Procudings of the Royal Society of Edinburgh, liSA, 1-18, 1990
Existence and stability of travelling wave solutions for an
evolutionary ecology model
Roger Lui
Department of Mathematical Sciences, Worcester Polytechnic Institute,
Worcester, MA 01609, U.S.A.*
(MS received 30 May 1989)
Synopsis
Monotone travelling wave solutions are known to exist for Fisher's equation which models the
propagation of an advantageous gene in a single locus, two alleles population genetics model. Fisher's
equation assumed that the population size is a constant and that the fitnesses of the individuals in the
population depend only on their genotypes. In this paper, we relax these assumptions and allow the
fitnesses to depend also on the population size. Under certain assumptions, we prove that in the
second heterozygote intermediate case, there exists a constant (}* > 0 such that monotone travelling
wave solutions for the reaction-diffusion system exist whenever (} > (}*. We also discuss the stability
properties of these waves.
1. Introduction
Consider the equation
+ h(u) in IR X [0, oo),
(1.1)
where hE C [0, 1], h(O) = 0 and h(1) = 0. In 1937, Fisher used this equation with
h(u) = u(1- u) to model the spatial spread of an advantageous gene in a
U1
= Uxx
1
population living in a homogeneous one-dimensional habitat [4]. Fisher assumed
that the individuals in the population carry a gene that occurs in two forms, A and
a. Then there are three genotypes: the homozygotes AA, aa and the heterozygote
Aa. Individuals in this population are classified according to the genotype to
which they belong. The ability of an individual to survive to adulthood depends
entirely on its genotype. The specific form of h that Fisher assumed means that
the fitness of the heterozygotes is between those of the homozygotes.
In 1975, Aronson and Weinberger studied (1.1), allowing h to have an
intermediate zero, denote by a, between 0 and 1 [1]. Under such an assumption,
there are two additional cases to consider: when h >0 on (0, a), h <0 on (a, 1)
and when h < 0 on (0, a), h > 0 on (a, 1). Aronson and Weinberger called
Fisher's model the heterozygote intermediate case, and the above two cases the
heterozygote superior and heterozygote inferior case, respectively. These last two
cases correspond to the situation when the heterozygotes are more fit and less fit
to survive than the homozygotes, respectively.
Fisher's model is based on assumptions of a highly idealised situation. The
author of this paper is interested in extending the results of Fisher's model to the
case when the fitnesses of the individuals also depend on the population size.
• Current address: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112,
U.S.A.
2
Roger Lui
Existence and stability o
Equation (1.1) then becomes a system of reaction-diffusion equations,
Pt =_Pxx + f(p, N)p(1 - p ),
{ Nr- N= + g(p, N)N,
2. Hy
(1.2)
where 0 ~p ~ 1 is allele A's frequency and N ~ 0 is the population size. The
function f(p, N) = T/A(p, N)- T/a(p, N), where T/A• Tla represent the fitnesses of
allele A and allele a, respectively. If we denote the fitnesses of genotypes AA, Aa
and aa by T/ 1(N), T/2(N) and r13(N), then T/A = PT/1 + (1- P)T/2 and Tla = PT/2 +
(1 - p )T/ 3. The second equation in (1.2) models the growth and spread of the
population. The function g is the specific growth-rate and may be thought of as
the fitness of the entire population. Thus g(p, N) = PT/A + (1 - p )T/a = p 2 11 1 +
2p(1- p )11 2 + (1- p ) 2 11 3 • It is important to observe the relation
gp(p, N)
= 2f(p, N).
(1.3)
The mathematical theory for equation (1.1) is very rich and well understood
(see [3] for details). One of the most intriguing properties of (1.1) is the existence
of travelling waves. A travelling wave solution of (1:1) with speed (} is a
non-constant function u(~) such that u(x + 8t) satisfies (1.1) for all x and t > 0. In
the heterozygote intermediate case, Kolmogorov, Petrovski and Piscunoff showed
in 1937 (among other things) that there exists (}* > 0 such that increasing
travelling waves u exist with the properties that u( -oo) = 0, u(oo) = 1 if and only if
(} ~ (}* [6]. For the heterozygote inferior case, Aronson and Weinberger showed
that if H h > 0, then there exists (}* > 0 such that increasing travelling wave
solutions connecting 0 and 1 exist if and only if 8 = 8*. For the heterozygote
superior case, since 0 and 1 are unstable solutions of the ordinary differential
equation u' = h(u), no travelling wave solution exists connecting 0 and 1.
For (1.2), it turns out that under appropriate assumptions on f and g, the
model may be classified into four cases: the first and second heterozygote
intermediate cases, the heterozygote superior case and the heterozygote inferior
case. The asymptotic behaviour of solutions to (1.2) with bounded initial-data in
all four cases have been investigated by the author in [7]. Now we are interested
in proving the existence of travelling waves that connect two critical points of
(1.2). This problem involves analysing the behaviour of trajectories of a
four-dimensional ordinary differential equation containing a parameter (the wave
speed). In this paper, we employ the shooting argument to prove the existence of
travelling wave solutions for the second intermediate case for sufficiently large
wave speed. We also prove the stability of these waves (in a certain sense) using
the maximum principles. We hope to investigate the existence and stability of
travelling wave solutions for the first intermediate and heterozygote inferior cases
in the future. For the inferior case, the techniques used will be substantially
different from those used in this paper, since travelling waves are expected to
exist only for a discrete set of values of the wave speed.
The organisation of the paper is as follows. In Section 2, we state the
hypotheses off and g and classify (1.2) into four cases. In Section 3, we prove the
existence of travelling waves for the second intermediate case and in Section 4,
we study their stability properties. The main results are Theorems 3.1 and 4.3.
Let f and g be twice continuously
region Q = {(p, N) I O~p ~ 1, N~O}.
fN>O,
8
and that there exists a positive constant
f(p, 0) < 0 and f(p
g(p, 0) > 0 and g(J
From (2.1) and the Implicit Function
such that f(p, N(p)) = 0 and g(p, N(p
graphs of Nand N in Q, respectively. '
more than once and if they do intersec1
are four basic types of behaviour for r1
r1 lies above r2;
r1 lies below r2;
r 1 intersects r 2 once with J
r 1 intersects r 2 once with J
Following the reasoning given in [7], w
heterozygote intermediate cases and (:<
inferior cases, respectively.
As mentioned in Section 1, our mo<
stronger than necessary and all we nee
that
signgP = s
From the definition of N, we have N'
f 1 is increasing in case (2.4), decreasin
(minimum) at the point of intersection '
A travelling wave solution to (1.2)
functions p(~), IV(~) such that p(x + 1
t > 0. This is equivalent to saying that p
p'=q
q' = (}q- 1
N'=M
{
M'
= 8M-
for -oo < ~ < oo (we dropped the tilda in
; - ± oo which must then be a root c
g(p, N)N=O.
Let us find the roots of the equation
For cases (2.4) to (2.7), there are fou1
They are (p, N) = (0, 0), (1, 0), (0, K 3 :
Existence and stability of travelling wave solutions
li
:tction-diffusion equations,
')p(1- p),
V)N,
2. Hypotheses
(1.2)
d N ~ 0 is the population size. The
tere TJA, TJa represent the fitnesses of
)te the fitnesses of genotypes AA, Aa
TJA = PT/1 + (1 - p )TJz and TJa = PTJz +
todels the growth and spread of the
rowth-rate and may be thought of as
2
: g(p, N) = PTJA + (1- P )TJa = P TJ1 +
observe the relation
(p, N).
