Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 38, pp. 1–24.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

TRANSITION FRONTS OF TWO SPECIES COMPETITION

LATTICE SYSTEMS IN RANDOM MEDIA

FENG CAO, LU GAO

Abstract. This article studies the existence and non-existence of transition

fronts for a two species competition lattice system in random media, and ex-

plores the influence of randomness of the media on the wave profiles and wave
speeds of such transition fronts. We first establish comparison principle for

sub-solutions and super-solutions of the related cooperative system. Next,

under some proper assumptions, we construct appropriate sub-solutions and
super-solutions for the cooperative system. Finally, we show that random tran-

sition fronts exist if their least mean speed is greater than an explicit threshold

and there is no random transition front with least mean speed less than the
threshold.

1. Introduction

This article studies the existence of transition fronts of the two species competi-
tion lattice random system

u̇i(t) = ui+1(t)− 2ui(t) + ui−1(t) + ui(t)(a1(θtω)

− b1(θtω)ui(t)− c1(θtω)vi(t)),

v̇i(t) = vi+1(t)− 2vi(t) + vi−1(t) + vi(t)(a2(θtω)

− b2(θtω)ui(t)− c2(θtω)vi(t)),

(1.1)

where i ∈ Z, t ∈ R, ω ∈ Ω, (Ω,F ,P) is a given probability space, θt is an ergodic
metric dynamical system on Ω, ai(·) : Ω→ R, bi(·) : Ω→ (0,∞), ci(·) : Ω→ (0,∞)
(i = 1, 2) are measurable, and for every ω ∈ Ω, aωi (t) := ai(θtω), bωi (t) := bi(θtω),
cωi (t) := ci(θtω) (i = 1, 2) are locally Hölder continuous in t ∈ R. Moreover, we
assume bi(θtω) > 0, ci(θtω) > 0 (i = 1, 2) for every ω ∈ Ω and t ∈ R.

System (1.1) is a spatial-discrete counterpart of the following two species com-
petition system with random dispersal,

∂tu = uxx + u(a1(θtω)− b1(θtω)u− c1(θtω)v),

∂tv = vxx + v(a2(θtω)− b2(θtω)u− c2(θtω)v),
(1.2)

Systems (1.1) and (1.2) are widely used to model the population dynamics of
competitive species when the movement or internal dispersal of the organisms oc-
curs between non-adjacent and adjacent locations, respectively (see, for example,

2010 Mathematics Subject Classification. 35C07, 34K05, 34A34, 34K60.
Key words and phrases. Transition fronts; competition systems; lattice systems;

random media.
c©2020 Texas State University.

Submitted December 16, 2019. Published April 26, 2020.

1



2 F. CAO, L. GAO EJDE-2020/38

[6, 21, 25, 26]). Note that system (1.2) often models the evolution of population
densities of competitive species in which the internal interaction or movement of
the organisms occurs randomly between adjacent spatial locations and is described
by the differential operator, referred to as the random dispersal operator. System
(1.1) arises in modeling the evolution of population densities of competitive species
in which the internal interaction or movement of the organisms occurs between
non-adjacent spatial locations and is described by the difference operator, referred
to as the discrete dispersal operator.

In (1.1) and (1.2), the functions a1, a2 represent the respective growth rates of
the two species, b1, c2 account for self-regulation of the respective species, and c1,
b2 account for competition between the two species. Two of the central dynamical
issues about (1.1) and (1.2) are spatial spreading speeds and traveling wave solu-
tions. A huge amount of research has been carried out toward the spatial spreading
speeds and traveling wave solutions of system (1.2) in spatially and temporally ho-
mogeneous media (see, for example, [7, 8, 13, 14, 15, 16, 17, 19, 20, 28]) or spatially
and/or temporally periodic media (see, for example, [9, 18, 29]). Recently, Bao,
Li, Shen and Wang in [2] studied the spatial spreading speeds and linear determi-
nacy of diffusive cooperative/competitive system in time recurrent environments.
Bao in [1] studied the spatial spreading speeds and generalized traveling waves of
competition system in general time heterogeneous media.

As for the lattice system arising in competition models, to the best of our knowl-
edge, there are only a few works on the related topics. The reader is referred
to [11, 12, 27] for the study on the spatial spreading speeds and traveling wave
solutions for competition lattice system in time independent habitats. We note
that Cao and Gao in [3] studied the existence and stability of random transition
fronts for KPP-type one species lattice random equations. The reader is referred to
[4, 5, 10, 24, 30] for the study on the spatial spreading speeds and traveling wave
solutions for KPP-type one species lattice equations in homogeneous or periodic or
time heterogeneous media.

In this article we study the traveling wave solutions of two species competition
lattice system with general time dependence. Since in nature, many systems are
subject to irregular influences arisen from various kind of noise, it is of great im-
portance to take the randomness of the environment into account and study the
existence and non-existence of random transition fronts of competition lattice sys-
tem in random media. Due to the lack of space regularity, we need finding new
approach to get the existence of transition fronts when dealing with spatial-discrete
system (1.1). We point out that the method used here can also be used to get the
existence and non-existence of transition fronts for two species competition lattice
system in general time dependent habitats. Besides, we will study the stability of
random transition fronts of competition lattice system elsewhere.

Let

l∞(Z) = {u = {ui}i∈Z : sup
i∈Z
|ui| <∞}

with norm ‖u‖ = ‖u‖∞ = supi∈Z |ui|, and

l∞,+(Z) = {u ∈ l∞(Z) : inf
i∈Z

ui ≥ 0}.

For u, v ∈ l∞(Z), we define

u ≥ v if u− v ∈ l∞,+(Z).



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 3

Then for any given (u0, v0) ∈ l∞(Z) × l∞(Z) , (1.1) has a unique (local) solution
(u(t;u0, v0, ω), v(t;u0, v0, ω)) = {(ui(t;u0, v0, ω), vi(t;u0, v0, ω))}i∈Z with initial da-
tum (u(0;u0, v0, ω), v(0;u0, v0, ω)) = (u0, v0). Note that, if u0 ∈ l∞,+(Z), v0 ∈
l∞,+(Z), then (u(t;u0, v0, ω), v(t;u0, v0, ω)) exists for all t ≥ 0 and u(t;u0, v0, ω) ∈
l∞,+(Z), v(t;u0, v0, ω) ∈ l∞,+(Z) for all t ≥ 0. A solution (u(t;ω), v(t;ω)) =
{ui(t;ω), vi(t;ω)}i∈Z of (1.1) is called spatially homogeneous if ui(t) = uj(t) and
vi(t) = vj(t) for all i, j ∈ Z.

Note that (1.1) contains the following two sub-systems,

u̇i(t) = ui+1(t)− 2ui(t) + ui−1(t) + ui(t)(a1(θtω)− b1(θtω)ui(t)), (1.3)

and

v̇i(t) = vi+1(t)− 2vi(t) + vi−1(t) + vi(t)(a2(θtω)− c2(θtω)vi(t)). (1.4)

First we give some notation and assumptions related to (1.1). Let

a(ω) = lim inf
t−s→∞

1

t− s

∫ t

s

a(θτω)dτ := lim
r→∞

inf
t−s≥r

1

t− s

∫ t

s

a(θτω)dτ,

a(ω) = lim sup
t−s→∞

1

t− s

∫ t

s

a(θτω)dτ := lim
r→∞

sup
t−s≥r

1

t− s

∫ t

s

a(θτω)dτ,

where a(ω) could be ai(ω), bi(ω), ci(ω) (i = 1 or 2) or any similar function. We
call a(·) and a(·) the least mean and the greatest mean of a(·), respectively. It is
easy to obtain

a(θtω) = a(ω), a(θtω) = a(ω) for all t ∈ R,

and

a(ω) = lim inf
t,s∈Q,t−s→∞

1

t− s

∫ t

s

a(θτω)dτ, a(ω) = lim sup
t,s∈Q,t−s→∞

1

t− s

∫ t

s

a(θτω)dτ.

Then a(ω) and a(ω) are measurable in ω. The ergodicity of the metric dynamical
system (Ω,F ,P, {θt}t∈R) implies that, there are a, a ∈ R and a measurable subset
Ω0 ⊂ Ω with P(Ω0) = 1 such that

θtΩ0 = Ω0 ∀t ∈ R

lim inf
t−s→∞

1

t− s

∫ t

s

a(θτω)dτ = a ∀ω ∈ Ω0

lim sup
t−s→∞

1

t− s

∫ t

s

a(θτω)dτ = a ∀ω ∈ Ω0,

That is, a(ω) and a(ω) are independent of ω in a subset of Ω of full measure (see
Lemma 2.3).

Throughout this paper, we assume that the trivial solution (0, 0) of (1.1) is
unstable with respect to perturbation in l∞(Z)× l∞(Z), i.e.

(H1) ai(ω) = lim inft−s→∞
1
t−s

∫ t
s
ai(θτω)dτ > 0 (i = 1, 2) for a.a. ω ∈ Ω.

Note that (H1) implies that (1.1) has two semi-trivial spatially homogeneous posi-
tive solutions (u∗(t;ω), 0) = (φ∗(θtω), 0) ∈ Int l∞,+(Z)×l∞,+(Z) and (0, v∗(t;ω)) =
(0, ψ∗(θtω)) ∈ l∞,+(Z)× Int l∞,+(Z) for some random equilibria φ∗ and ψ∗, where
u∗(t;ω) = φ∗(θtω) is the unique spatially homogeneous positive solution of (1.3),
and v∗(t;ω) = ψ∗(θtω) is the unique spatially homogeneous positive solution of
(1.4) (see [4, Theorem 1.1] and [22, Theorem A]).



4 F. CAO, L. GAO EJDE-2020/38

We also assume that

(H2) (0, v∗(t;ω)) is linearly unstable in l∞,+(Z)× l∞,+(Z), that is,

a1(ω)− c1(ω)v∗(·;ω) > 0.

