DOI: 10.1515/auom-2016-0010
An. Şt. Univ. Ovidius Constanţa
Vol. 24(1),2016, 189–200
Drift perturbation’s influence on traveling wave
speed in KPP-Fisher system
Fathi DKHIL and Bechir MANNOUBI
Abstract
This paper dressed the drift perturbation effects on the traveling
wave speed in a reaction-diffusion system. We prove the existence of
a traveling front solution of a KPP-Fisher equation and we show an
asymptotic expansion of her speed. Finally, we discuss according all
parameters of our system regions of the plane in which the traveling
wave speed increases or decreases as a function of a small parameter ε.
1
Introduction
Front propagation is a phenomenon that has many scientific applications ; such
as : the sprawl of epidemics and diseases, biological invasions and collective
behavior, ecology, population dynamics, reaction kinetics, the flow in porous
materials, etc.
This phenomenon is generally modeled by a reaction-diffusion equation of the
following form

 Ut = ∇.(A∇U ) + f (U )

U (0, x) = U0 (x)
where the diffusivity A is a positive definite matrix and the reaction term f is
a C 2 nonlinear function.
Key Words: Reaction-diffusion equations, heterogeneous media, traveling wave speed,
drift perturbation, KPP-Fisher equations.
2010 Mathematics Subject Classification: Primary 35B20, 35B27, 35B50; Secondary
35C20, 35K55, 35K57, 41A60.
Received: 17.02.2014
Revised: 3.05.2014
Accepted: 12.05.2014.
189
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
190
Understanding the mechanism of the settings of perturbation on the diffusion
and (or) reaction influences the traveling fronts like configuring their location,
profile, and their speed is one of the most fundamental issues. In fact, variational principles for the speed of front propagation for KPP-type nonlinearities,
respectively Fisher’s population genetic model as well as various other implications of the qualitative behavior of propagating waves, were given by Hadeler
and Rothe in [7] and Gärtner and Freidlin in [6]. Paper [2] is used to handle the speed of traveling fronts of reactions-diffusion equations of bistable or
combustion type with rapidly oscillating diffusion and drift coefficients. In the
monostable case (KPP-Fisher type), the variation of the traveling wave speed
was treated in [3] for nonlocally perturbed reaction-diffusion equations. Compare the survey papers ([14],[16],[15]) in order to have an excellent reference
on propagation phenomenas mathematical results especially on the existence
and stability of traveling waves, and other references that has not already been
cited here.
In this paper we consider a particular kind of reaction-diffusion system with
drift perturbation. The type of system we are dealing with is the following

 ut = α∆u + γu(1 − u − v)
(1)

vt = β∆v + δb∇v + ηv(1 − u − v)
with:
x = (x1 , x2 , ...., xn ) ∈ D = R × Ω ⊂ Rn , t > 0,
b = b(x2 ...xn ), β = β(ε) −→ α, η = η(ε) −→ γ and δ = δ(ε) −→ 0 as ε −→ 0.
The cross section of the cylinder Ω ⊂ Rn−1 is bounded with C 1,λ boundary.
For simplicity, we will consider here traveling waves in direction e1 = (1, 0, ..., 0).
A very important question is how to study the variation of the traveling
wave speed in discussing the influence of drift perturbation and all parameters α, γ, β, δ, η of system (1).
Existence of unique monotone and stable traveling waves were shown in [5],
[9], [10] and [13] for monotonic systems. A variational characterization given
in [12] allows to prove an asymptotic expansion for the traveling wave speed
solution of system (1).
The paper is organized as follows. In section 2, we rescale system (1) to obtain
a monotone system and we show the existence of unique monotone and stable
traveling wave up to translation and we give a variational characterization of
the wave speed. In section 3, we prove that the traveling wave has an asymptotic expansion. Finally in section 4, we determine regions of the plane in
which the traveling wave speed solution of system (1) increases or decreases
as a function of ε.
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
2
191
The variational characterization
We rescale the system (1). So, with no loss of generality, we can suppose that
α = γ = 1 and our system becomes

 ut = ∆u + u(1 − u − v)
(2)

vt = β∆v + δb∇v + ηv(1 − u − v)
with b = b(x2 ...xn ), β = β(ε) −→ 1, η = η(ε) −→ 1 and δ = δ(ε) −→ 0 as
ε −→ 0.
We pose w = 1 − u. Thereby system (2) becomes

 wt = ∆w − (1 − w)(w − v)
(3)

vt = β∆v + δb∇v + ηv(w − v).
We can easily see that system (3) is monotone. Therefore there exists a unique
monotone traveling wave (cε , wε , vε ) in direction e1 solution of (3) connecting
two zeros of its nonlinearities; this wave is stable with respect to some subset
Is of initial data. For more detail, see for example [5], [9], [10] and [13].
The traveling wave solution (cε , wε , vε ) of system (3) will satisfy the following
boundary conditions
wε , vε (−∞) = 1, 0
wε , vε (+∞) = 12 , 21 .
and
Let (c0 , w0 , v0 ) be the traveling wave solution of the system (3) when ε = 0
(i.e. β = 1, η = 1 and δ = 0).
Let W0 = 1 + v0 − w0 = u0 + v0 , then W0 is a traveling front solution of the
following KPP-Fisher equation

c0 W00 = W000 + W0 (1 − W0 )