3
(1.3)
.. 1) is very rich and well understood
ing properties of (1.1) is the existence
olution of (1:1) with speed 8 is a
9t) satisfies (1.1) for all x and t > 0. In
;orov, Petrovski and Piscunoff showed
exists (J* > 0 such that increasing
hat u( -oo) = 0, u(oo) = 1 if and only if
tse, Aronson and Weinberger showed
such that increasing travelling wave
only if (J = (J*. For the heterozygote
solutions of the ordinary differential
tion exists connecting 0 and 1.
opriate assumptions on f and g, the
: the first and second heterozygote
10r case and the heterozygote inferior
; to (1.2) with bounded initial-data in
~author in [7]. Now we are interested
es that connect two critical points of
the behaviour of trajectories of a
tion containing a parameter (the wave
.ng argument to prove the existence of
1termediate case for sufficiently large
these waves (in a certain sense) using
~stigate the existence and stability of
tediate and heterozygote inferior cases
techniques used will be substantially
ince travelling waves are expected to
:wave speed.
follows. In Section 2, we state the
, four cases. In Section 3, we prove the
td intermediate case and in Section 4,
in results are Theorems 3.1 and 4.3.
Let f and g be twice continuously differentiable in a neighbourhood of the
region Q = {(p, N) J O~p ~ 1, N~O}. We assume that
fN>O,
gN<O
in Q
(2.1)
and that there exists a positive constant M such that
f(p,O)<O
and f(p,M)>O
for0~p~1,
g(p, 0) > 0 and g(p, M) < 0 for
O~p ~
1.
(2.2)
(2.3)
From (2.1) and the Implicit Function Theorem, there exist functions N and N
such thatf(p, N(p))=O and g(p, N(p))=O for O~p~l. Let r 1 and r 2 be the
graphs of Nand N in Q, respectively. We assume that r 1 and r 2 do not intersect
more than once and if they do intersect, they do so non-tangentially. Then there
are four basic types of behaviour for rl and rz:
rl lies above r2;
(2.4)
rl lies below rz;
(2.5)
r 1 intersects r 2 once with N(O) > N(O)
r 1 intersects r 2 once with N(O) < N(O)
and N(1) < N(1);
(2.6)
and N(1) > N(1).
(2. 7)
Following the reasoning given in [7], we call (2.4) and (2.5) the first and second
heterozygote intermediate cases and (2.6), (2.7) the heterozygote superior and
inferior cases, respectively.
As mentioned in Section 1, our model implies that gP = 2f. This condition is
stronger than necessary and all we need to assume for our theorems is the fact
that
signgP =sign[ in Q.
(2.8)
From the definition of N, we haveN'= -(gp/gN)· Therefore (2.8) implies that
increasing in case (2.4), decreasing in case (2.5) and has a local maximum
(minimum) at the point of intersection with r 2 in case (2.6) (case (2.7)).
A travelling wave solution to (1.2) with speed 8 is a pair of non-constant
functions p(;), N(;) such that p(x + 8t), N(x + Ot) satisfy (1.2) for all x and
t > 0. This is equivalent to saying that p, N satisfy the system
r 1 is
p'=q
q' = 8q- f(p, N)p(1- p)
N'=M
{
M' = 8M- g(p, N)N
< oo (we dropped the tilda in (2.9)). (p, N) should also have a limit as
± oo which must then be a root of the equations f(p, N)p(1- p) = 0 and
g(p, N)N=O.
Let us find the roots of the equations f(p, N)p(1- p) = 0 and g(p, N)N = 0.
For cases (2.4) to (2. 7), there are four roots which lie on the boundary of Q.
They are (p, N) = (0, 0), (1, 0), (0, K 3 ) and (1, K 1). The last two roots are the
for
-oo <;
(2.9)
;--+
4
Roger Lui
endpoints of r 1 in Q. There is also a root (p *, N*) in the interior of Q in cases
(2.6) and (2.7). All these roots are equilibrium solutions of (1.2) with or without
diffusion. We can analyse their local stability properties without diffusion by
finding the eigenvalues of the linearised matrix at the respective equilibrium
points. Doing so, we find that (0, 0) and (1, 0) are always unstable. For (2.4),
(1, K1) is stable and (0, K 3 ) is unstable while the opposite is true for (2.5). For
(2.6), all the boundary equilibrium points are unstable and (p*, N*) is stable. For
(2.7), (0, K 3 ) and (1, K 1) are stable while (p*, N*) is unstable.
We now return to the problem of finding travelling waves. Let (p 1 , N1),
(p 2 , N2 ) be two equilibrium points mentioned above. If p 1 = p 2 = 0 or 1 or if
N1 = N 2 = 0, then finding a solution of (2.9) connecting the critical points
(p 1 , 0, N1 , 0) and (p 2 , 0, N 2 , 0) concerns only two of the equations of (2.9). The
methods and results for such smaller systems are well known (see the discussion
in Section 1). Here we are interested in connecting critical points which lie on
opposite sides of p = 0 and p = 1. We are able to do so for the case (2.5) and for
that we need the additional assumption that
(2.10)
r 2 lies above the lineN= K 3 •
The result will then be the existence of monotone travelling wave solutions
connecting (1, 0, 0, 0) to (0, 0, K 3 , 0) for sufficiently large 8. There may exist
non-monotone travelling waves connecting the same critical points so that we do
not have uniqueness.
There are a number of papers in the recent past which proved the existence of
travelling wave solutions for a reaction-diffusion system using techniques similar
to the one used in this paper (see [2, 8, 10]). In fact, some of the ideas of the
author in this paper grew out of reading a very nice paper by Dunbar [2]. Dunbar
treated a more difficult problem than (2.5) in that, in his paper, the critical point
at -oo has a three-dimensional unstable manifold, whereas in (2.5), the critical
point (1, 0, 0, 0) has a four-dimensional unstable manifold.
3. Existence of travelling waves
Existence and stability ,
statement and proof may be found in
and prove our main theorem.
THEOREM 3.1. Suppose f and g are
hood of Q and (2.1)-(2.3), (2.5
max ( 81, 82 , 8 3 , 84). Then travelling H
8 > 8*. Furthermore, p' < 0, N' > 0,
~- -oo and it approaches (0, 0, K 3 , 0)
Proof. We first find the dimensions
critical points A= (1, 0, 0, 0) and B = 1
at A, denoted by JA, has four posit
(1, 0), 4g(l, 0)), which is satisfied. A1
eigenvalues. Therefore the unstable n
stable manifold at B is two-dimensiona
Since JA and 18 have real eigenvalu
travelling waves. Since f is negative be
should either be decreasing and movir
assume that 8 > 0 so that p should be d
be increasing. We then try to look
{(p, q, N, M) IO<p < 1, q <0, O<N
section, y(~, y0 ) will denote the solutio
If Yo E W, then y( ;, y0 ) either rema
w-. w- contains part of the boundari
or N = K 3 or M = 0 in the definition of
{q = 0}, {N = K 3 } and {M = 0}, respc
boundary {p = 0, q = 0} = {(p, q, N, .t.
it is contained in the invariant manifolc
exit at the boundary {N = K 3 , M = 0}.
Wplus {p=O,M=O}, {M=O,q=O
the exit set w-. A good way to look ;
opposite sides do not intersect.
An important observation here is
Consider the equation p, = Pxx + f(p, O)p(1 - p ). Let r = 1 - p. Then r satisfies
1
r, = rxx- /(1- r, O)r(1- r). Let h(r) = - /(1- r, O)r(1- r). Then he C [0, 1],
h(O) = 0, h(1) = 0, h > 0 on (0, 1) and h '(0) = - /(1, 0) > 0. Therefore there
exists 8 1 ?;. 2V-f(1, 0) such that decreasing travelling wave solutions connecting
p = 1 to p = 0 exist for p, = Pxx + f(p, O)p(1 - p) if 8?;. 8 1 • We define
8 2 ?;. 2V -/(1, K 3 ) similarly for the equation p, = Pxx + f(p, K 3 )p(1 - p ). Next
consider the equation N, = Nxx + g(1, N)N. Let h(N) = g(1, N)N. Then he
C 1[0, K 1], h(O) = 0, h(K 1 ) = 0, h > 0 on (0, K 1) and h'(O) = g(1, 0) >0.