Note that (u∗(t;ω), 0) is linearly and globally stable in l∞,+(Z)× l∞,+(Z), that is,

a2(ω)− b2(ω)u∗(·;ω) < 0,

and for any (u0, v0) ∈ l∞,+(Z) × l∞,+(Z) with u0 6= 0 and a.a. ω ∈ Ω, ui(t;u0, v0,
θt0ω) − u∗(t + t0;ω) → 0 and vi(t;u0, v0, θt0ω) → 0 as t → ∞ uniformly in i ∈ Z
and t0 ∈ R.

We remark that if a1(ω)− c1(ω)v∗(·;ω) > 0, then (0, v∗(t;ω)) is unstable in

l∞,+(Z) × l∞,+(Z), and if a2(ω)− b2(ω)u∗(·;ω) < 0, then (u∗(t;ω), 0) is locally

stable in l∞,+(Z)× l∞,+(Z), and if ai(ω) > 0 (i = 1, 2), aω1L >
cω1Ma

ω
2M

cω2L
and aω2M ≤

aω1Lb
ω
2L

bω1M
for any ω ∈ Ω, then (u∗(t;ω), 0) is globally stable and (0, v∗(t;ω)) is unstable

in l∞,+(Z) × l∞,+(Z), where aωiL = inft∈R ai(θtω), aωiM = supt∈R ai(θtω) and bωiL,
bωiM , cωiL, cωiM are defined similarly (This can be proved similarly as [1, Proposition
2.4]).

Now we present the third standing hypothesis.

(H3) For any ω ∈ Ω, inft∈R b2(θtω) > 0, bi(θtω) ≥ ci(θtω) (i = 1, 2) and

a1(θtω)− c1(θtω)v∗(t;ω) ≥ a2(θtω)− 2c2(θtω)v∗(t;ω) + b2(θtω)v∗(t;ω)

for t ∈ R.

Under the assumptions (H1)–(H3), one of the most interesting dynamical prob-
lems is to study the existence of random transition front (generalized traveling
wave) solutions connecting (u∗(t;ω), 0) and (0, v∗(t;ω)) for (1.1). To do so, we first
transform (1.1) to a cooperative system via the standard change of variables,

ũi = ui, ṽi = v∗(t;ω)− vi.
Dropping the tilde, (1.1) is transformed into

u̇i = Hui + ui(a1(θtω)− b1(θtω)ui − c1(θtω)(v∗(t;ω)− vi)),
v̇i = Hvi + b2(θtω)(v∗(t;ω)− vi)ui + vi(a2(θtω)

− 2c2(θtω)v∗(t;ω) + c2(θtω)vi),

(1.5)

where

Hui(t) := ui+1(t)− 2ui(t) + ui−1(t), i ∈ Z, t ∈ R.
It is clear that (1.5) is cooperative in the region ui(t) ≥ 0 and 0 ≤ vi(t) ≤

v∗(t;ω), and the trivial solution (0, 0) of (1.1) becomes (0, v∗(t;ω)), the semitrivial
solutions (0, v∗(t;ω)) and (u∗(t;ω), 0) of (1.1) becomes (0, 0) and (u∗(t;ω), v∗(t;ω)),
respectively. To study the random transition front solutions of (1.1) connecting
(u∗(t;ω), 0) and (0, v∗(t;ω)) is then equivalent to study the random transition front
solutions of (1.5) connecting (u∗(t;ω), v∗(t;ω)) and (0, 0).

We denote (u(t;u0, v0, ω), v(t;u0, v0, ω)) = {(ui(t;u0, v0, ω), vi(t;u
0, v0, ω))}i∈Z

as the solution of (1.5) with (u(0;u0, v0, ω), v(0;u0, v0, ω)) = (u0, v0) ∈ l∞(Z) ×
l∞(Z). For any (u1, u2) ∈ l∞(Z) × l∞(Z) and (v1, v2) ∈ l∞(Z) × l∞(Z), the re-
lation (u1, u2) < (v1, v2) ((u1, u2) ≤ (v1, v2) resp.) is to be understood compo-
nentwise: ui < vi (ui ≤ vi) for each i. Other relations like “max”, “min”, “sup”,
“inf” can be similarly understood. Then it is clear that, if (u0, v0) ≥ (0, 0), then



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 5

(u(t;u0, v0, ω), v(t;u0, v0, ω)) exists for all t ≥ 0 and (u(t;u0, v0, ω), v(t;u0, v0, ω)) ≥
(0, 0) for all t ≥ 0 (see Proposition 2.1). A solution (u(t;ω), v(t;ω)) of (1.5) is called
an entire solution if it is a solution of (1.5) for t ∈ R.

Definition 1.1 (Random transition front). An entire solution (u(t;ω), v(t;ω)) is
called a random transition front or a random generalized traveling wave of (1.5)
connecting (0, 0) and (u∗(t;ω), v∗(t;ω)) if for a.a. ω ∈ Ω,

(ui(t;ω), vi(t;ω)) = (Φ(i−
∫ t

0

c(s;ω)ds, θtω), Ψ(i−
∫ t

0

c(s;ω)ds, θtω))

for some Φ(x, ω), Ψ(x, ω) (x ∈ R) and c(t;ω), where Φ(x, ω), Ψ(x, ω) and c(t;ω) are
measurable in ω, and for a.a. ω ∈ Ω,

(0, 0) < (Φ(x, ω), Ψ(x, ω)) < (u∗(t;ω), v∗(t;ω)),

lim
x→−∞

(Φ(x, θtω), Ψ(x, θtω)) = (u∗(t;ω), v∗(t;ω)),

lim
x→∞

(Φ(x, θtω), Ψ(x, θtω)) = (0, 0) uniformly in t ∈ R.

Suppose that (u(t;ω), v(t;ω)) = {(ui(t;ω), vi(t;ω))}i∈Z with (ui(t;ω), vi(t;ω)) =

(Φ(i−
∫ t
0
c(s;ω)ds, θtω), Ψ(i−

∫ t
0
c(s;ω)ds, θtω)) is a random transition front of (1.5).

If Φ(x, ω) and Ψ(x, ω) are non-increasing with respect to x for a.a. ω ∈ Ω and all
x ∈ R, then (u(t;ω), v(t;ω)) is said to be a monotone random transition front. If
there is cinf ∈ R such that for a.a. ω ∈ Ω,

lim inf
t−s→∞

1

t− s

∫ t

s

c(τ ;ω)dτ = cinf ,

then cinf is called its least mean speed.
Note that the ergodicity of the metric dynamical system (Ω,F ,P, {θt}t∈R) im-

plies a1(ω)− c1(ω)v∗(·;ω) = lim inft−s→∞
1
t−s

∫ t
s

(a1(θτω)− c1(θτω)v∗(τ ;ω))dτ is
independent of ω in a subset Ω0 ⊂ Ω of full measure. We denote

λ = a1(ω)− c1(ω)v∗(·;ω)

for ω ∈ Ω0. For given µ > 0, let

c0 := inf
µ>0

eµ + e−µ − 2 + λ

µ
,

By [4, Lemma 5.1], there is a unique µ∗ > 0 such that

c0 =
eµ
∗

+ e−µ
∗ − 2 + λ

µ∗

and for any γ > c0, the equation γ = eµ+e−µ−2+λ
µ has exactly two positive solutions

for µ.
Now we are in a position to state the main results on the existence and non-

existence of random transition fronts of two species cooperative lattice systems in
random media.

Theorem 1.2. Assume (H1)–(H3) hold. Then we have:
(i) For any given γ > c0, there is a monotone random transition front of (1.5)

with least mean speed cinf = γ. More precisely, for any given γ > c0, let 0 < µ < µ∗

be such that eµ+e−µ−2+λ
µ = γ. Then (1.5) has a monotone random transition front

(u(t;ω), v(t;ω)) = {(ui(t;ω), vi(t;ω))}i∈Z with ui(t;ω) = Φ(i−
∫ t
0
c(s;ω)ds, θtω) and



6 F. CAO, L. GAO EJDE-2020/38

vi(t;ω) = Ψ(i −
∫ t
0
c(s;ω)ds, θtω), where c(t;ω) = eµ+e−µ−2+a1(θtω)−c1(θtω)v∗(t;ω)

µ

and hence cinf = eµ+e−µ−2+λ
µ = γ. Moreover, for any ω ∈ Ω0,

lim
x→−∞

(Φ(x, θtω), Ψ(x, θtω)) = (u∗(t;ω), v∗(t;ω)),

lim
x→∞

(Φ(x, θtω), Ψ(x, θtω)) = (0, 0)

uniformly in t ∈ R.
(ii) There is no random transition front of (1.5) with least mean speed less than

c0.

Remark 1.3. (i) When ai, bi, ci (i = 1, 2) are constants, our existence result of the
transition front is consistent with the result obtained in [12, Theorems 1, 4]. Also,
we obtain the non-existence result of the transition front.

(ii) We leave as an open problem the case cinf = c0, that is, the existence of
random transition front of (1.5) with least mean speed cinf = c0.

The rest of this article is organized as follows. In Section 2, we establish the
comparison principle for sub-solutions and super-solutions of (1.5) and prove some
basic properties and fundamental lemmas to be used in later section. We prove the
existence and non-existence of random transition fronts after constructing appro-
priate sub-solutions and super-solutions of (1.5) in Section 3.

2. Preliminaries

In this section, we present some preliminary material to be used in later sections.
We first present a comparison principle for sub-solutions and super-solutions of (1.5)
and prove the convergence of solutions on compact subsets. Next, we present some
useful lemmas including a technical lemma.

Consider first the following space continuous version of (1.5),

∂tu = Hu+ u(a1(θtω)− b1(θtω)u− c1(θtω)(v∗(t;ω)− v)),

∂tv = Hv + b2(θtω)(v∗(t;ω)− v)u+ v(a2(θtω)

− 2c2(θtω)v∗(t;ω) + c2(θtω)v),

(2.1)

where

u = u(x, t), v = v(x, t),

Hu(x, t) := u(x+ 1, t) + u(x− 1, t)− 2u(x, t), x ∈ R, t ∈ R.
Let

l∞(R) = {u : R→ R : sup
x∈R
|u(x)| <∞}

with the norm ‖u‖ = supx∈R |u(x)|, and

l∞,+(R) = {u ∈ l∞(R) : inf
x∈R

u(x) ≥ 0}.

