying (4.16) and in view of Lemma 4.5, we have

sup
ξ∈R

∥F−1[J1](ξ, t)∥ = sup
ξ∈R

∥∥ 1

2π

∫ +∞

−∞
eiξηeA(η)(t+τ)eB(η)t

τ Ẑ0(η,−τ)dη
∥∥

≤ Ce−ετσ1t sup
η∈R

∥Ẑ0(η,−τ)∥
1

2π

∫ +∞

−∞
e−ετm(η)tdη

≤ Ct−1/2e−ετσ1t
2∑

i=1

∥Zi0(·,−τ)∥L1(R).

(4.17)

Note that

sup
η∈R

∥A(η)Ẑ0(η, s)∥ ≤ C

2∑
i=1

∥Zi0(·, s)∥W 1,1(R).

Similarly, we can obtain

sup
ξ∈R

∥F−1[J2](ξ, t− s)dη∥

= sup
ξ∈R

∥∥ 1

2π

∫ +∞

−∞
eiξηeA(η)(t−s)eB(η)(t−s−τ)

τ

[
∂sẐ0(η, s)−A(η)Ẑ0(η, s)

]
dη

∥∥
≤ C

1

2π

∫ +∞

−∞
e−ετσ1(t−s)e−ετm(η)(t−s)∥∂sẐ0(η, s)−A(η)Ẑ0(η, s)∥dη

≤ Ce−ετσ1teετσ1s
1

2π
sup
η∈R

∥∂sẐ0(η, s)−A(η)Ẑ0(η, s)∥
∫ +∞

−∞
e−ετm(η)(t−s)dη.

Furthermore,∫ 0

−τ

sup
ξ∈R

∥F−1[J2](ξ, t− s)ds∥

≤ Ce−ετσ1t
1

2π

∫ 0

−τ

eετσ1s sup
η∈R

∥∂sẐ0(η, s)−A(η)Ẑ0(η, s)∥
∫ +∞

−∞
e−ετm(η)(t−s)dηds

≤ Ct−1/2e−ετσ1t

∫ 0

−τ

(
∥∂sZ0(η, s)∥L1(R) + ∥Z0(η, s)∥W 1,1(R)

)
ds

≤ Ct−1/2e−ετσ1t
(
∥∂sZ0(η, s)∥L1([−τ,0];L1(R)) + ∥Z0(η, s)∥L1([−τ,0];W 1,1(R))

)
.

(4.18)



16 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

Substituting (4.17) and (4.18) into (4.13), we obtain the decay rate

∥U(·, t)∥L∞(R) + ∥V (·, t)∥L∞(R) ≤ Ct−1/2e−ετσ1t for c ≥ c∗. (4.19)

When ξ ≤ ξ0, one can see that eλ
∗(ξ−ξ0) ≤ 1. This together with (4.8) yields

0 ≤ U1(ξ, t) ≤ U+
1 (ξ, t) = eλ

∗(ξ−ξ0)U(ξ, t) ≤ U(ξ, t), (4.20)

0 ≤ V1(ξ, t) ≤ V +
1 (ξ, t) = eλ

∗(ξ−ξ0)V (ξ, t) ≤ V (ξ, t) (4.21)

for all ξ ∈ I and t > 0. It then follows from (4.19), (4.20) and (4.21) that (4.14)
holds. The proof is complete. □

Next, we present the decay rate for U1(ξ, t) and V1(ξ, t) in R \ I = (ξ0,+∞).

Lemma 4.7. It holds that

∥U1(·, t)∥L∞(R\I) + ∥V1(·, t)∥L∞(R\I) ≤ Ct−1/2e−γt for t > 0,

where γ > 0 is a small constant satisfying 0 < γ < min{ετσ1, δ1} with

δ1 = min
{
r1(1− b1), r2(b2 − 1),

1

τ
ln

1

b1

}
, (4.22)

when c > c∗, and γ = 0, when c = c∗.

Proof. By (4.3), we can see that (U1, V1) for c ≥ c∗ satisfies

U1t + cU1ξ ≤ d1D[U1] + r1(1− ϕ)U1 + r1ϕ(b1
∑
i∈Z

g1(i)V
τ
1 − U1),

ξ ∈ R \ I, t > 0,

V1t + cV1ξ ≤ d2D[V1] + r2(1− b2
∑
i∈Z

g2(i)ϕ
τ )V1 + r2(1− φ)(b2

∑
i∈Z

g1(i)U
τ
1 − V1),

ξ ∈ R \ I, t > 0,

U1|ξ=ξ0 ≤ C0(1 + t)−1/2e−ετσ1t, t > 0,

V1|ξ=ξ0 ≤ C0(1 + t)−1/2e−ετσ1t, t > 0,

U1|t=s = U10(ξ, s), V1|t=s = V10(ξ, s), ξ ∈ R \ I, s ∈ [−τ, 0].
(4.23)

where σ1 is defined in (4.15), and C0 = C1 when c > c∗, and C0 = C2 when
c = c∗. We choose ξ0 and t∗ large enough and γ small enough satisfying 0 < γ <
min{ετσ1, δ1} such that