(4)

W0 (−∞) = 0, W0 (+∞) = 1
with W0 (t, x) = W0 (x + c0 te1 ) and W00 represent the derivative of W0 with
respect to the first component x1 of x.
Let
Wε = 1 + vε − wε = vε + uε ,
(5)
then Wε is a traveling wave solution of the following equation

 cε Wε0 = ∆Wε + Wε (1 − Wε ) + (β − 1)∆vε + δb∇vε + (η − 1)vε (1 − Wε )

Wε (−∞) = 0, Wε (+∞) = 1.
(6)
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
192
Let
(
K=
)
y ∈ C 1 (R, C 2 (D))∂t y(t, x) > 0, 0 < y(t, x) < 1, y(0, .) ∈ Is ,
y(−∞) = 0, y(+∞) = 1
be the sets of admissible comparison functions. For y ∈ K define
ψ(y) =
∆y + y(1 − y) + (β − 1)∆vε + δb∇vε + (η − 1)vε (1 − y)
.
∂t y
In [5] a variational characterization of the wave speed was given for more
general situations. Here we will state a specific version of the result as a
lemma.
LEMMA 2.1. [5] Suppose that there exists a unique stable traveling wave for
problem (6) then the traveling wave speed cε is given by
sup
inf
y∈K (t,x)∈(R×D)
ψ(y(t, x)) = cε = inf
sup
y∈K (t,x)∈(R×D)
ψ(y(t, x)).
(7)
The proof of this minimax characterization of the wave speed cε is based
on the maximum principle and relates technically to results given by Vol’pert
et al in [12] for monotone systems of ODE’s.
3
Asymptotic expansion
THEOREM 3.1. The traveling wave (cε , wε , vε ) solution of the system (3)
has the following expansion:
cε
vε
wε
Proof. We note β 0 =
= c0 + εc1 + o(ε)
= v0 + εv1 + o(ε)
= w0 + εw1 + o(ε).
(8)
dη
dδ
dβ
, η0 =
and δ 0 =
then equation
dε |ε=0
dε |ε=0
dε |ε=0
(6) becomes
Wεt = ∆Wε + Wε (1 − Wε ) + ε [β 0 ∆vε + δ 0 b∇vε + η 0 vε (1 − Wε )] + o(ε).
Since Wε and vε are regular and bounded functions, then the comparison
principle implies that
Wε = W0 + εW1 + o(ε).
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
Furthermore, we have

 uεt
=
∆uε + uε (1 − uε − vε )
u0t
=
∆u0 + u0 (1 − u0 − v0 )

193
We substract these two equations to get
(uε − u0 )t − ∆(uε − u0 ) − (uε − u0 )(1 − u0 − v0 ) = u(W0 − Wε ).
Using the comparison principle, we obtain
uε = u0 + εu1 + o(ε).
Therefore, we deduce from (5) that
wε = w0 + εw1 + o(ε)
and
vε = v0 + εv1 + o(ε)
which justify our expansion for vε and wε .
To show the asymptotic expansion of cε , we consider the test function:
y(x) = W0 + εy1 (x),
where y1 will be determined below.
An easy computation gives that
ψ(y)
=
∆y + y(1 − y) + (β − 1)∆vε + δb∇vε + (η − 1)vε (1 − y)
∂t y
=
c0 W00 + ε [y100 + (1 − 2W0 )y1 + η 0 v0 (1 − W0 ) + β 0 v000 + δ 0 bv00 ]
+ o(ε)
W00 + εy10
ε
[y 00 − c0 y10 + (1 − 2W0 )y1 + η 0 v0 (1 − W0 ) + β 0 v000 + δ 0 bv00 ]
W00 1
+o(ε).
= c0 +
We choose y1 such that the coefficient of ε in ψ(y) is constant. Thus y1 solves
the following equation
 00
 y1 − c0 y10 + (1 − 2W0 )y1 = −c1 W00 − η 0 v0 (1 − W0 ) − β 0 v000 − δ 0 bv00