For u, v ∈ l∞(R), we define

u ≥ v if u− v ∈ l∞,+(R).

Recall that for any (u0, v0) ∈ l∞(Z)× l∞(Z),

(u(t;u0, v0, ω), v(t;u0, v0, ω)) = {(ui(t;u0, v0, ω), vi(t;u
0, v0, ω))}i∈Z

is the solution of (1.5) with (ui(0;u0, v0, ω), vi(0;u0, v0, ω)) = (u0i , v
0
i ) for i ∈ Z. For

any (u0, v0) ∈ l∞(R)× l∞(R), let (u(x, t;u0, v0, ω), v(x, t;u0, v0, ω)) be the solution



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 7

of (2.1) with (u(x, 0;u0, v0, ω), v(x, 0;u0, v0, ω)) = (u0(x), v0(x)). For any (u1, u2),
(v1, v2) ∈ l∞(R)× l∞(R), the relation (u1, u2) < (v1, v2) ((u1, u2) ≤ (v1, v2) resp.)
is also to be understood componentwise: ui < vi (ui ≤ vi) for each i.

Let

f(t, u, v, ω) = u(a1(θtω)− b1(θtω)u− c1(θtω)(v∗(t;ω)− v)),

g(t, u, v, ω) = b2(θtω)(v∗(t;ω)− v)u+ v(a2(θtω)− 2c2(θtω)v∗(t;ω) + c2(θtω)v).

A pair of function (u(x, t;ω), v(x, t;ω)) on R × [0, T ) which is continuous in t is
called a super-solution or sub-solution of (2.1) (resp. (1.5)) if for a.a. ω ∈ Ω and
any given x ∈ R (resp. x ∈ Z), u(x, t;ω) and v(x, t;ω) are absolutely continuous in
t ∈ [0, T ), and

ut(x, t;ω) ≥ Hu(x, t;ω) + f(t, u, v, ω)

vt(x, t;ω) ≥ Hv(x, t;ω) + g(t, u, v, ω)

for a.a. t ∈ [0, T ), or

ut(x, t;ω) ≤ Hu(x, t;ω) + f(t, u, v, ω)

vt(x, t;ω) ≤ Hv(x, t;ω) + g(t, u, v, ω)

for a.a. t ∈ [0, T ).
A pair of function is said to be a generalized super-solution (resp. sub-solution)

if it is the infimum (resp. supremum) of a finite number of super-solutions (resp.
sub-solutions).

Now we are in a position to present a comparison principle for solutions of (2.1),
the comparison principle for solutions of (1.5) can be proved similarly.

Proposition 2.1 (Comparison principle). (1) Suppose that (u1(x, t;ω), v1(x, t;ω))
is a bounded sub-solution of (2.1) on [0, T ) and that (u2(x, t;ω), v2(x, t;ω)) is a
bounded super-solution of (2.1) on [0, T ) and (ui(x, t;ω), vi(x, t;ω)) ∈ [0, u∗(t;ω)]×
[0, v∗(t;ω)] (i = 1, 2) for x ∈ R and t ∈ [0, T ). If

(u1(·, 0;ω), v1(·, 0;ω)) ≤ (u2(·, 0;ω), v2(·, 0;ω)),

then

(u1(·, t;ω), v1(·, t;ω)) ≤ (u2(·, t;ω), v2(·, t;ω)) for t ∈ [0, T ).

(2) Suppose that (ui(x, t;ω), vi(x, t;ω)) ∈ [0, u∗(t;ω)]× [0, v∗(t;ω)] (i = 1, 2) are
bounded and satisfy that for any given x ∈ R, ui(x, t;ω), vi(x, t;ω) (i = 1, 2) are
absolutely continuous in t ∈ [0,∞), and

∂tu2(x, t;ω)− (Hu2(x, t;ω) + f(t, u2, v2, ω))

> ∂tu1(x, t;ω)− (Hu1(x, t;ω) + f(t, u1, v1, ω)),

∂tv2(x, t;ω)− (Hv2(x, t;ω) + g(t, u2, v2, ω))

> ∂tv1(x, t;ω)− (Hv1(x, t;ω) + g(t, u1, v1, ω))

for t > 0. Moreover, suppose that

(u2(·, 0;ω), v2(·, 0;ω)) ≥ (u1(·, 0;ω), v1(·, 0;ω)).

Then (u2(·, t;ω), v2(·, t;ω)) > (u1(·, t;ω), v1(·, t;ω)) for t > 0.



8 F. CAO, L. GAO EJDE-2020/38

Proof. (1) Let

Q1(x, t;ω) = ect(u2(x, t;ω)−u1(x, t;ω)), Q2(x, t;ω) = ect(v2(x, t;ω)−v1(x, t;ω)),

where c := c(ω) is to be determined later. Then there is a measurable subset Ω̄ of
Ω with P(Ω̄) = 0 such that for any ω ∈ Ω \ Ω̄, Q1(x, t;ω) and Q2(x, t;ω) satisfy

∂tQ1 ≥ Q1(x+ 1, t;ω) +Q1(x− 1, t;ω) + a1(x, t;ω)Q1 + b1(x, t;ω)Q2,

∂tQ2 ≥ Q2(x+ 1, t;ω) +Q2(x− 1, t;ω) + a2(x, t;ω)Q1 + b2(x, t;ω)Q2,
(2.2)

where

a1(x, t;ω) = c− 2 + fu(t, u∗1, v
∗
1 , ω), b1(x, t;ω) = fv(t, u

∗
1, v
∗
1 , ω),

a2(x, t;ω) = gu(t, u∗2, v
∗
2 , ω), b2(x, t;ω) = c− 2 + gv(t, u

∗
2, v
∗
2 , ω)

for some u∗i = u∗i (x, t;ω) (i = 1, 2) between u1(x, t;ω) and u2(x, t;ω) and some
v∗i = v∗i (x, t;ω) (i = 1, 2) between v1(x, t;ω) and v2(x, t;ω).

Since (2.1) is cooperative in [0, u∗(t;ω)]× [0, v∗(t;ω)], we have b1(x, t;ω) ≥ 0 and
a2(x, t;ω) ≥ 0. By the boundedness of ui(x, t;ω) and vi(x, t;ω) (i = 1, 2), we can
choose c = c(ω) > 0 such that b2(x, t;ω) ≥ 0 and a1(x, t;ω) ≥ 0.

We claim that Qi(x, t;ω) ≥ 0 (i = 1, 2) for x ∈ R and t ∈ [0, T ]. Let p0(ω) :=
maxi=1,2 max(x,t)∈R×[0,T ]{ai(x, t;ω), bi(x, t;ω)}. It suffices to prove the claim for

x ∈ R and t ∈ (0, T0] with T0 = min{T, 1
2(1+p0(ω))

}. Assume that there are some

x̃ ∈ R and t̃ ∈ (0, T0] such that Q1(x̃, t̃;ω) < 0 or Q2(x̃, t̃;ω) < 0. Then there is
t0 ∈ (0, T0) such that

Qinf
1 (ω) := inf

(x,t)∈R×[0,t0]
Q1(x, t;ω) < 0 or Qinf

2 (ω) := inf
(x,t)∈R×[0,t0]

Q2(x, t;ω) < 0.

Without loss of generality, we assume that Qinf
1 (ω) ≤ Qinf

2 (ω). Observe that there
are xn ∈ R and tn ∈ (0, t0] such that

Q1(xn, tn;ω)→ Qinf
1 (ω) as n→∞.

By (2.2) and the fundamental theorem of calculus for Lebesgue integrals, we obtain

Q1(xn, tn;ω)−Q1(xn, 0;ω)

≥
∫ tn

0

[Q1(xn + 1, t;ω) +Q1(xn − 1, t;ω) + a1(xn, t;ω)Q1(xn, t;ω)

+ b1(xn, t;ω)Q2(xn, t;ω)]dt

≥
∫ tn

0

[2Qinf
1 (ω) + a1(xn, t;ω)Qinf

1 (ω) + b1(xn, t;ω)Qinf
2 (ω)]dt

≥
∫ tn

0

[2Qinf
1 (ω) + 2p0(ω)Qinf

1 (ω)]dt

≥ 2(1 + p0(ω))t0Qinf
1 (ω) for n ≥ 1.

Note that Q1(xn, 0;ω) ≥ 0, we then have

Q1(xn, tn;ω) ≥ 2(1 + p0(ω))t0Qinf
1 (ω) for n ≥ 1.

Letting n→∞, we obtain

Qinf
1 (ω) ≥ 2(1 + p0(ω))t0Qinf

1 (ω) > Qinf
1 (ω).



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 9

A contradiction. Hence Qi(x, t;ω) ≥ 0 (i = 1, 2) for x ∈ R and t ∈ [0, T ], which
implies that (u1(x, t;ω), v1(x, t;ω)) ≤ (u2(x, t;ω), v2(x, t;ω)) for ω ∈ Ω \ Ω̄, x ∈ R
and t ∈ [0, T ].

(2) Since (2.1) is cooperative in [0, u∗(t;ω)] × [0, v∗(t;ω)], then for ω ∈ Ω \ Ω̄,
by the similar arguments as getting (2.2), we can find c(ω), µ(ω) > 0 such that for
any given x ∈ R,

∂tw(x, t;ω) > w(x+ 1, t;ω) + w(x− 1, t;ω) + µ(ω)w(x, t;ω) for t > 0,

where w(x, t;ω) = ec(ω)t(u2(x, t, ω)− u1(x, t, ω)). Thus we have that for any given
x ∈ R,

w(x, t;ω) > w(x, 0;ω) +

∫ t

0

[w(x+ 1, s;ω) + w(x− 1, s;ω) + µ(ω)w(x, s;ω)]ds.