Therefore, there exists 8 3 ?;. 2Vg(1, 0) such that increasing travelling wave
solutions joining N = 0 to N = K 1 exist for N, = Nxx + g(1, N)N if 8?;. 8 3 • We
define 8 4 ?;. 2Vg(O, 0) similarly for the equation N, = N= + g(O, N)N. We denote
these four monotone travelling waves by w1 , w2 , w3 , w4 , respectively.
We assume the reader is familiar with the basic results on stable and unstable
manifolds and the LaSalle-Lyapunov Invariant Principle [5]. We also need to
Wazewski's Theorem which is a formalisation of the shooting argument.
\
Figure 1.
Existence and stability of travelling wave solutions
N*) in the interior of Q in cases
solutions of (1.2) with or without
r properties without diffusion by
rix at the respective equilibrium
I) are always unstable. For (2.4),
:he opposite is true for (2.5). For
llStable and (p*, N*) is stable. For
N*) is unstable.
~ travelling waves. Let (p 1 , Nt),
l above. If p 1 = p 2 = 0 or 1 or if
)) connecting the critical points
two of the equations of (2.9). The
tre well known (see the discussion
tecting critical points which lie on
to do so for the case (2.5) and for
N=K 3 •
(2.10)
)notone travelling wave solutions
iciently large 8. There may exist
: same critical points so that we do
past which proved the existence of
lon system using techniques similar
. In fact, some of the ideas of the
r nice paper by Dunbar [2]. Dunbar
that, in his paper, the critical point
ifold, whereas in (2.5), the critical
>le manifold.
Ding waves
- p ). Let r = 1 - p. Then r satisfies
1
l - r, O)r(1- r). Then hE C [0, 1],
~) = - f(1, 0) > 0. Therefore there
ravelling wave solutions connecting
l)p(1- p) if 8?; 8 1 • We define
t1 p, = P= + f(p, K3)p(1 - p ). Next
Let h(N) = g(1, N)N. Then h e
1 (O,K 1 )
and h'(O)=g(1,0)>0.
h that increasing travelling wave
N, = N= + g(1, N)N if 8?; 8 3 • We
lon N, = N= + g(O, N)N. We denote
w2 , w3, W4, respectively.
basic results on stable and unstable
nt Principle [5]. We also need to use
tion of the shooting argument. Its
5
statement and proof may be found in [2, Section 2]. We are now ready to state
and prove our main theorem.
THEOREM 3.1. Suppose f and g are continuously differentiable in a neighbourhood of Q and (2.1)-(2.3), (2.5), (2.8) and (2.10) hold. Let (J* =
max(8 1 , 8 2 , 8 3, 8 4 ). Then travelling wave solutions p(;), N(;) for (1.1) exist if
8>8*. Furthermore, p'<O, N'>O, (p,p',N,N') approaches (1,0,0,0) as
;~ -oo and it approaches (0, 0, K 3, 0) as ;~ oo.
Proof We first find the dimensions of the stable and unstable manifolds at the
critical points A = (1, 0, 0, 0) and B = (0, 0, K 3, 0). The Jacobian matrix for (2.9)
at A, denoted by JA, has four positive eigenvalues as long as 8 2 >max ( -4f
(1, 0), 4g(1, 0)), which is satisfied. At B, ] 8 has two positive and two negative
eigenvalues. Therefore the unstable manifold at A is four-dimensional and the
stable manifold at B is two-dimensional.
Since JA and ] 8 have real eigenvalues, it is reasonable to look for monotone
travelling waves. Since f is negative below K 3, the wave p(;) connecting 0 and 1
should either be decreasing and moving left or increasing and moving right. We
assume that (J > 0 so that p should be decreasing. Likewise, the wave N(;) should
be increasing. We then try to look for solutions of (2.9) in the set W =
{(p, q, N, M) IO<p < 1, q <0, O<N < K 3 , M >0} in IR 4 • For the rest of this
section, y(;, y0 ) will denote the solution to (2.9) that equals Yo at ; = 0.
If y0 E W, then y( ;, y0 ) either remains in W for all ; ~ 0 or exits W through
w-. w- contains part of the boundaries of w obtained by setting p = 0 or q = 0
or N = K3 or M = 0 in the definition of W. We denote these four sets by {p = 0},
{q = 0}, {N = K 3 } and {M = 0}, respectively. The solution y cannot exit at the
boundary {p =0, q =0} = {(p, q, N, M) IP =0, q =0, O<N<K3 , M>O} since
it is contained in the invariant manifold p = q = 0 and y0 E W. Similarly, y cannot
exit at the boundary {N = K 3 , M = 0}. The union of the above four boundaries of
Wplus {p=O,M=O}, {M=O,q=O}, {q=O,N=K3 } and {N=K3 ,p=O} is
the exit set w-. A good way to look at w- is shown in Figure 1, where sets on
opposite sides do not intersect.
An important observation here is that if Yo E w and y( ;*, Yo) E w-' then
(
{p=O}
\
{N = K3 }
\
{M=O}
{q= 0}
Figure 1. The set
/
w-.
6
Roger Lui
Existence and sta
y( ;, y0 ) fi W (closure of W) for ; immediately after .;*. This is obvious if y( .;, y0 )
lands on {p=O} or {N=K3 }. Ify(;*,y0 )E{q=O}, thenq'=-f(p,N)p(1-p)
is positive at ;* since 0 < p( ;*) < 1 and (2.10) holds. Similarly, if y( ;*, y0 ) E
{M = 0}, then M'(.;*) < 0 from the last equation in (2.9). The immediate exit set
of W is defined as W* = {y0 E aw I y(;, y0 ) fi W for ; > 0, .; sufficiently small}.
(aW means the boundary of W.) It is clear from above that w- c W*. In fact,
W* is just w- union the following sets: {p = 0, N = 0}, {p = 1, N = K 3 },
{p=1,M=O}, {q=O,N=O}, {p=O or 1, q=O, N=K3 }, {p=O or 1,
q=O,M=O}, {N=OorK3 , M=O,p=O}, {N=OorK3 , M=O, q=O}. Note
that a trajectory of (2.9) with initial data in W cannot reach any of these sets in
forward .;. w- will play an important role in our proof.
We claim that
for (} > (}*, there exists a y0 E W such that y*( .;) = y( .;, y0 ) remains in
W for all; and y*(.;)~A as;~- oo.
31
<· )
Suppose for the moment that (3 .1) is true. Then y* = (p *, q *, N*, M *) must
be bounded in IR 4 • To see this, let C be chosen such that 1-f(p, N)p(1- p)l ~ C
for 0 ~p ~ 1 and 0 ~ N ~ K 3 . If q*(.;*) < -C/ (}at some .;*, then [q*]'(.;*) < 0 so
that q*(.;) < -C/(} for.;~.;*. This implies that p*(.;) = 0 for some finite value of
.;, a contradiction. Therefore q* ~ -C/(} = -C 1 • Similarly, we can show that M*
is bounded above by some positive constant c2.