y1 (±∞)
=
0.
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
194
Near ±∞ all derivatives of W0 have the same exponential decay rate as W00 .
Therefore with our choice of y we have ∂t y > 0 for small ε. Thus y is an
admissible function for the minimax characterization (7) and we have
cε = c0 + εc1 + o(ε).
This ends the proof of Theorem 3.1.
4
Sign of c1
In this section, we discuss according to derivative perturbations parameters
how c1 may change sign.
PROPOSITION 4.1. Let (cε , wε , vε ) be the traveling wave solution of system (3), then the variation of cε is as follows.
p
I I
c0 + c20 − 4
0
0
0
0
c0 β 0 + δ 0 b
• If c0 β + δ
b ≤ 0 and η − β ≤ −
2
then cε decreases as a function of ε.
•
If
0
c0 β + δ
0
I
b≥0
and
0
0
η −β ≥−
c0 +
p
c20 − 4
2
c0 β 0 + δ 0
I
b
then cε increases as a function of ε.
Proof. We note w0 and v 0 the derivatives of w and v with respect to the
first variable then we have

 cε wε0 = ∆wε − (1 − wε )(wε − vε )
cε vε0
=
β∆vε + δbvε0 + ηvε (wε − vε )
this can be writen as

(c0 + εc1 )(w0 + εw1 )0





=
∆(w0 + εw1 ) − (1 − w0 − εw1 )(w0 + εw1 − v0
−εv1 ) + o(ε)


(c0 + εc1 )(v0 + εv1 )0



=
β∆(v0 + εv1 ) + δb(v0 + εv1 )0
+η(v0 + εv1 )(w0 + εw1 − v0 − εv1 ) + o(ε).

DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
195
Since (c0 , w0 , v0 ) is a solution of the system (3) when ε = 0 then we have

c1 w00 + c0 w10 = ∆w1 − (1 − w0 )(w1 − v1 ) + w1 (w0 − v0 )



c1 v00 + c0 v10



= β 0 ∆v0 + ∆v1 + η 0 v0 (w0 − v0 ) + δ 0 bv00 + v0 (w1 − v1 )
+v1 (w0 − v0 )
which implies that