By the arguments in (1), w(x, t;ω) ≥ 0 for all x ∈ R and t ≥ 0. It then follows that
w(x, t;ω) > w(x, 0;ω) ≥ 0 and and hence u2(x, t;ω) > u1(x, t;ω) for ω ∈ Ω \ Ω̄,
x ∈ R and t > 0. Similarly, we can get that v2(x, t;ω) > v1(x, t;ω) for ω ∈ Ω \ Ω̄,
x ∈ R and t > 0. �

Proposition 2.2. Suppose that (un, vn) ∈ l∞,+(R) × l∞,+(R) (n = 1, 2, . . . ) and
(u0, v0) ∈ l∞,+(R) × l∞,+(R) with {‖un‖}, {‖vn‖} bounded. If (un(x), vn(x)) →
(u0(x), v0(x)) as n → ∞ uniformly in x on bounded sets, then for each t > 0,
(u(x, t;un, vn, θt0ω), v(x, t;un, vn, θt0ω)) → (u(x, t;u0, v0, θt0ω), v(x, t;u0, v0, θt0ω))
as n→∞ uniformly in x on bounded sets and t0 ∈ R.

Proof. Fix any ω ∈ Ω, and let

un(x, t; θt0ω) = u(x, t;un, vn, θt0ω)− u(x, t;u0, v0, θt0ω),

vn(x, t; θt0ω) = v(x, t;un, vn, θt0ω)− v(x, t;u0, v0, θt0ω).

Then

∂tu
n = Hun + an1 (x, t; θt0ω)un + bn1 (x, t; θt0ω)vn,

∂tv
n = Hvn + an2 (x, t; θt0ω)un + bn2 (x, t; θt0ω)vn,

where

an1 (x, t; θt0ω) = fu(t, un1 (x, t; θt0ω), vn1 (x, t; θt0ω), θt0ω),

bn1 (x, t; θt0ω) = fv(t, u
n
1 (x, t; θt0ω), vn1 (x, t; θt0ω), θt0ω),

an2 (x, t; θt0ω) = gu(t, un2 (x, t; θt0ω), vn2 (x, t; θt0ω), θt0ω),

bn2 (x, t; θt0ω) = gv(t, u
n
2 (x, t; θt0ω), vn2 (x, t; θt0ω), θt0ω),

for un1 (x, t; θt0ω), un2 (x, t; θt0ω) between u(x, t;un, vn, θt0ω) and u(x, t;u0, v0, θt0ω),
and vn1 (x, t; θt0ω), vn2 (x, t; θt0ω) between v(x, t;un, vn, θt0ω) and v(x, t;u0, v0, θt0ω).

Take ρ > 0, and let

X(ρ) = {(u, v) : R→ R2 : (u(·)e−ρ|·|, v(·)e−ρ|·|) ∈ l∞(R)× l∞(R)}
with the norm ‖(u, v)‖X(ρ) = supx∈R(|u(x)|+ |v(x)|)e−ρ|x|. Observe that (H,H) :
X(ρ)→ X(ρ), given by

(H,H)(u, v) = (Hu,Hv),

is a bounded linear operator. Note also that ani (x, t; θt0ω) and bni (x, t; θt0ω) are
uniformly bounded (i = 1, 2). Then there are M > 0 and α > 0 such that

‖e(H,H)t‖X(ρ) ≤Meαt



10 F. CAO, L. GAO EJDE-2020/38

and |ani (x, t; θt0ω)| ≤M , |bni (x, t; θt0ω)| ≤M . Hence,

(un(·, t; θt0ω), vn(·, t; θt0ω))

= e(H,H)t(un(·, 0; θt0ω), vn(·, 0; θt0ω))

+

∫ t

0

e(H,H)(t−τ)[an1 (·, τ ; θt0ω)un(·, τ ; θt0ω) + bn1 (·, τ ; θt0ω)vn(·, τ ; θt0ω),

an2 (·, τ ; θt0ω)un(·, τ ; θt0ω) + bn2 (·, τ ; θt0ω)vn(·, τ ; θt0ω)]dτ

and then

‖(un(·, t; θt0ω), vn(·, t; θt0ω))‖X(ρ)

≤Meαt‖(un(·, 0; θt0ω), vn(·, 0; θt0ω))‖X(ρ)

+M2

∫ t

0

eα(t−τ)‖(un(·, τ ; θt0ω), vn(·, τ ; θt0ω))‖X(ρ)dτ.

By Gronwall’s inequality,

‖(un(·, t; θt0ω), vn(·, t; θt0ω))‖X(ρ) ≤ e(α+M
2)tM‖(un(·, 0; θt0ω), vn(·, 0; θt0ω))‖X(ρ).

Note that ‖(un(·, 0; θt0ω), vn(·, 0; θt0ω))‖X(ρ) → 0 as n → ∞ uniformly in t0 ∈ R.
It then follows that

(un(x, t; θt0ω), vn(x, t; θt0ω))→ (0, 0) as n→∞

uniformly in x on bounded sets and t0 ∈ R. The proof is complete. �

Now we present some lemmas including the technical results.

Lemma 2.3. a(·), a(·), a(·) ∈ L1(Ω,F ,P). Also a(ω) and a(ω) are independent of
ω for a.a. ω ∈ Ω.

The proof of the above lemma follows from [23, Lemma 2.1].

Lemma 2.4. Suppose that for ω ∈ Ω, aω(t) = a(θtω) ∈ C(R, (0,∞)). Then for
a.a. ω ∈ Ω,

a = sup
A∈W 1,∞

loc (R)∩L∞(R)
ess inft∈R(A′ + aω)(t).

The proof of the above lemma follows from [23, Lemma 2.2] and Lemma 2.3.
Note that by (H3) there is a strictly positive solution h(t;ω) of

dv

dt
−(a2(θtω)−2c2(θtω)v∗(t;ω))v−b2(θtω)v∗(t;ω) = −(a1(θtω)−c1(θtω)v∗(t;ω))v.

Denote

c(t;ω, µ) =
eµ + e−µ − 2 + a1(θtω)− c1(θtω)v∗(t;ω)

µ
.

Lemma 2.5. Let ω ∈ Ω0 and 0 < σ � 1. Then for any µ, µ̃ with 0 < µ <
µ̃ < min{2µ, µ∗}, there exist {tk}k∈Z with tk < tk+1 and limk→±∞ tk = ±∞,

Aω ∈ W 1,∞
loc (R) ∩ L∞(R) with Aω(·) ∈ C1((tk, tk+1)) for k ∈ Z, and dω > 0 such

that for any d ≥ dω the functions

ũ(x, t, ω) := e−µ(x−
∫ t
0
c(s;ω,µ)ds) − de(

µ̃
µ−1)Aω(t)−µ̃(x−

∫ t
0
c(s;ω,µ)ds),

ṽ(x, t, ω) := σe−µ(x−
∫ t
0
c(s;ω,µ)ds)h(t;ω)− σde(

µ̃
µ−1)Aω(t)−µ̃(x−

∫ t
0
c(s;ω,µ)ds)h(t;ω)



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 11

satisfy

∂tũ ≤ Hũ+ ũ(a1(θtω)− b1(θtω)ũ− c1(θtω)(v∗(t;ω)− ṽ)),

∂tṽ ≤ Hṽ + b2(θtω)(v∗(t;ω)− ṽ)ũ+ ṽ(a2(θtω)− 2c2(θtω)v∗(t;ω) + c2(θtω)ṽ),

for t ∈ (tk, tk+1), x ≥
∫ t
0
c(s;ω, µ)ds+ ln d

µ̃−µ + Aω(t)
µ , k ∈ Z.

Proof. For a given ω ∈ Ω0 and 0 < µ < µ̃ < min{2µ, µ∗}, by the arguments in

the proof of [4, Lemma 5.1] we can get that eµ̃+e−µ̃−2+λ
µ̃ < eµ+e−µ−2+λ

µ , and hence

λ > µ(eµ̃+e−µ̃−2)−µ̃(eµ+e−µ−2)
µ̃−µ . Let 0 < δ � 1 be such that

(1− δ)λ > µ(eµ̃ + e−µ̃ − 2)− µ̃(eµ + e−µ − 2)

µ̃− µ
.

It follows from Lemma 2.4 that there exist T > 0 and Aω ∈ W 1,∞
loc (R) ∩ L∞(R)

such that Aω(·) ∈ C1((tk, tk+1)) with tk = kT for k ∈ Z, and

(1− δ)(a1(θtω)− c1(θtω)v∗(t;ω)) +A′ω(t) ≥ µ(eµ̃ + e−µ̃ − 2)− µ̃(eµ + e−µ − 2)

µ̃− µ
(2.3)

for all t ∈ (tk, tk+1), k ∈ Z.
Now we fix the above δ > 0 and Aω(t). Let

ξ(x, t;ω) = x−
∫ t

0

c(s;ω, µ)ds,

ũ(x, t, ω) = e−µξ(x,t;ω) − de(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω),

ṽ(x, t, ω) = σe−µξ(x,t;ω)h(t;ω)− σde(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)h(t;ω)

with d > 1 to be determined later. Then we have

∂tũ− [Hũ+ ũ(a1(θtω)− b1(θtω)ũ− c1(θtω)(v∗(t;ω)− ṽ))]

= µc(t;ω, µ)e−µξ(x,t;ω) + d[−(
µ̃

µ
− 1)A′ω(t)− µ̃c(t;ω, µ)]e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

− [(eµ + e−µ − 2)e−µξ(x,t;ω) − d(eµ̃ + e−µ̃ − 2)e(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)]

− ũ[a1(θtω)− b1(θtω)ũ− c1(θtω)(v∗(t;ω)− ṽ)]

= d[−(
µ̃

µ
− 1)A′ω(t)− µ̃c(t;ω, µ) + eµ̃ + e−µ̃ − 2 + a1(θtω)− c1(θtω)v∗(t;ω)]

× e(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω) + ũ[b1(θtω)ũ− c1(θtω)ṽ]

(2.4)
Recall that

c(t;ω, µ) =
eµ + e−µ − 2 + a1(θtω)− c1(θtω)v∗(t;ω)

µ
.