Consider the function V(p, q, N, M) = g(1- p, N)- q + (}p which is defined
and differentiable on W. Then V = -gP(1- p, N)q + f(p, N)p(1- p) + gN(1p, N)M where V means the directional derivative of V in the direction given by
the vector whose components are the right-hand side of (2.9). From (2.8), (2.1)
and (2.10), each term in the above expression is non-positive so that V ~ 0 in W.
Thus Vis a Lyapunov function on W. Let E = {(p, q, N, M) E WI V = 0}. Since
y* lies in W for .; ~ 0 and is bounded, the Invariant Principle implies that y* must
converge to the largest invariant set inside E. Such a set contains only two
elements, (0, 0, 0, 0) or (0, 0, K 3 , 0). For y* to approach (0, 0, 0, 0), it must
intersect the stable manifold at (0, 0, 0, 0) which has dimension one and lies inside
the invariant manifold N = 0, M = 0. Since y* never intersects this invariant
manifold, y* must approach (0, 0, K 3 , 0) as ;~ oo. Therefore the proof of
Theorem 3.1 is complete once we prove (3.1).
To prove (3.1), we shall define a set ~ E IR 4 with the following properties
(a)-(d):
(a) ~is a simply connected, compact subset of W.
(b) If a trajectory of (2.9) intersects ~' then it must intersect w- before it
intersects ~ again in forward ;.
(c) If y0 E~ and y(.;,y0 )EW but y(.;,y0 )fiW*, then there exists an open
neighbourhood of y(.;, y0 ) in W which is disjoint from W*.
Suppose such a set ~ can be found. Then there exists y0 E ~ such that
y(.;, y0 ) E W for .; ~ 0. To see this, assume the contrary. Then Wazewski's
Theorem (which uses (b) and (c) above) implies that the solution map of (2.9) is a
homeomorphism between ~ and its image on w-. We shall also show that the
image of a~ forms a closed loop around B = (0, 0, K 3 , 0) in w- (see Fig. 1).
These and the fact that ~ is simi
w- is
Therefore there exists y0 E ~ su1
(3.1) is completed by showing th
(d) Let YoE~. Then y(.;,y0;
;~- oo.
The rest of this paper is devo1
proving (3.2).
Let (} > (}* and let w1 , w2 , w3 ,
the beginning of this section. C
q' = (}q- f(p, O)p(1- p). Let z1
(1, 0) to the origin in the pha:
.
eigenvalues
of the matrix [
0
/(1, 0
need (} > (} 1). The phase plane fc
is similar to Figure 2, except thl
llt > ll2 > 0 are the eigenvalues <
.
~
dz2
smce JN > 0 and- ( -oo) = ,u 2 (h
dw2
Let
Xt = [Xn], x2 = [x2IJ be tVI
X12
X22
X22
0 <-
x21
q
0
Figure 2. Phase plane diagram for the
-[;J
is tangent to the curve (w1 , z1) at (1
Existence and stability of travelling wave solutions
fter ;*. This is obvious if y(;, y0 )
=0}, thenq'=-f(p,N)p(1-p)
)) holds. Similarly, if y( ;*, y0) E
n in (2.9). The immediate exit set
V for ; > 0, ; sufficiently small}.
1m above that w- c W*. In fact,
{p=O,N=O}, {p=1,N=K3 },
, q = 0, N = K 3}, {p = 0 or 1,
'V = 0 or K3 , M = 0, q = 0}. Note
cannot reach any of these sets in
ur proof.
y*(;)=y(;, Yo) remains in
( _)
31
Then y* = (p*, q*, N*, M*) must
that 1-f(p, N)p(1- p)l ~ C
~at some ;*, then [q*]'(;*) < 0 so
t p*(;) = 0 for some finite value of
:t. Similarly, we can show that M*
1 such
These and the fact that
+ 8p which is defined
N)q + f(p, N)p(1- p) + gN(1 tive of V in the direction given by
nd side of (2.9). From (2.8), (2.1)
is non-positive so that V ~ 0 in W.
= {(p, q, N, M) E WI V = 0}. Since
riant Principle implies that y* must
E. Such a set contains only two
to approach (0, 0, 0, 0), it must
:h has dimension one and lies inside
y* never intersects this invariant
s ;-oo. Therefore the proof of
CJ,
is simply connected contradict the fact that
w- is not simply connected.
(3.2)
Therefore there exists Yo E ~ such that y( ;, y0) E W for all ; ~ 0. The proof of
(3.1) is completed by showing the following:
(d) Let YoE~. Then y(;,y0 )EW for all -oo<;~o and y(;,y0 )-A as
; - - oo.
The rest of this paper is devoted to finding ~ with the desired properties and
proving (3.2).
Let 8 > ()* and let Wv w2 , w3 , w4 be the monotone travelling waves defined at
the beginning of this section. Consider the phase plane of the system p' = q,
q' = 8q- f(p, O)p(1- p). Let Zt = w~. Then (wt(;), Zt(;)) traces a curve joining
(1, 0) to the origin in the phase plane (see Fig. 2). Let At> A2 > 0 be the
0
1
)
eigenvalues of the matrix [ (
It is known that dzt ( -oo) = A2 (here we
f 1, 0 8
dwt
need 8 > Ot). The phase plane for the system p' = q, q' = 8q- f(p, K 3 )p(1- p)
is similar to Figure 2, except that we have to replace At, A2 by J-tt, J,t 2 , where
0
1
p, 1 > J,t 2 > 0 are the eigenvalues of the matrix [ (
)
Note that l'z < Az
d
f 1, K 3 8
since fN > 0 and dZz ( -oo) = J,t 2 (here we need 8 > 8 2 ).
J.
J.
Wz
~-
- p, N)- q
~
7
Let
Xt
. . umt. vectors such that
= [Xu] , x = [Xzt] be two positive
Xzz
2
X12
Xzz
X12
Xzt
Xu
0 < - < l'z < Az <- < At.
(3.3)
q
0
p
- [;]_ It- x,
(w1 ,z 1 )
-[,:J -[;,]
: IR 4 with the following properties
t of W.
len it must intersect
w-
before it
1It W*, then there exists an open
s disjoint from W*.
'hen there exists Yo E ~ such that
e the contrary. Then Wazewski's
es that the solution map of (2.9) is a
11 w-. We shall also show that the
= (0, 0, K3 , 0) in w- (see Fig. 1).
Figure 2. Phase plane diagram for the system p' = q, q' = fJq- f(p, O)p(l- p ). The vector
-[~]is tangent to the curve (w 1 , z1)
at (1, 0).
Roger Lui
8
Existence and stab
data
M
where a, {3, y and (J are nonnega1
easily verified:
(i) If f3 = y = (J = 0, then y(;
M = N = 0. From Figure ~
y( ;, Yo) intersects p = 0 firs1
(ii) If a = r = (J = 0, then y(
M = N = 0 and if {3 > 0 is l
sects q = 0 first, before p =
(iii) If a= f3 = (J = 0, then y(;
p = 1, q = 0. From Figure
y( ;, Yo) intersects M = 0 bel
(iv) If a=f3=y=O, then y(;
p = 1, q = 0 and if (J > 0
intersects N = K 1 before M
M' and M will continue tot
reaches N = K 3 before M =
s
~
0
Figure 3. Phase plane diagram for the system N'
[;J
=
M,
M' =OM- g(l, N)N.
The vector
is tangent to the curve ( w3 , z3 ) at (0, 0).
This condition is equivalent to choosing x 1 , x2 at certain relative positions to the
[;J, [;J [:J
eigenvectors
and
as indicated in Figure 2.
Next we consider the phase plane of the system N' = M, M' = fJM- g(l, N)N.