r1ϕ(ξ1)
(
1− b1

(
1 +

τ

1 + t

)1/2
eγτ

)
+ r1(ϕ(ξ1)− 1)− 1

2
(1 + t+ τ)−1 − γ ≥ 0,

r2(φ(ξ1)− 1)
(
b2
(
1 +

τ

1 + t

)1/2
eγτ − 1

)
+ r2(b2ϕ

τ (ξ1)− 1)− 1

2
(1 + t+ τ)−1 − γ ≥ 0

for ξ1 ≥ ξ0 and t > t∗. When c > c∗, we let

U1(ξ, t) = V 1(ξ, t) = C̃(1 + t+ τ)−1/2e−γt for t > 0,

where C̃ is large enough such that (U1(ξ, t), V 1(ξ, t)) ≥ (U1(ξ, t), V1(ξ, t)) for (ξ, t) ∈
R×[0, t∗]. By a direct computation, we can verify that (U1, V 1) is an upper solution



EJDE-2025/02 WAVEFRONTS FOR A DISCRETE DIFFUSIVE COMPETITION SYSTEM 17

to (4.23), i.e.,

U1t + cU1ξ ≥ d1D[U1] + r1(1− ϕ)U1 + r1ϕ(b1
∑
i∈Z

g1(i)V
τ

1 − U1),

ξ ∈ R \ I, t > 0,

V 1t + cV 1ξ ≥ d2D[V 1] + r2(1− b1
∑
i∈Z

g2(i)ϕ
τ )V 1

+ r2(1− φ)(b2
∑
i∈Z

g1(i)U
τ

1 − V 1), ξ ∈ R \ I, t > 0,

U1|ξ=ξ0 ≥ C1(1 + t)−1/2e−ετσ1t, t > 0,

V 1|ξ=ξ0 ≥ C1(1 + t)−1/2e−ετσ1t, t > 0,

U1|t=s ≥ U10(ξ, s), V 1|t=s ≥ V 10(ξ, s), ξ ∈ R \ I, s ∈ [−τ, 0].

Hence, for c > c∗, we obtain

0 ≤ U1(ξ, t) ≤ U1(ξ, t) = C̃(1 + t+ τ)−1/2e−γt for t > 0, ξ ∈ R \ I, (4.24)

0 ≤ V1(ξ, t) ≤ V 1(ξ, t) = C̃(1 + t+ τ)−1/2e−γt for t > 0, ξ ∈ R \ I. (4.25)

When c = c∗, we let

U1(ξ, t) = V 1(ξ, t) = C̃(1 + t+ τ)−1/2 for t > 0.

Similarly, we can obtain that for c = c∗,

0 ≤ U1(ξ, t) ≤ U1(ξ, t) = C̃(1 + t+ τ)−1/2 for t > 0, ξ ∈ R \ I, (4.26)

0 ≤ V1(ξ, t) ≤ V 1(ξ, t) = C̃(1 + t+ τ)−1/2 for t > 0, ξ ∈ R \ I. (4.27)

Thus, Lemma 4.7 can be immediately obtained by (4.24)-(4.27). The proof is
complete. □

Lemma 4.8. It holds that

∥U1(·, t)∥L∞(R) + ∥V1(·, t)∥L∞(R) ≤ Ct−1/2e−ετσt for t > 0, c > c∗,

∥U1(·, t)∥L∞(R) + ∥V1(·, t)∥L∞(R) ≤ Ct−1/2 for t > 0, c = c∗,

where 0 < σ < min{σ1, δ1/ετ}, σ1 := −∆(λ∗, c) > 0 with c > c∗, and δ1 is given by
(4.22).

According to the definition of U1 and V1, we have the following convergence of
the solution (U+(x, t), V +(x, t)) to (ϕ(x+ ct), φ(x+ ct)).

Lemma 4.9. It holds that for c > c∗,

sup
x∈R

|U+(x, t)− ϕ(x+ ct)| ≤ Ct−1/2e−ετσt, ∀t > 0,

sup
x∈R

|V +(x, t)− φ(x+ ct)| ≤ Ct−1/2e−ετσt, ∀t > 0,

and for c = c∗,

sup
x∈R

|U+(x, t)− ϕ(x+ ct)| ≤ Ct−1/2, ∀t > 0,

sup
x∈R

|V +(x, t)− φ(x+ ct)| ≤ Ct−1/2, ∀t > 0,



18 Z.-J. YANG, G.-B. ZHANG, J. HE EJDE-2025/02

where 0 < σ < min{σ1, δ1/ετ}, σ1 := −∆(λ∗, c) > 0 with c > c∗, and δ1 is given by
(4.22).

Acknowledgments. B. G. Zhang was supported by the NSF of China [12261081],
the NSF of Gansu Province [21JR7RA121], and the Outstanding Youth Science
Fund of Gansu Province [20JR10RA111].

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Zhi-Jiao Yang

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu

730070, China
Email address: 1493801034@qq.com

Guo-Bao Zhang (corresponding author)
College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu

730070, China

Email address: zhanggb2011@nwnu.edu.cn

Juan He

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu
730070, China

Email address: 1099912126@qq.com


	1. Introduction
	2. Preliminaries and main results
	3. Existence
	4. Global stability
	Acknowledgments

	References