12 F. CAO, L. GAO EJDE-2020/38

Then by (2.4) we obtain

∂tũ− [Hũ+ ũ(a1(θtω)− b1(θtω)ũ− c1(θtω)(v∗(t;ω)− ṽ))]

= d(
µ̃

µ
− 1)[

µ(eµ̃ + e−µ̃ − 2)− µ̃(eµ + e−µ − 2)

µ̃− µ
− (1− δ)(a1(θtω)− c1(θtω)v∗(t;ω))−A′ω(t)]e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

+ ũ[b1(θtω)ũ− c1(θtω)σh(t;ω)ũ]

− δd(
µ̃

µ
− 1)(a1(θtω)− c1(θtω)v∗(t;ω))e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

≤ d(
µ̃

µ
− 1)[

µ(eµ̃ + e−µ̃ − 2)− µ̃(eµ + e−µ − 2)

µ̃− µ
− (1− δ)(a1(θtω)− c1(θtω)v∗(t;ω))

−A′ω(t)]e(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω) + b1(θtω)ũ2

− δd(
µ̃

µ
− 1)(a1(θtω)− c1(θtω)v∗(t;ω))e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

= d(
µ̃

µ
− 1)[

µ(eµ̃ + e−µ̃ − 2)− µ̃(eµ + e−µ − 2)

µ̃− µ
− (1− δ)(a1(θtω)− c1(θtω)v∗(t;ω))−A′ω(t)]e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

− [dδ(
µ̃

µ
− 1)e(

µ̃
µ−1)Aω(t)(a1(θtω)− c1(θtω)v∗(t;ω))

− b1(θtω)e−(2µ−µ̃)ξ(x,t;ω)]e−µ̃ξ(x,t;ω) + d[−2e−µξ(x,t;ω)

+ de(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)]e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)b1(θtω)

(2.5)

for t ∈ (tk, tk+1). Note that

∂tṽ − [Hṽ + b2(θtω)(v∗(t;ω)− ṽ)ũ+ ṽ(a2(θtω)− 2c2(θtω)v∗(t;ω) + c2(θtω)ṽ)]

= σ[(a2(θtω)− 2c2(θtω)v∗(t;ω))h(t;ω) + b2(θtω)v∗(t;ω)

− (a1(θtω)− c1(θtω)v∗(t;ω))h(t;ω)]ũ+ σh(t;ω)∂tũ− σh(t;ω)Hũ

− b2(θtω)v∗(t;ω)ũ+ σh(t;ω)ũ2b2(θtω)

− σh(t;ω)ũ(a2(θtω)− 2c2(θtω)v∗(t;ω) + c2(θtω)ṽ)

= σh(t;ω){∂tũ− [Hũ+ ũ(a1(θtω)− b2(θtω)ũ− c1(θtω)v∗(t;ω) + c2(θtω)ṽ)]}
+ b2(θtω)v∗(t;ω)ũ(σ − 1).

Then by similar arguments as for proving (2.5), we obtain

∂tṽ − [Hṽ + b2(θtω)(v∗(t;ω)− ṽ)ũ+ ṽ(a2(θtω)− 2c2(θtω)v∗(t;ω)

+ c2(θtω)ṽ)]

≤ {d(
µ̃

µ
− 1)[

µ(eµ̃ + e−µ̃ − 2)− µ̃(eµ + e−µ − 2)

µ̃− µ
− (1− δ)(a1(θtω)− c1(θtω)v∗(t;ω))−A′ω(t)]e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

+ [b2(θtω)e−(2µ−µ̃)ξ(x,t;ω)

− dδ( µ̃
µ
− 1)e(

µ̃
µ−1)Aω(t)(a1(θtω)− c1(θtω)v∗(t;ω))]e−µ̃ξ(x,t;ω)



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 13

+ d[−2e−µξ(x,t;ω) + de(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)]e(

µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω)

× b2(θtω)}σh(t;ω) + b2(θtω)v∗(t;ω)ũ(σ − 1) (2.6)

for t ∈ (tk, tk+1). Let

d ≥ dω = max{ max
i∈{1,2},t∈R

{ bi(θtω)

a1(θtω)− c1(θtω)v∗(t;ω)
}µe

−( µ̃µ−1)‖Aω‖∞

δ(µ̃− µ)
,

e(
µ̃
µ−1)‖Aω‖∞}.

Then we have

dδ(
µ̃

µ
− 1)e(

µ̃
µ−1)Aω(t)(a1(θtω)− c1(θtω)v∗(t;ω)) ≥ bi(θtω) (i = 1, 2).

For this choice of d, if ξ(x, t;ω) = x −
∫ t
0
c(s;ω, µ)ds ≥ ln d

µ̃−µ + Aω(t)
µ , which is

equivalent to ũ(x, t, ω) ≥ 0 and ṽ(x, t, ω) ≥ 0, then ξ(x, t;ω) ≥ 0 and

de(
µ̃
µ−1)Aω(t)−µ̃ξ(x,t;ω) ≤ e−µξ(x,t;ω).

From this and (2.3), we obtain that each term the right hand side of (2.5) and (2.6)
is less than or equal to zero. The lemma thus follows. �

For a given function t 7→ u(t) ∈ l∞(Z) and c ∈ R, we define

lim sup
|i|≤ct,t→∞

ui(t) = lim sup
t→∞

sup
i∈Z,|i|≤ct

ui(t).

Lemma 2.6. Let (u0, v0) ∈ l∞,+(Z) × l∞,+(Z). If there is a positive constant
c(ω) > 0 such that

lim inf
s∈R,|i|≤c(ω)t,t→∞

ui(t;u
0, v0, θsω) = lim inf

t→∞
inf

s∈R,i∈Z,|i|≤c(ω)t
ui(t;u

0, v0, θsω) > 0,

(2.7)
then for any 0 < c < c(ω),

lim sup
|i|≤ct,t→∞

[|ui(t;u0, v0, θsω)−u∗(t+s;ω)|+|vi(t;u0, v0, θsω)−v∗(t+s;ω)|] = 0 (2.8)

uniformly in s ∈ R.

Proof. Let ω ∈ Ω0 and c(ω) satisfy (2.7). We denote

δ0 = lim inf
s∈R,|i|≤c(ω)t,t→∞

ui(t;u
0, v0, θsω).

Then there is T � 1 such that

inf
|i|≤c(ω)t

ui(t;u
0, v0, θsω) ≥ δ0

2
, ∀s ∈ R, t ≥ T. (2.9)

Suppose by contradiction that there is 0 < c0 < c(ω) such that (2.8) does not hold.
Then there are ε0 > 0, sn ∈ R, in ∈ Z, tn > 0 such that |in| ≤ c0tn, tn →∞, and

|uin(tn;u0, v0, θsnω)− u∗(tn + sn;ω)|
+ |vin(tn;u0, v0, θsnω)− v∗(tn + sn;ω)| ≥ ε0.

(2.10)

Let (ũ0, ṽ0) = {(ũ0i , ṽ0i )} and (û0, v̂0) = {(û0i , v̂0i )}, where ũ0i = δ0
2 , ṽ0i = 0, û0i =

‖u0‖ and v̂0i = ‖v0‖ for all i ∈ Z. By the global stability of (u∗(t;ω), v∗(t;ω)), there

is T̃ ≥ T such that

|ui(t; ũ0, ṽ0, θsnω)− u∗(t+ s;ω)|+ |vi(t; ũ0, ṽ0, θsnω)− v∗(t+ s;ω)| < ε0
4

(2.11)



14 F. CAO, L. GAO EJDE-2020/38

for all i ∈ Z, s ∈ R, t ≥ T̃ , and

ui(t;u
0, v0, θsω) ≤ ui(t; û0, v̂0, θsω) < u∗(t+ s;ω) +

ε0
2
,

vi(t;u
0, v0, θsω) ≤ vi(t; û0, v̂0, θsω) < v∗(t+ s;ω) +

ε0
2

(2.12)

for all i ∈ Z, s ∈ R, t ≥ T̃ . Observe that (c(ω)−c0)(tn− T̃ )−2c0T̃ →∞ as n→∞.
Hence there is N such that

(c(ω)− c0)(tn − T̃ )− 2c0T̃ ≥ T, ∀n ≥ N.
For every n ≥ N , let ũn = {ũni } ∈ l∞(Z) with ‖ũn‖ ≤ δ0

2 and

ũni =

{
δ0
2 , |i| ≤ (c(ω)− c0)(tn − T̃ )− 2c0T̃ ,

0, |i| ≥ (c(ω)− c0)(tn − T̃ )− c0T̃ ,
ṽn ≡ 0.

(2.13)

Since |i| ≤ (c(ω)− c0)(tn − T̃ )− c0T̃ implies that |i+ in| ≤ c(ω)(tn − T̃ ) for every
n ≥ N , it follows from (2.9) and (2.13) that

ũni ≤ ui+in(tn − T̃ ;u0, v0, θsnω), ∀i ∈ Z, ∀n ≥ N.
Note that

ṽni = 0 ≤ vi+in(tn − T̃ ;u0, v0, θsnω), ∀i ∈ Z, ∀n ≥ N.
Then by the comparison principle, we have

ui(t; ũ
n, ṽn, θs̃nω) ≤ ui+in(t+ tn − T̃ ;u0, v0, θsnω), (2.14)

vi(t; ũ
n, ṽn, θs̃nω) ≤ vi+in(t+ tn − T̃ ;u0, v0, θsnω), (2.15)

for all i ∈ Z, t > 0, and n ≥ N , where s̃n = sn + tn − T̃ . It follows from the
definition of (ũn, ṽn) that

lim
n→∞

(ũn, ṽn) = (ũ0, ṽ0) locally uniformly in i ∈ Z.