Let z3 = w~. Then (w3 (;), z3 (;)) traces out a curve joining the origin to (K 1 , 0) in
the phase plane (see Fig. 3). Let A3 > A4 > 0 be the eigenvalues of the matrix
0
1
dz3 ( -oo) = A ( fJ > fJ ). The phase plane of the
4
3
[ -g ( 1, 0) fJ . It is known that -d
w3
system N' = M, M' = fJM- g(O, N)N is similar to Figure 3, except that we have
to replace K 1 by K 3 and A3 , A4 by !J 3 , !J 4 , where !J 3 > !J 4 > 0 are the eigenvalues of
. [
dz4 (-oo)=!J (fJ>fJ ) and A <!J <-<A
fJ
the matnx
(0 ) 1 J. Also -d
4
4
4
4
3,
-g 0, 0
fJ
w4
2
since gP(p, 0) < 0.
One of the problems with choo
Yo f1 W. The following lemmas say tl
exit W in the right way. Recall the
LEMMA
X3t] , x = [X41] be positive
. . umt
. vectors sueh t h at
{
p'=q,
q'
= fJq
Plot the solution (p(;), q(;)) on
(p(O), q(O)) lies below the curve
Suppose (p(;), q(;)) crosses this cu
dw1 dq
Then at ( w 1*, z 1*) , -~-which
4
X32
= 0 and let y0 ~
Proof Think of N( ;) as a given
J
Let x3 = [
3.2. Let f3
y(;, Yo) reaches {p = 0} first before
dz1 dp
'
f(p(;*), N(;*)) ~f(wr, O). This c
X42
(3.4)
Therefore (p(;), q(;)) remains bel
first before {q = 0}. This completes
This condition is equivalent to choosing x3 , x4 at certain relative positions to the
The following lemmas may be I
Lemma 3.3, we have to use the
eigenvectors
[;J, [:J [:J.
Having defined
and
X;,
See Figure 3.
i = 1, 2, 3, 4, we consider the trajectories of (2.9) with initial
x22
- < ll-2· In Lemma 3.4, we have to
X21
Existence and stability of travelling wave solutions
9
data
y,~ m
-~[~'] -p[~'] +y[n +b[l]'
(3.5)
where a, {3, y and () are nonnegative constants. The following statements can be
easily verified:
•· z3)
----...;;;:::a.-t--N
K,
{' = M,
•2
M'
= OM- g(l, N)N.
The vector
at certain relative positions to the
din Figure 2.
stem N' = M, M' = ()M- g(1, N)N.
curve joining the origin to (K 1 , 0) in
0 be the eigenvalues of the matrix
(i) If {3 = y = b = 0, then y( ;, y0 ) lies on the invariant manifold
M = N = 0. From Figure 2, if a> 0 is sufficiently small, then
y( ;, y0 ) intersects p = 0 first, before q = 0.
(ii) If a= y = b = 0, then y(;, y0 ) lies in the invariant manifold
M = N = 0 and if {3 > 0 is sufficiently small, then y(;, y0 ) intersects q = 0 first, before p = 0.
(iii) If a= {3 = b = 0, then y( ;, y0 ) lies in the invariant manifold
p = 1, q = 0. From Figure 3, if y > 0 is sufficiently small, then
y(;, y0 ) intersects M = 0 before N = K 1 •
(iv) If a= {3 = y = 0, then y(;, y0 ) lies in the invariant manifold
p = 1, q = 0 and if () > 0 is sufficiently small, then y(;, y0 )
intersects N = K 1 before M = 0. Once y(;, y0 ) crosses N = K 1 ,
M' and M will continue to be positive (see (2.9)) so that y(;, y0 )
reaches N = K 3 before M = 0.
One of the problems with choosing initial data y0 according to (3.6) is that
y0 It W. The following lemmas say that we may choose y0 E W and y( ;, y0 ) will still
exit Win the right way. Recall the set w-.
LEMMA 3.2. Let {3 = 0 and let y0 satisfy (3.5) with a> 0 sufficiently small. Then
y(;, y0 ) reaches {p = 0} first before {q = 0} as long as N > 0.
Proof. Think of N(;) as a given function and consider the system
A. 4 ( () > () 3 ). The phase plane of the
lar to Figure 3, except that we have
:re f1 3 > Jl 4 > 0 are the eigenvalues of
=
Jl 4 ( () > () 4 ) and A4 < /14 <
()
2 < A3,
.x4z<A. •
3
X41
~
(3.4)
at certain relative positions to the
e 3.
~r
p' =q,
{ q' = ()q- f(p, N)p(1- p).
the trajectories of (2.9) with initial
(3.7)
Plot the solution (p(;), q(;)) on the phase plane in Figure 2. Since {3 = 0,
(p(O), q(O)) lies below the curve representing the travelling wave (wt> z1).
Suppose (p(~), q(~)) crosses this curve at ~ = ~*. Let (wf, zD = (p(~*), q(~*)).
Then at (wt,
vectors such that
(3.6)
zn,
ddwl
z1
~ddq
p
which, according to (3.7) and (2.9), implies that
f(p(~*), N(~*))~f(wf, 0). This contradicts the fact that fN>O if N(;*)>O.
Therefore (p(~), q(~)) remains below (wl> z1 ) for ;~o and it reaches {p =0}
first before {q
= 0}.
This completes the proof of Lemma 3.2.
D
The following lemmas may be proved in the same way as Lemma 3.2. In
Lemma 3.3, we have to use the travelling wave (w2 , z2 ), and the fact that
22
x < f.lz· In Lemma 3.4, we have to use the travelling wave ( w3 , z3 ) in the phase
Xz1
Existence and stab
Roger Lui
32
plane Figure 3, and the fact that x < A. 4. Finally, in Lemma 3.5, we have to use
X31
42
the travelling wave (w4 , z4 ) and the fact that 11 4 < x •
X41
10
a(s)
3.3. Let a= 0 and {3 > 0 be sufficiently small. Then y(;, y0 ) reaches
{q = 0} first before {p = 0} as long as N < K 3 •
LEMMA
LEMMA 3.4. Let b = 0 and y > 0 be sufficiently small. Then y(;, y0 ) reaches
{M = 0} first before {N = K 3 } as long asp< 1.
LEMMA
{N = K 3 }
Pis)
0
3.5. Let y = 0 and b > 0 be sufficiently small. Then y(;, y0 ) reaches
first before {M = 0} as long asp> 0.
1r
2
We now define a closed loop a in W which will become the boundary of~.
Recall that we are still trying to define ~. Let a= {y0 (s) I 0 ~ s ~ 2:r}, where
y(s)
0
1r
2
The loop a has the following properties:
(i) a, {3, y and b are nonnegative continuous functions defined on)
the interval [0, 2:r ].
(ii) y0 (s 1) = y0 (s 2) if and only if s 1 = s 2 or s 1 = 0 and s 2 = 2:r.
(iii) ac W.
(iv) a lies on the ellipsoid.
{
Y= (x,y,z,w)
6(s)
0
l
1r
2
(3.9)
Figure 4. The graphs of a(s), p(s), y(s)
6(0) = 6(2n).
2
lP(x- 1)2
lPz
}
+2y 2 + --+2w 2 =e 2 .
2
2
":--~-------......_,
(3.10)
The constant e > 0 in (3.10) is chosen to be sufficiently small so that if
(p, q, N, M) E Y, then 8 2 + 4f(p, N)p > 0 and 8 2 - 4g(p, N) > 0. This is possible
because 8 > 8 1 ~ 2V-f(1, 0) and 8 > 8 3 ~ 2Vg(1, 0). We also choose e so small
that if y0 EY, then y(;,y0)~A as ;~-oo. Recall that JA has four positive
eigenvalues.