0
0

 ∆w1 − c0 w1 − (1 − w0 )(w1 − v1 ) + w1 (w0 − v0 ) = c1 w0




∆v1 − c0 v10 + v0 (w1 − v1 ) + v1 (w0 − v0 ) = c1 v00 − β 0 ∆v0 − η 0 v0 (w0 − v0 )
−δ 0 bv00 .
We substract these two equations to obtain
∆(v1 − w1 ) − c0 (v1 − w1 )0 − (v1 − w1 )(1 + 2(v0 − w0 ))
= c1 (v0 − w0 )0 − β 0 ∆v0 − η 0 v0 (w0 − v0 ) − δ 0 bv00 .
We pose φ = v1 − w1 then this equation is equivalent to
∆φ − c0 φ0 − φ(1 − 2W0 ) = c1 W00 − β 0 ∆v0 − η 0 v0 (w0 − v0 ) − δ 0 bv00 .
(9)
Using that W0 is a solution of (4) and the Fredholm alternative, the solvability
condition of the equation (9) can be writen as
R
R
R
2
c1 R×Ω W00 e−c0 ξ = β 0 R×Ω ∆v0 W00 e−c0 ξ dx + η 0 R×Ω v0 (1 − W0 )W00 e−c0 ξ
+δ 0
R
R×Ω
bv00 W00 e−c0 ξ .
Since v0 satisfies ∆v0 = c0 v00 − v0 (1 − W0 ) and using the fuct that v0 and W0
are independent on (x2 ..xn ) ∈ Ω then the solvability condition is equivalent to
R
R
R
2
c1 R W00 e−c0 ξ = c0 β 0 R v00 W00 e−c0 ξ + (η 0 − β 0 ) R v0 (1 − W0 )W00 e−c0 ξ
+δ 0
H R 0 0 −c ξ
b R v0 W 0 e 0 .
After an integration by part we obtain that
Z
Z I
2
c1 W00 e−c0 ξ =
(c0 β 0 + δ 0 b)W0 + (η 0 − β 0 )W00 v0 (1 − W0 )e−c0 ξ .
R
R
We know that v0 (1 − W0 ) > 0, so that if
c0 β 0 + δ 0
I
b W0 + (η 0 − β 0 )W00
does not change sign then we can deduce the sign of c1 . Here we have three
cases.
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
First case:
c0 β 0 + δ 0
I
196
b and (η 0 − β 0 ) have the same sign then it is the
sign of c1 .
Second case: one of
0
c0 β + δ
0
I
b
and (η 0 − β 0 ) is zero then the sign of c1
is the sign of the non zero term.
I 0
0
Third case: c0 β + δ
b and (η 0 −β 0 ) have an opposite sign. In this case
we need a good estimate on W00 .
Equation (4) is equivalent to
W000 − (c0 − a)W00 = aW00 − W0 (1 − W0 ) ∀a ∈ R.
Since 0 < W0 < 1 then we have
1
W000 − (c0 − a)W00 > a(W00 − W0 ) ∀a 6= 0.
a
We choose a such that c0 − a =
(see [1] and [11]).
We note
1
which is possible since we have c20 ≥ 4,
a
c0 ±
a± =
(10)
p
c20 − 4
2
> 0.
1
After multiplication by e− a ξ and integrating the inequality (10) on
(ξ, ∞) for any ξ ∈ R we obtain that
W00 < aW0
with a = a− =
c0 −
p
c20 − 4
.
2
We discuss two subcases
0
0
0
0
I
1. η − β > 0 and c0 β + δ
b < 0. We have that
I I
0
0
0
0
0
0
0
0
0
c0 β + δ
b W0 + (η − β )W0 < c0 β + δ
b + a− (η − β ) W0
<
c0 0
(η + β 0 ) + δ 0
2
I
b−
η0 − β 0
2
q
c20 − 4 W0 .
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
197
I
c0 0
(η + β 0 ) + δ 0 b < 0 then c1 < 0.
2
If not, we have
I
c0 0
(η + β 0 ) + δ 0 b > 0,
2
therefore an easy computation gives that
I
q
η0 − β 0
c0 0
0
0
b−
(η + β ) + δ
c20 − 4 ≤ 0,
2
2
So that if
if and only if
p
I −c0 + c20 − 4 c0 β 0 + δ 0 b ≤ η 0 − β 0
2
and
η0 − β 0 ≤ −
c0 +
p
I c20 − 4 0
0
c0 β + δ
b .
2
Hence, in this subcase we have c1 < 0 and then cε decreases as a
function of ε, for ε small enough.
I
2. η 0 − β 0 < 0 and c0 β 0 + δ 0 b > 0. We have that
I I
0
0
0
0
0
0
0
0
0
c0 β + δ
b W0 + (η − β ) W0 >
c0 β + δ
b + a− (η − β ) W0
I
q
η0 − β 0
c0 0
(η + β 0 ) + δ 0 b −
c20 − 4 W0 .
2
2
I
c0 0
(η + β 0 ) + δ 0 b > 0 then c1 > 0.
So that if
2
If not, we have
I
c0 0
(η + β 0 ) + δ 0 b < 0,
2
therefore an easy computation gives that
I
q
c0 0
η0 − β 0
0
0
(η + β ) + δ
b−
c20 − 4 ≥ 0,
2
2
>
if and only if
−
c0 +
p
I c20 − 4
0
0
c0 β + δ
b ≤ η0 − β 0
2
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
and
0
0
η −β ≤
−c0 +
198
p
I c20 − 4
0
0
c0 β + δ
b .
2
Hence, in this subcase we have c1 > 0 and then cε increases as a
function of ε, for ε small enough.
This ends the proof of proposition 4.1.
Return now to the full system (1). A simple scaling argument allow to conclude the result in a general way. We note c∗ε the traveling wave speed solution
of (1). Then we have the following
THEOREM 4.2. Let (c∗ε , u∗ε , vε∗ ) be the traveling wave solution of system (1)
cε
and has the following variations.
for ε small enough. Then c∗ε =
α
p
I
I c0 + c20 − 4αγ c0 0
c0 0 0
0
0
• If
β +δ
b ≤ 0 and αη −γβ ≤ −
β + δ0 b
α
2
α
then c∗ε decreases as a function of ε.
•
If
c0 0 0
β +δ
α
I
b≥0
and
αη 0 −γβ 0 ≥ −
c0 +
p
I c20 − 4αγ c0 0
β + δ0 b
2
α
then c∗ε increases as a function of ε.
Where cε is the traveling wave speed solution of (3).
Figure 1 presents the different regions of the plane
I
c0
X = β 0 + δ 0 b, Y = αη 0 − γβ 0
α
in which c∗ε increases or decreases as a function of ε.
REMARK 4.3. In the regions
!
p
c0 + c20 − 4αγ
X > 0, Y < −
X and
2
X < 0, Y > −
c0 +
!
p
c20 − 4αγ
X
2
of Figure 1, the determination of the sign of c1 rest an open problem.
DRIFT PERTURBATION’S INFLUENCE ON TRAVELING WAVE SPEED IN
KPP-FISHER SYSTEM
199
Figure 1: c∗ε variation.
Acknowledgment
The authors are very grateful to Hatem Zaag for his helpful comments.
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Fathi DKHIL,
Département de Mathématiques,
Institut Supérieur d’Informatique,
Université Tunis El-Manar,
2 rue Abou Raihan Bayrouni, 2080 Ariana, Tunisia.
Email: fathi.dkhil@gmail.com
Bechir MANNOUBI
Département de Mathématiques,
Institut Supérieur d’Informatique,
Université Tunis El-Manar,
2 rue Abou Raihan Bayrouni, 2080 Ariana, Tunisia.
Email: bechir ca@live.com