Therefore, from Proposition 2.2 we have that for every t > 0,

lim
n→∞

[|ui(t; ũn, ṽn, θs̃nω)− ui(t; ũ0, ṽ0, θs̃nω)|

+ |vi(t; ũn, ṽn, θs̃nω)− vi(t; ũ0, ṽ0, θs̃nω)|] = 0
(2.16)

locally uniformly in i ∈ Z. It then follows from (2.11), (2.14), (2.15) and (2.16)
that

u∗(sn + tn;ω)− ε0
2
< u0(T̃ ; ũn, ṽn, θs̃nω) ≤ uin(tn;u0, v0, θsnω),

v∗(sn + tn;ω)− ε0
2
< v0(T̃ ; ũn, ṽn, θs̃nω) ≤ vin(tn;u0, v0, θsnω) for n� 1.

Note that by (2.12) we have

uin(tn;u0, v0, θsnω) < u∗(sn + tn;ω) +
ε0
2
,

vin(tn;u0, v0, θsnω) < v∗(sn + tn;ω) +
ε0
2

for n� 1. Then

|uin(tn;u0, v0, θsnω)− u∗(sn + tn;ω)|+ |vin(tn;u0, v0, θsnω)− v∗(sn + tn;ω)| < ε0

for n� 1, which contradicts (2.10). Hence (2.8) holds. �



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 15

3. Random transition fronts

In this section, we study the existence and non-existence of random transition
fronts, and prove Theorem 1.2.

For any γ > c0, let 0 < µ < µ∗ be such that eµ+e−µ−2+λ
µ = γ, where λ =

a1(ω)− c1(ω)v∗(·;ω) for ω ∈ Ω0. For every ω ∈ Ω, denote

c(t;ω, µ) =
eµ + e−µ − 2 + (a1(θtω)− c1(θtω)v∗(t;ω))

µ

and ûµ(x, t;ω) = e−µ(x−
∫ t
0
c(s;ω,µ)ds). Then ûµ(x, t;ω) satisfies

∂tû
µ(x, t;ω)−Hûµ(x, t;ω)− (a1(θtω)− c1(θtω)v∗(t;ω))ûµ(x, t;ω)

= ûµ(x, t;ω)[µc(t;ω, µ)− (eµ + e−µ − 2) + (a1(θtω)− c1(θtω)v∗(t;ω))] = 0

for x ∈ R, t ∈ R. Then we have

∂tû
µ −Hûµ − ûµ(a1(θtω)− b1(θtω)ûµ − c1(θtω)(v∗(t;ω)− ûµ))

= ûµ[µc(t;ω, µ)− (eµ + e−µ − 2)− (a1(θtω)− c1(θtω)v∗(t;ω))]

+ ûµ(b1(θtω)− c1(θtω))ûµ

= ûµ(b1(θtω)− c1(θtω))ûµ ≥ 0,

and

∂tû
µ −Hûµ − b2(θtω)(v∗(t;ω)− ûµ)ûµ

− ûµ(a2(θtω)− 2c2(θtω)v∗(t;ω) + c2(θtω)ûµ)

= µc(t;ω, µ)ûµ − (eµ + e−µ − 2)ûµ − b2(θtω)v∗(t;ω)ûµ + ûµb2(θtω)ûµ

− (a2(θtω)− 2c2(θtω)v∗(t;ω))ûµ − ûµc2(θtω)ûµ

= [a1(θtω)− c1(θtω)v∗(t;ω)− (a2(θtω)− 2c2(θtω)v∗(t;ω) + b2(θtω)v∗(t;ω))]ûµ

+ ûµ(b2(θtω)− c2(θtω))ûµ ≥ 0 for x ∈ R, t ∈ R.

Hence, (ûµ(x, t;ω), ûµ(x, t;ω)) = (e−µ(x−
∫ t
0
c(s;ω,µ)ds), e−µ(x−

∫ t
0
c(s;ω,µ)ds)) is a super-

solution of (2.1). Denote

(uµ(x, t;ω), vµ(x, t;ω)) = min{(u∗(t;ω), v∗(t;ω)), (ûµ(x, t;ω), ûµ(x, t;ω))}.
Then (uµ(x, t;ω), vµ(x, t;ω)) is a generalized super-solution of (2.1).

Lemma 3.1. For ω ∈ Ω0, we have

u(x, t− t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≤ uµ(x, t;ω),

v(x, t− t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≤ vµ(x, t;ω),

for all x ∈ R, t ≥ t0, t0 ∈ R.

Proof. For any constant C, (Û(x, t;ω), V̂ (x, t;ω)) := (eCtûµ(x, t;ω), eCtûµ(x, t;ω))
satisfies

∂tÛ(x, t;ω) = (∂tû
µ(x, t;ω) + Cûµ(x, t;ω))eCt

≥ HÛ(x, t;ω) + CÛ(x, t;ω) + eCtf(t, û, û, ω),

and

∂tV̂ (x, t;ω) = (∂tû
µ(x, t;ω) + Cûµ(x, t;ω))eCt

≥ HV̂ (x, t;ω) + CV̂ (x, t;ω) + eCtg(t, û, û, ω).



16 F. CAO, L. GAO EJDE-2020/38

Hence,

Û(x, t;ω) ≥ Û(x, t0;ω) +

∫ t

t0

(HÛ(x, τ ;ω) + CÛ(x, τ ;ω) + eCτf(τ, û, û, ω))dτ,

V̂ (x, t;ω) ≥ V̂ (x, t0;ω) +

∫ t

t0

(HV̂ (x, τ ;ω) + CV̂ (x, τ ;ω) + eCτg(τ, û, û, ω))dτ.

Denote (U(x, t;ω), V (x, t;ω)) := (eCtuµ(x, t;ω), eCtvµ(x, t;ω)). Then we have

U(x, t;ω) ≥ U(x, t0;ω) +

∫ t

t0

(HU(x, τ ;ω) + CU(x, τ ;ω) + eCτf(τ, u, v, ω))dτ,

V (x, t;ω) ≥ V (x, t0;ω) +

∫ t

t0

(HV (x, τ ;ω) + CV (x, τ ;ω) + eCτg(τ, u, v, ω))dτ.

Let Q1(x, t;ω) = eCt(uµ(x, t;ω) − u(x, t − t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω)) and
Q2(x, t;ω) = eCt(vµ(x, t;ω)− v(x, t− t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω)). Then

Q1(x, t;ω)−Q1(x, t0;ω)

≥
∫ t

t0

(HQ1(x, τ ;ω) + a1(x, τ ;ω)Q1(x, τ ;ω) + b1(x, τ ;ω)Q2(x, τ ;ω))dτ,

and

Q2(x, t;ω)−Q2(x, t0;ω)

≥
∫ t

t0

(HQ2(x, τ ;ω) + a2(x, τ ;ω)Q1(x, τ ;ω) + b2(x, τ ;ω)Q2(x, τ ;ω))dτ,

where

a1(x, t;ω) = C + fu(t, u∗1, v
∗
1 , ω), b1(x, t;ω) = fv(t, u

∗
1, v
∗
1 , ω),

a2(x, t;ω) = gu(t, u∗2, v
∗
2 , ω), b2(x, t;ω) = C + gv(t, u

∗
2, v
∗
2 , ω).

Since (2.1) is cooperative, we know that b1(x, t;ω) ≥ 0 and a2(x, t;ω) ≥ 0. By the
boundedness of ūµ(x, t;ω), v̄µ(x, t;ω), u(x, t − t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) and
v(x, t−t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω), we can choose C > 0 such that b2(x, t;ω) ≥
0 and a1(x, t;ω) ≥ 0 for all t ≥ t0, x ∈ R and a.a. ω ∈ Ω. By the arguments of
Proposition 2.1, we have that

Qi(x, t;ω) ≥ Qi(x, t0;ω) = 0, i = 1, 2,

and hence for ω ∈ Ω0, we have that u(x, t − t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≤
uµ(x, t;ω) and v(x, t − t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≤ vµ(x, t;ω) for all x ∈ R,
t ≥ t0, t0 ∈ R. �

Next, we construct a sub-solution of (2.1). Let µ̃ > 0 be such that µ < µ̃ <
min{2µ, µ∗} and ω ∈ Ω0. Let Aω and dω be given by Lemma 2.5, and let

xω(t) =

∫ t

0

c(s;ω, µ)ds+
ln dω + ln µ̃− lnµ

µ̃− µ
+
Aω(t)

µ
.

Recall that

ũ(x, t, ω) = e−µ(x−
∫ t
0
c(s;ω,µ)ds) − de(

µ̃
µ−1)Aω(t)−µ̃(x−

∫ t
0
c(s;ω,µ)ds),

ṽ(x, t, ω) = σe−µ(x−
∫ t
0
c(s;ω,µ)ds)h(t;ω)− σde(

µ̃
µ−1)Aω(t)−µ̃(x−

∫ t
0
c(s;ω,µ)ds)h(t;ω)



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 17

By calculations we have that for any given t ∈ R,

(ũ(xω(t), t, ω), ṽ(xω(t), t, ω)) = (sup
x∈R

ũ(x, t, ω), sup
x∈R

ṽ(x, t, ω))

=
(
e−µ(

ln dω
µ̃−µ +

Aω(t)
µ )e−µ

ln µ̃−lnµ
µ̃−µ (1− µ

µ̃
),

σh(t;ω)e−µ(
ln dω
µ̃−µ +

Aω(t)
µ )e−µ

ln µ̃−lnµ
µ̃−µ (1− µ

µ̃
)
)
.

(3.1)

Define

(uµ(x, t; θt0ω), vµ(x, t; θt0ω))

=

{
(ũ(x, t+ t0, ω), ṽ(x, t+ t0, ω)), if x ≥ xω(t+ t0),

(ũ(xω(t+ t0), t+ t0, ω), ṽ(xω(t+ t0), t+ t0, ω)), if x ≤ xω(t+ t0).