The best way to describe a is draw the graphs of a, {3, y and b. From Figure
4, (3.9)(i) and (ii) are obvious, (3.9)(iii) follows because a(s), {3(s) or y(s),
b(s) do not vanish simultaneously. Finally, (3.9)(iv) is equivalent to choosing a,
{3, y and b to satisfy the equation
2
2
2
2
+ i: ) 2 _ 2
+ R ) + 8 ( yx31 + bx4t) + 2(
8 ( ax 11 + {3x21) + 2 (
a'X12 pX22 2
YX32
ux 42 - e .
2
2
(3.11)
At the same time as we are defining y 0 (s), we also show that assuming that
y( ;, y0 (s)) exits W for all 0 ~ s ~ 2:r, its image on w- forms a closed loop around
B = (0, 0, K 3 , 0). (If for some s, y(;, y0 (s)) remains in W for all ; ~ 0, then
we can show that y(;, y0 (s)) approaches A in Was ;~ -oo and (3.1) is valid.)
We introduce the notation
(p(;), q(;), N(;), M(;)) temporari
Consider first the interval r ~ s
~(s) > 0 be constant and {3(s) =
(mdependent of s ). Suppose y, {)
N(;) = 0, M(;) = 0 for all ; andY'
; = T. Let Et > 0 be sufficiently s
ne~ghbourhood of y(T + e1 , Yo) wh
neighbourhood U of y0 such that if y
1
0<
E2
< K3, we can choose y(s ), b(J
small so that y0(s) E U and JN(;)J ~
above, y(T + e 1 , y0 (s)) E V. From I
exited W through {q = 0}. Since 0
could not have first exited W througl
first exit W through {p
= 0} U {M =
1). This fact continues to hold even
Existence and stability of travelling wave solutions
y, in Lemma 3.5, we have to use
a(s)
X42
<-.
~
0
~
r
!!. -
r
2
X41
!!. !!. + T
2
:=....
n - r
.
1
n n +r
2
11
,.,......-:::::
, 1
3 TC - r 3n 3n + r 2n - r 2n
2
2 2
ntly small. Then y(;, y0 ) reaches
~
ntly small. Then y(;, y0 ) reaches
Pis)
0
Jr
-
ntly small. Then y( ;, y0 ) reaches
t
1
will become the boundary of ~.
= {yo(s) I0 ~ s ~ 2n}, where
Y('{!] + 6(')UJ (3.8)
t
2
~·'
o(sl
1ous functions defined onl
0
--=-
~
Jr
':::
2
__._
"0
===-
Jr
•
Jr
= 0 and s2 = 2n.
(3.9)
::=.
......::::
2n
I
3n
2
Jr
2
t
3n
2
Jr
_/
3n
2
2n
"
2n
Figure 4. The graphs of a(s), p(s), y(s) and D(s). a(O) = a(2n), P(O) = P(2n), y(O) = y(2n),
6(0) = 6(2n).
(3.10)
We introduce the notation y0 (s) = (p 0 , q 0 , N0 , M0 ) and y( ;, y0 (s)) =
(p(;), q(;), N(;), M(;)) temporarily for the next three pargraphs.
> be sufficiently small so that if
Consider first the interval r ~ s ~ ~- r where r appears in Figure 4. Let
2
lFz
y2+2+2w2=
e2 } .
(f- 4g(p, N) > 0. This is possible
/g(1, 0). We also choose e so small
l. Recall that JA has four positive
i
tphs of a, /3, y and lJ. From Figure
lllows because a(s), J3(s) or y(s),
3.9)(iv) is equivalent to choosing a,
a(s) > 0 be constant and /3(s) = 0 on this interval in the definition of y0
(independent of s). Suppose y, {) also equal to zero. Then 3.6(i) implies that
N(;) = 0, M(;) = 0 for all ; and y(;, y0 ) first exits W through p = 0, say when
~ = T. Let e 1 > 0 be sufficiently small. Then y(T + E 1 , y0 ) tt W. Let V be a
neighbourhood of y(T + e 1 , y0 ) which is disjoint from W. Then there exists a
neighbourhood U of Yo such that if y1 E U, then y(T + E 1 , y1) E V. Therefore given
0< e2 < K 3 , we can choose y(s), lJ(s) on [ r,
31
J1.
)2 + 2( yx + DX42)2e2 •
+ uX41
32
2
(3.11)
;), we also show that assuming that
~e on w- forms a closed loop around
') remains in W for all ; ~ 0, then
in W as ; - -oo and (3.1) is valid.)
~- r J to be positive and sufficiently
small so that y0 (s) E U and IN(;)I ~ E 2 , IM(;)I ~ E 2 for 0 ~; ~ T + E 1 . From the
above, y(T + E 1 , y0 (s)) E V. From Lemma 3.1, y(;, y0 (s)) could not have first
exited W through {q = 0}. Since 0 < N(;) < K 3 for 0 ~; ~ T + E 1 , y(;, y0 (s))
could not have first exited W through {N = K 3 } either. Therefore y(;, y0 (s)) must
first exit Wthrough {p
= 0} U {M = 0} U {p = 0, M = 0} if r~s ~~- r (see Fig.
1). This fact continues to hold even for the interval ~- r < s ~ ~ + r because of
Roger Lui
12
Existence and stabl
Lemmas 3.2, 3.4 and Figure 4. Using the same idea, we can conclude the
following about how y(;, y0 (s)) exits W.
(i)
S E [ T,
~+T
l
y(;, y0 (s)) exits through {p = 0} U {M = 0} U {p = 0, M = 0};
(ii)
S
E[~ + T,
3
;-
S E [
T],
3
; - T, 2Jr- T
shows that there exists -r3 E ( n + -r
3
{q=O} if -r3<s< ; --r. We defi
-q-, z- wh"1ch.1s a solution to the
p- 1
At > A2 > 0 and the vector x1 which
(3.12)
y(;, y0 (s)) exits through {M=O} U {q=O} U {M=O, q=O};
(iii)
for sin this interval, y(;, Yo(s)) m\1
l
We now define ~ and show tha1
the unit sphere in ~J\~1 4 and let
cp(x, y, z, w) = (~
y(;, y0 (s)) exits through {q=O} U {N= K 3} U {q=O, N= K3};
(iv) s E [2n- -r, -r],
y(;, y0 (s)) exits through {N = K 3} U {p = 0} U {N = K3, p = 0}.
It is best to refer to Figure 1 when studying (3.12). Note that if a(s) = 0,
fJ(s) = 0, then p(;) = 1 and q(;) = 0 for all ;.
3
3
On the intervals [ -r,
-r
+ -r, :Jr - -r
n + -r, ; - -r and [ ; +
~- J, [~
J, [
J
Then cp: Y' ~ S 3 is a homeomorph
octant of ~J\~1 4 • A point (x', y', z', w
(x', y', z') inside the unit ball in IR
this map be 1/J and let a' = 1/J o cp1
loop inside P. P is defined to be tht
~J\~13.
Consider the following open com
-r, 2n - -r], one of the four functions a, {3, y and 6 is a constant, one is zero
and the other two are sufficiently small, monotone and arranged to satisfy (3.11)
(decreasing e > 0 if necessary). On the rest of [0, 2n], two of the above functions
are zero, the other two are arranged to be monotone and satisfy (3.11). We now
3
show that there exists -r 1 E ( -r,
-r ), -r2 E
+ -r, :Jr- -r ), -r3 E ( :Jr + -r, ; - -r)
and
1"4
c;
E
~-
+
T,
2Jr-
T)
(~
such that y(;, Yo(s)) first exits W through {p = 0} if
-r < s < -rv first exits W through {M = 0}
if~+ -r < s < -r2 , first exits
3
{q = 0} if -r3 <s < ; - -r and first exits W through {N = K 3} if
T4
u = {(i, y, i) E p I~
where w = v'1 - x 2 - y2 - z 2 • u is
(i, j, i) E U, let w be defined as a
smre
W through
<s < 2n- -r.