Then (uµ(x, t;ω), vµ(x, t;ω)) is a generalized sub-solution of (2.1). It is clear
that

(0, 0) < (uµ(·, t; θt0ω), vµ(·, t; θt0ω))

< (uµ(·, t; θt0ω), vµ(·, t; θt0ω))

≤ (u∗(t+ t0;ω), v∗(t+ t0;ω))

for all t, t0 ∈ R, and there exists σ̃ > 0 such that

lim
x→∞

sup
t∈R,t0∈R

uµ(x, t; θt0ω)

uµ(x, t; θt0ω)
= 1, lim

x→∞
sup

t∈R,t0∈R

vµ(x, t; θt0ω)

vµ(x, t; θt0ω)
= σ̃. (3.2)

Note that by the similar arguments as in Lemma 3.1, we can prove that

u(x, t− t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≥ uµ(x, t;ω),

v(x, t− t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≥ vµ(x, t;ω)

for x ∈ R, t ≥ t0 and a.a. ω ∈ Ω. Now we are in a position to prove the main
Theorem.

Proof of Theorem 1.2. (i) By Lemma 3.1 we have

u(x, t− t0;uµ(·, t0;ω), vµ(·, t0;ω), θt0ω) ≤ uµ(x, t;ω)

It then follows that

u(x, τ2 − τ1;uµ(·,−τ2;ω), vµ(·,−τ2;ω), θ−τ2ω) ≤ uµ(x,−τ1;ω)

for x ∈ R and τ2 > τ1. Then we obtain

u
(
x, t+ τ1;u(·, τ2 − τ1;uµ(·,−τ2;ω), vµ(·,−τ2;ω), θ−τ2ω),

v(·, τ2 − τ1;uµ(·,−τ2;ω), vµ(·,−τ2;ω), θ−τ2ω), θ−τ1ω
)

≤ u(x, t+ τ1;uµ(·,−τ1;ω), vµ(·,−τ1;ω), θ−τ1ω)

for x ∈ R, t ≥ −τ1, τ2 > τ1, and hence

u
(
x, t+ τ2;uµ(·,−τ2;ω), vµ(·,−τ2;ω), θ−τ2ω

)
≤ u

(
x, t+ τ1;uµ(·,−τ1;ω), vµ(·,−τ1;ω), θ−τ1ω

)
for x ∈ R, t ≥ −τ1, τ2 > τ1.



18 F. CAO, L. GAO EJDE-2020/38

Therefore limτ→∞ u(x, t+τ ;uµ(·,−τ ;ω), vµ(·,−τ ;ω), θ−τω) exists. Similarly, we
can get that limτ→∞ v(x, t+ τ ;uµ(·,−τ ;ω), vµ(·,−τ ;ω), θ−τω) exists. Define

U(x, t;ω) := lim
τ→∞

u(x, t+ τ ;uµ(·,−τ ;ω), vµ(·,−τ ;ω), θ−τω),

V (x, t;ω) := lim
τ→∞

v(x, t+ τ ;uµ(·,−τ ;ω), vµ(·,−τ ;ω), θ−τω)

for x ∈ R, t ∈ R, ω ∈ Ω0. Then (U(x, t;ω), V (x, t;ω)) is non-increasing in x ∈ R
and by dominated convergence theorem we know that (U(x, t;ω), V (x, t;ω)) is a
solution of (2.1).

We claim that, for every ω ∈ Ω0,

lim
x→−∞

(
U(x+

∫ t

0

c(s;ω, µ)ds, t;ω), V (x+

∫ t

0

c(s;ω, µ)ds, t;ω)
)

= (u∗(t;ω), v∗(t;ω)) uniformly in t ∈ R.
(3.3)

In fact, fixing any ω ∈ Ω0, and letting x̂ω = ln dω+ln µ̃−lnµ
µ̃−µ − ‖Aω‖∞µ , from (3.1),

inft∈R h(t;ω) > 0 and (uµ(x, t;ω), vµ(x, t;ω)) ≤ (U(x, t;ω), V (x, t;ω)) it follows
that

0 < (1− µ

µ̃
)e−µ(

ln dω+ln µ̃−lnµ
µ̃−µ +

‖Aω‖∞
µ ) ≤ inf

t∈R
U(x̂ω +

∫ t

0

c(s;ω, µ)ds, t;ω),

and

0 < σ inf
t∈R

h(t;ω)(1−µ
µ̃

)e−µ(
ln dω+ln µ̃−lnµ

µ̃−µ +
‖Aω‖∞

µ ) ≤ inf
t∈R

V (x̂ω +

∫ t

0

c(s;ω, µ)ds, t;ω).

Let (u0(x), v0(x)) ≡ (u0, v0), where

(u0, v0) := (inf
t∈R

U(x̂ω +

∫ t

0

c(s;ω, µ)ds, t;ω), inf
t∈R

V (x̂ω +

∫ t

0

c(s;ω, µ)ds, t;ω)),

and (ũ0(x), ṽ0(x)) be uniformly continuous such that (ũ0(x), ṽ0(x)) = (u0(x), v0(x))
for x < x̂ω−1 and (ũ0(x), ṽ0(x)) = (0, 0) for x ≥ x̂ω. Then limn→∞(ũ0(x−n), ṽ0(x−
n)) = (u0(x), v0(x)) locally uniformly in x ∈ R. Note that by (H2), we have

lim
t→∞

(u(x, t;u0, v0, θt0ω)− u∗(t+ t0;ω), v(x, t;u0, v0, θt0ω)− v∗(t+ t0;ω)) = (0, 0)

uniformly in t0 ∈ R and x ∈ R. Then for any ε > 0, there is T := T (ε) > 0 such
that

u∗(t0 + T ;ω) > u(x, T ;u0, v0, θt0ω) > u∗(t0 + T ;ω)− ε, ∀t0 ∈ R, x ∈ R.
Therefore, from the definition of c(t, ω, µ) we know that,

u∗(t0 + T ;ω) > u(x+

∫ T

0

c(s; θt0ω, µ)ds, T ;u0, v0, θt0ω) > u∗(t0 + T ;ω)− ε

for all t0 ∈ R and x ∈ R. By Proposition 2.2, there is N := N(ε) > 1 such that

u∗(t0 + T ;ω) > u
(∫ T

0

c(s; θt0ω, µ)ds, T ; ũ0(· −N), ṽ0(· −N), θt0ω
)

> u∗(t0 + T ;ω)− 2ε, ∀t0 ∈ R.
That is,

u∗(t0 + T ;ω) > u(

∫ T

0

c(s; θt0ω, µ)ds−N,T ; ũ0(·), ṽ0(·), θt0ω)

> u∗(t0 + T ;ω)− 2ε, ∀t0 ∈ R.



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 19

Note that

U(x+

∫ t−T

0

c(s;ω, µ)ds, t− T ;ω) ≥ ũ0(x), ∀t ∈ R, x ∈ R,

V (x+

∫ t−T

0

c(s;ω, µ)ds, t− T ;ω) ≥ ṽ0(x), ∀t ∈ R, x ∈ R,∫ t

0

c(s;ω, µ)ds =

∫ T

0

c(s; θt−Tω, µ)ds+

∫ t−T

0

c(s;ω, µ)ds.

Then we obtain

u∗(t;ω) > U(x+

∫ t

0

c(s;ω, µ)ds, t;ω)

= u
(
x+

∫ T

0

c(s; θt−Tω, µ)ds, T ;U(·+
∫ t−T

0

c(s;ω, µ)ds, t− T ;ω),

V (·+
∫ t−T

0

c(s;ω, µ)ds, t− T ;ω), θt−Tω
)

> u∗(t;ω)− 2ε, ∀t ∈ R, x ≤ −N,

and hence limx→−∞ U(x+
∫ t
0
c(s;ω, µ)ds, t;ω) = u∗(t;ω) uniformly in t ∈ R. Sim-

ilarly, we can derive limx→−∞ V (x +
∫ t
0
c(s;ω, µ)ds, t;ω) = v∗(t;ω) uniformly in

t ∈ R. Thus (3.3) follows.
Note that by (3.2) we have that for every ω ∈ Ω0,

lim
x→∞

sup
t∈R

U(x+

∫ t

0

c(s;ω, µ)ds, t;ω) = 0,

lim
x→∞

sup
t∈R

V (x+

∫ t

0

c(s;ω, µ)ds, t;ω) = 0.

Set

(Φ̃(x, t;ω), Ψ̃(x, t;ω)) =
(
U(x+

∫ t

0

c(s;ω, µ)ds, t;ω), V (x+

∫ t

0

c(s;ω, µ)ds, t;ω)
)
,

(Φ(x, ω), Ψ(x, ω)) = (Φ̃(x, 0;ω), Ψ̃(x, 0;ω)).

We now claim that (Φ̃(x, t;ω), Ψ̃(x, t;ω)) is stationary ergodic in t, that is, for a.a.
ω ∈ Ω,

(Φ̃(x, t;ω), Ψ̃(x, t;ω)) = (Φ̃(x, 0; θtω), Ψ̃(x, 0; θtω)).

In fact, note that for ω ∈ Ω,∫ t

−τ
c(s;ω, µ)ds =

∫ t

−τ

eµ + e−µ − 2 + a1(θsω)− c1(θsω)v∗(s;ω)

µ
ds

=
eµ + e−µ − 2

µ
(t+ τ) +

∫ t

−τ

a1(θsω)− c1(θsω)v∗(s;ω)

µ
ds

(3.4)



20 F. CAO, L. GAO EJDE-2020/38

and ∫ 0

−(t+τ)
c(s; θtω, µ)ds

=

∫ 0

−(t+τ)

eµ + e−µ − 2 + a1(θs ◦ θtω)− c1(θs ◦ θtω)v∗(s; θtω)

µ
ds

=
eµ + e−µ − 2

µ
(t+ τ) +

∫ 0

−(t+τ)

a1(θs+tω)− c1(θs+tω)v∗(s+ t;ω)

µ
ds

=
eµ + e−µ − 2

µ
(t+ τ) +

∫ t

−τ

a1(θsω)− c1(θsω)v∗(s;ω)

µ
ds.

(3.5)

Combining (3.4) with (3.5), we derive
∫ t
−τ c(s;ω, µ)ds =

∫ 0

−(t+τ) c(s; θtω, µ)ds for

τ ≥ 0 and t ∈ R. Recall that

(uµ(x, t;ω), vµ(x, t;ω)) = min {(u∗(t;ω), v∗(t;ω)), (û(x, t;ω), û(x, t;ω))} ,

(û(x, t;ω), û(x, t;ω)) = (e−µ(x−
∫ t
0
c(s;ω,µ)ds), e−µ(x−

∫ t
0
c(s;ω,µ)ds)).

Then we have

Φ̃(x, t;ω)

= lim
τ→∞

u
(
x+

∫ t

0

c(s;ω, µ)ds, t+ τ ;uµ(·,−τ ;ω), vµ(·,−τ ;ω), θ−τω
)

= lim
τ→∞

u
(
x, t+ τ ;uµ(·+

∫ t

0

c(s;ω, µ)ds,−τ ;ω), vµ(·+
∫ t

0

c(s;ω, µ)ds,

− τ ;ω), θ−τω
)

= lim
τ→∞

u
(
x, t+ τ ;uµ(·,−(t+ τ); θtω), vµ(·,−(t+ τ); θtω), θ−τω

)
= lim
τ→∞

u
(
x, t+ τ ;uµ(·,−(t+ τ); θtω), vµ(·,−(t+ τ); θtω), θt−(t+τ)ω

)
= lim
τ→∞

u
(
x, τ ;uµ(·,−τ ; θtω), vµ(·,−τ ; θtω), θt−τω

)
= Φ̃(x, 0; θtω).

Similarly, we can get Ψ̃(x, t;ω) = Ψ̃(x, 0; θtω), and hence (Φ̃(x, t;ω), Ψ̃(x, t;ω)) =

(Φ̃(x, 0; θtω), Ψ̃(x, 0; θtω)). The claim thus follows and we obtain the desired random
profile (Φ(x, ω), Ψ(x, ω)).

(ii) Let

c∗(ω) = sup
{
c : lim sup
|i|≤ct,t→∞

[|ui(t;u0, v0, θsω)− u∗(t+ s;ω)|

+ |vi(t;u0, v0, θsω)− v∗(t+ s;ω)|] = 0

uniformly in s ∈ R for all (u0, v0) ∈ l∞0 (Z)× l∞0 (Z)
}
,

where

l∞0 (Z) = {u = {ui}i∈Z ∈ l∞(Z) : ui ≥ 0 for all i ∈ Z, ui = 0 for |i| � 1, {ui} 6= 0}.
Recall that

λ = lim inf
t−s→∞

1

t− s

∫ t

s

(a1(θτω)− c1(θτω)v∗(τ ;ω))dτ,



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 21

c0 := inf
µ>0

eµ + e−µ − 2 + λ

µ
.

We claim that c∗(ω) = c0 for ω ∈ Ω0. In fact, we consider

u̇i(t) = ui+1(t)− 2ui(t) + ui−1(t) + ui(t)(a1(θtω)− c1(θtω)v∗(t;ω)− b1(θtω)ui(t))
(3.6)

For any u0 ∈ l∞,+(Z), let u−(t;u0, ω) be the solution of (3.6) with u−(0;u0, ω) =
u0. By comparison principle, for any (u0, v0) ∈ l∞,+(Z)× l∞,+(Z), we have

ui(t;u
0, v0, ω) ≥ u−i (t;u0, ω), ∀t ≥ 0. (3.7)

By [3, Remark 1.1 (1)], for any c(ω) with 0 < c(ω) < c0,

lim inf
s∈R,|i|≤c(ω)t,t→∞

u−i (t;u0, θsω) > 0.

With (3.7), we then have

lim inf
s∈R,|i|≤c(ω)t,t→∞

ui(t;u
0, v0, θsω) > 0.

Then by Lemma 2.6, for any 0 < c < c(ω),

lim sup
|i|≤ct,t→∞

[|ui(t;u0, v0, θsω)− u∗(t+ s;ω)|+ |vi(t;u0, v0, θsω)− v∗(t+ s;ω)|] = 0

uniformly in s ∈ R. which implies that c∗(ω) ≥ c0.
Assume that c∗(ω) > c0 for some ω ∈ Ω0. Fix γ, c′ and c′′ such that

c0 < γ < c′ < c′′ < c∗(ω).

Observe that c0 > 0. For any (u0, v0) ∈ l∞0 (Z)× l∞0 (Z),

lim sup
|i|≤c′′t,t→∞

[|ui(t;u0, v0, θsω)− u∗(t+ s;ω)|

+ |vi(t;u0, v0, θsω)− v∗(t+ s;ω)|] = 0
(3.8)

uniformly in s ∈ R.

Let (ui(t;ω), vi(t;ω)) = (Φ(i −
∫ t
0
c(s;ω)ds, θtω), Ψ(i −

∫ t
0
c(s;ω)ds, θtω)) be as

in (i) with c̄inf = γ. Let

usi = Φ(i−
[ ∫ s

0

c(τ ;ω)dτ
]
, θsω), vsi = Ψ(i−

[ ∫ s

0

c(τ ;ω)dτ
]
, θsω), ∀s ∈ R.

By (i), there is (u0, v0) ∈ l∞0 (Z)× l∞0 (Z) such that

(u0, v0) ≤ (us, vs), ∀s ∈ R.

Hence

ui(t;u
0, v0, θsω) ≤ ui(t;us, vs, θsω),

vi(t;u
0, v0, θsω) ≤ vi(t;us, vs, θsω)

for i ∈ Z, s ∈ R and t ≥ 0. This together with (3.8) implies that

lim sup
|i|≤c′′t,t→∞

[|ui(t;us, vs, θsω)− u∗(t+ s;ω)|

+ |vi(t;us, vs, θsω)− v∗(t+ s;ω)|] = 0
(3.9)



22 F. CAO, L. GAO EJDE-2020/38

uniformly in s ∈ R. Note that
∫ t+s
0

c(τ ;ω)dτ =
∫ s
0
c(τ ;ω)dτ +

∫ t
0
c(τ ; θsω)dτ . By

(i) again, we have

ui(t;u
s, vs, θsω) = Φ

(
i−
∫ t

0

c(τ ; θsω)dτ −
[ ∫ s

0

c(τ ;ω)dτ
]
, θt+sω

)
≤ Φ

(
i−
∫ t+s

0

c(τ ;ω)dτ, θt+sω
)
,

and

vi(t;u
s, vs, θsω) = Ψ

(
i−
∫ t

0

c(τ ; θsω)dτ −
[ ∫ s

0

c(τ ;ω)dτ
]
, θt+sω

)
≤ Ψ

(
i−
∫ t+s

0

c(τ ;ω)dτ, θt+sω
)
.

Then

lim sup
i≥(c′′−c′)(t+s)+

∫ t+s
0

c(τ ;ω)dτ,t→∞
[ui(t;u

s, vs, θsω) + vi(t;u
s, vs, θsω)] = 0 (3.10)

uniformly in s ∈ R. It follows from (3.9) and (3.10) that

c̄inf ≥ c′ > γ,

which is a contradiction. Therefore, c∗(ω) = c0.
Suppose that (u(t;ω), v(t;ω)) = {(ui(t;ω), vi(t;ω))}i∈Z with (ui(t;ω), vi(t;ω)) =

(Φ(i −
∫ t
0
c(s;ω)ds, θtω), Ψ(i −

∫ t
0
c(s;ω)ds, θtω)) is a random transition front of

(1.5) connecting (u∗(t;ω), v∗(t;ω)) and (0, 0). We prove that its least mean speed
cinf ≥ c0. Observe that infx≤z infs∈R Φ(x, θsω) > 0 and infx≤z infs∈R Ψ(x, θsω) > 0
for all z ∈ R. Therefore, we can choose (u0ω, v

0
ω) ∈ l∞0 (Z) × l∞0 (Z) such that

(u0ω, v
0
ω) ≤ (Φ(x, θsω), Ψ(x, θsω)) for all s ∈ R. Let 0 < ε� 1. Then by c∗(ω) = c0

and the comparison principle, we have

lim sup
t→∞

sup
s∈R

[|u[(c0−ε)t](t;u
0
ω, v

0
ω, θsω)− u∗(t+ s;ω)|

+ |v[(c0−ε)t](t;u
0
ω, v

0
ω, θsω)− v∗(t+ s;ω)|] = 0,

and

lim inf
t→∞

inf
s∈R
{u[(c0−ε)t](t;u

0
ω, v

0
ω, θsω) + v[(c0−ε)t](t;u

0
ω, v

0
ω, θsω)}

≤ lim inf
t→∞

inf
s∈R
{u[(c0−ε)t](t;Φ(·, θsω), Ψ(·, θsω), θsω)

+ v[(c0−ε)t](t;Φ(·, θsω), Ψ(·, θsω), θsω)}

= lim inf
t→∞

inf
s∈R
{Φ([(c0 − ε)t]−

∫ t

0

c(τ ; θsω)dτ, θt+sω)

+ Ψ([(c0 − ε)t]−
∫ t

0

c(τ ; θsω)dτ, θt+sω)}.

From this and
∫ t+s
0

c(τ ;ω)dτ =
∫ s
0
c(τ ;ω)dτ +

∫ t
0
c(τ ; θsω)dτ , we know that there

is a M(ω) such that (c0 − ε)t ≤
∫ t+s
0

c(τ ;ω)dτ −
∫ s
0
c(τ ;ω)dτ +M(ω) for all t > 0,

s ∈ R. Hence,

cinf = lim inf
t→∞

inf
s∈R

∫ t+s
0

c(τ ;ω)dτ −
∫ s
0
c(τ ;ω)dτ

t
≥ c0 − ε.

By the arbitrariness of ε > 0, we obtain cinf ≥ c0. �



EJDE-2020/38 COMPETITION LATTICE SYSTEMS IN RANDOM MEDIA 23

Acknowledgments. Feng Cao was supported by NSF of China No. 11871273, and
by the Fundamental Research Funds for the Central Universities No. NS2018047.

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Feng Cao

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nan-
jing, Jiangsu 210016, China

Email address: fcao@nuaa.edu.cn

Lu Gao

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nan-

jing, Jiangsu 210016, China
Email address: gaolunuaa@163.com


	1. Introduction
	2. Preliminaries
	3. Random transition fronts
	Acknowledgments

	References