These facts plus (3.12) and Figure 1 imply that the first exit points of y(;, y0 (s))
on w- form a closed loop around B as s increases from 0 to 2n.
M
To define -r1 , let z = N. Then z'- ()z + z 2 = -g(p, N) > -g(O, 0) as long as
2
p > 0 and 0 < N < K 3. Let z satisfy the equation z' = ()z - z - g(O, 0). It is easy
to see that if z(O) > 114 , then z( ;) > 0 and lim z( ;) = 11 3. The constants
1;-+oo
2
113 > 11 4 > 0 are defined in the paragraph after (3.3) and satisfy 811- 11 g(O, 0) = 0. Let u = z - i. Then u satisfies the inequality u' - Ou + (z + i)u > 0
so that u(;) > 0 for;> 0 if u(O) > 0. If we choose z(O) > z(O) > /1 4 , then z(;) > 0
(hence M(;) > 0) as long as p(;) > 0 and 0 < N(;) < K 3. From (3.5), z(O) =
Mo X42
•
X42
•
-~- as y~O. From Ftgure 3, - > 11 4 • Therefore from Ftgure 4, there
No X41 (
)
M.
X41
exists -r 1 E -r, ~- -r such that N: > 11 4 if -r ~ s < -r 1 • Putting all the facts together,
[i']. rn [!,]. [l:
[ ] in the fonn of the rigbt-ha1
inequalities satisfied by i, y, z, win
6 in (3.5) are positive.
Let ~, be a compact, simply com
Let~= (1/J o cp)- 1 (~'). Then~ is co
as its boundary. Condition (a) is th<
l:, then O<x < 1, y <0, z >0, w >
interior point of ~ in the form of (3
last property of ~ will be useful late1
To prove condition (b), let (p, q
()2
d(;) =
2
2
()2
(p -1)2 + 2q2 +2 N2+ 2~
q[0 + 4f(p, N)p] + MN[0 2 - 4g(p,.
1, 2 and ;1 < ; 2. Then from (3.10
Existence and stability of travelling wave solutions
me idea, we can conclude the
13
for sin this interval, y(s, y0 (s)) must exit W through {p = 0}. A similar argument
3
shows that there exists T3 e
+ T, ; - T) such that y(s, y0 (s)) exits W through
3
{q = 0} if r 3 < s < ; - T. We define T2 and T4 similarly using the functions z =
(Jt
J} U {p = 0, M = 0};
(3.12)
U{M=O,q=O};
_q_' i which is a solution to the equation i' = 8i- i 2 + /(1, 0), the constants
p-1
A. 1> A.2 > 0 and the vector x1 which appears in Figure 2.
We now define l:: and show that (3.2) and conditions (a)-(d) hold. Let S3 be
the unit sphere in IR 4 and let
cp(x,y,z,w)= (
I} u {q=O, N= K3};
= 0} U {N = K 3 , p
= 0}.
1g (3.12). Note that if a(s) = 0,
l[
T
1
3
1r + T, ; - T
3
and [ ; +
8(1-x) -Viy 8z Viw)
Vie ,-e-'Vie'_e_ ·
Then cp: Y~ S 3 is a homeomorphism; cp( a) lies in S 3 intersecting the positive
octant of IR 4. A point (x', y', z', w') e S 3, w' ~ 0, may be identified with a point
(x', y', z') inside the unit ball in IR 3, with distant v'1- (w'f from the origin. Let
this map be 1J1 and let a' = 1J1 o cp( a). Then a' is a closed non-self-intersecting
loop inside P. P is defined to be the unit ball intersecting the positive quadrant in
IR3.
Consider the following open connected set
and {) is a constant, one is zero
U=
{<x, y, i) e PI x22< 8~ <x12, X32< 8~ <X42},
2i
:one and arranged to satisfy (3.11)
:o, 21r], two of the above functions
notone and satisfy (3.11). We now
3
+ T, 1r- T), T3 E ( 1r + T, ; - T)
where w= v'1 - i - y - i • U is non-empty because of (3.3) and (3.4). Let
(i,y,i)eU, let w be defined as above and let (x,y,z, w)=(1J!ocp)- 1(i,y,i).
1) first exits W through {p = 0} if
Sinre [
+ T < s < r 2, first exits W through
[~]
~
rough {N=K3} if r4<s<2Jr-T.
at the first exit points of y(s, Yo(s))
!ases from 0 to 21r.
2 = -g(p, N) > -g(O, 0) as long as
ion i' = 8i - i 2 - g(O, 0). It is easy
md lim i(s) = ,_, 3. The constants
b"'
2
after (3.3) and satisfy o,_, - ,_, tie inequality u'- 8u + (z + i)u > 0
loose z(O) > i(O) > ,_, 4, then z(s) > 0
O<N(s)<K3. From (3.5), z(O)=
4. Therefore from Figure 4, there
s < r 1. Putting all the facts together,
x 21
2
2
x 11 x 31
2z
x 41
2
n[n[lJ [n
Me linearly independent, we can write
in the form of the right-hand side of (3.5). Fwm (3.3), (3.4) and the
inequalities satisfied by i, y, i, win the definition of U, we find that a, {3, y and
{J in (3.5) are positive.
Let l::' be a compact, simply connected surface in U with a' as its boundary.
Let~= ( 1J1 o cp )- 1(l::'). Then l:: is compact, simply connected, lies in Y and has a
as its boundary. Condition (a) is therefore satisfied. In addition, if (x, y, z, w) e
~. then 0 <x < 1, y < 0, z > 0, w > 0 (assuming that Vie< 8). If we express an
interior point of l:: in the form of (3.5), then a, {3, y and 6 are all positive. This
last property of l:: will be useful later.
To prove condition (b), let (p, q, N, M) be a trajectory of (2.9) and define
82
82
d(s) =--z (p -1) 2 + 2q 2 +
N 2 +2M 2.
Then d'(s) = 48q 2 +48M 2 - (1- p)
2
q[8 2 +4f(p, N)p] +MN[8 2 -4g(p, N)]. Suppose (p, q, N, M)(s;) el:: for i=
1, 2 and s 1 < s 2. Then from (3.10) and the remarks after it, d(s;) = e 2 and
Roger Lui
14
Existence and sta
2
d'(s;) > 0. Therefore there exists St < s* < Sz such that d(s*) = E and d'(s*) ~0.
d(s*) = e 2 implies that the terms inside the square brackets of d'(s*) are
positive. Therefore, either (1- p(s*))q(s*) > 0 or M(s*)N(s*) < 0. This forces
(p, q, N, M)(s*) to be outside Wand proves condition (b).
To prove condition (c), suppose y(s, y0 ) ft aw. Then there exists a neighbourhood containing y( y0 ) which is disjoint from W* c oW. On the other hand, the
discussion following the proof of Theorem 3.1 implies that if y0 E ~ and
y(s, Yo) E aw, then y(s, Yo) E W*. Therefore condition (c) is satisfied.
We now prove (3.2) by showing that a closed loop around B = (0, 0, K 3 , 0) in
w- (like the a we have just described) cannot be continuously deformed to a
point while remaining in w-. Assume the contrary and suppose z(s, t): [0, 1] x
[0, 1]~ w- is a continuous deformation such that z(s, 0) is the closed loop
around Bin w- and z(s, 1) is a point which lies in, say, {p = 1}. Then for some
t, the loop z(s, t) has to leave {q = 0} (see Fig. 1) and this forces the set
{M = 0} n {N = K 3 } to be non-empty, a contradiction. We leave the technical
details to the reader.
Finally we have to prove condition (d). We already know that if Yo E ~ c Y,
then y(s, y0)~A as;~
Thus we only need to show that y(s, y0 ) e W for all
s ~ 0. From the second and last equation of (1.10), we see that if q( s*) = 0 or
M(s*) = 0, then since y(s*, Yo) is near A= (1, 0, o, 0), q'(s*) > o and M'(s*) <
0. Therefore y(s, y0 ) leaves W for s < 0 only through {p = 1} or {N = 0}.
To show that y( y0 ) cannot reach {p = 1}, recall that At > A2 > 0 are the
4. Stabi
We first state two results whicl
THEOREM
4.1. Let u,v be boun
s,
-oo.
s,
eigenvalues of the matrix
[f( 1~ O) ~].
trajectory ([J, q) to the system fi'
proaches (1, 0) as
It is known that there is exactly one
= q, q' = (Jq- f([J,
;~ -oo tangent to the eigenvector
O)fi(1- p) which ap-
[;J
{
where f is a decreasing function OJ
a, fJ, Y and Dare bounded and s,
(a(x, 0), fJ(x, 0)) ~ (u(x, 0), v(x, j
(y(x, t), D(x, t)). Suppose also thG
y,- dJ]
(),- dz(
{3,- dzp
on 1R
X
[0, T). Then (a, {3) ~ (u, v
Proof See the proof in [9].
LEMMA 4.2. Let w(x, t) satisjj
Conditions w(x, 0) = Ct if X> 0 WI
are positive constants. Let o'< 6
o[exp (A- :z)tJas t~ oo uniform,
corresponding to the
Proof We have
eigenvalue At· We may assume that ([J, q) lies in the sector between -xt and
-[;J
shown in Figure 2. Recall that
~
intersect
implies that f(p, N) ~f(p, 0), a contradiction. Thus y(s, y0 ) never reaches
{p = 1} if N > 0 for ~ 0. A similar argument may be used to show that y( ;, y0 )
never reaches {N = 0} if p < 1 and ~ 0. Let (IV, M) be the solution toN'= M,
M' = (JM- g(1, N)N, approaching (0, 0) as ;~
tangent to the eigenvector
s
[;J
s
[-g(1,0 0) 1]() . Then (N,- M)- prevents y(s, y
c
_t_
.~
v2j{
from reaching {N
3). The proof of condition (d) is complete and so is Theorem 3.1.
= 0} (see Fig.
D
l"'
e-z2/2dz-
-xlvZt
(- means ratio goes to one). Simi!
Czebx l-xtv2i
'~
V 2j{
X
If
+2bt
Vii
Write X =
0)
v4.7rt
-oo
-oo
corresponding to the largest positive eigenvalue A3 of the matrix
10 ,r:.-=e-<x
1
(x- y)
Let z - - Vii ; then the second
the same figure, in order for y(s, y0 ) to reach {p = 1}, (p, q) of (2.9) must
1]
d
d- [At and hence ([J, q). At the point of intersection, d; ~ d; which
=
e -At w ( x, t )
was chosen so that if Yo E ~. then
y0 = (p 0 , q 0 , N 0 , M0 ) is of the form (3.5) with a, {3, y and () nonnegative. This
implies that (p 0 , q 0 ) lies in the sector between -xt and -x2 in Figure 2. From
u, = dtuxx +
v, = dzVxx +
2
e-z 12+v2ibz
-oo
l
~ 00 , then the above in
Xo-
(Jt, Where 0 <f)< 2b,
Existence and stability of travelling wave solutions
;h that d(s*) = e and d'(s*) ~0.
square brackets of d'(s*) are
or M(s*)N(s*) < 0. This forces
ndition (b).
7• Then there exists a neighbourV* caw. On the other hand, the
3.1 implies that if Yo E ~ and
tdition (c) is satisfied .
. loop around B = (0, 0, K 3 , 0) in
t be continuously deformed to a
rary and suppose z(s, t): [0, 1] x
that z(s, 0) is the closed loop
>in, say, {p = 1}. Then for some
Fig. 1) and this forces the set
adiction. We leave the technical
already know that if Yo E ~ c Y,
E W for all
l.10), we see that if q(s*) = 0 or
), 0, 0), q'(s*) > 0 and M'(s*) <
rough {p = 1} or {N = 0}.
f, recall that A1 > A2 > 0 are the
d to show that y(s, y0)
known that there is exactly one
= 8ij - f(p, O)p(1 - p) which ap-
1vector [ ;
]
We first state two results which are needed to prove our stability theorem:
THEOREM
) from reaching {N = 0} (see Fig.
so is Theorem 3.1.
0
X
[O, T),
D1- d2Dxx- g(x, t, y, D)~ 0,
(4.1)
(4.2)
C¥1- dta'xx -!(x, t, C¥, o) ~ 0,
fJ1- dzf3xx - g(x, t, C¥, {3) ~ 0,
on IR x [0, T). Then (C¥, fJ) ~ (u, v) ~ (y, D) on~ x [0, T).
Proof See the proof in [9].
LEMMA 4.2. Let w(x, t) satisfy the equation W 1= wxx + Aw and the initial
conditions w(x, 0) = C 1 if x > 0, w(x, 0) = C 2ebx if x ~ 0, where A is real and b, C;
are positive constants. Let 0 < 8 < 2b. Then for each L > 0, w(x 0 - 8t, t) =
2
o[ exp (A- ~ )t] as t-oo uniformly for x
0 E [ -L
Vi, L].
Proof We have
=
e-Atw(x, t)
1S chosen so that if y0 E ~. then
eigenvalue A3 of the matrix
~
in
Y1- d1 Yxx -J(x, t, y, {3) ~ 0,
s in the sector between -x 1 and
~
U1: d1Uxx + ~(x, t, u, v)
VI- d2Vxx + g(x, t, u, v)
where f is a decreasing function of v and g is an increasing function of u. Suppose
£¥, {3, y and {) are bounded and smooth functions defined on ~ X [0, T) such that
(C¥(X, 0), {J(x, 0)) ~ (u(x, 0), v(x, 0)) ~ (y(x, 0), D(x, 0)) and (C¥(X, t), {J(x, t)) ~
(y(x, t), o(x, t)). Suppose also that£¥, {3, y and o satisfy the inequalities:
corresponding to the
ln. Thus y(s, y0 ) never reaches
may be used to show that y( ;, y0 )
:R, M) be the solution to N' = M,
- -co tangent to the eigenvector
4.1. Let u, v be bounded solutions of the reaction-diffusion system
{
1
C¥, {3, y and {) nonnegative. This
t -x1 and -x2 in Figure 2. From
:h {p = 1}, (p, q) of (2.9) must
.
f .
.
dq < dij h"
mt o mtersectton, dp = dp w 1ch
15
4. Stability of travelling waves
2
f v'4Jti
o --e-<x-y)'I41Czeby
1
dy
+
Let z =
(x- y)
I"' _l_e-<x-y)2t41
C
dy.
0
-00
v'4Jti
.
- V2t ; then the second mtegral equals
1
-C-
Vfir
f"'
Y2i 2;41 as---oo.
X
e-z 2;2 dz~-C 1 -e-x
-x/Vzt
X
Wt
( ~ means ratio goes to one). Similarly, the first integral equals
c ebx f-xtVzi e-z2f2+Vztbz dz = c2 ebx+b 21f-(x+2b1)1Yzt e-v 2/2 dv.
_2_
v'2ir
v'2ir
-oo
-oo
+ 2bt
.
.
.
CzWt
2
, M. then the above mtegral IS asymptotic to ---e-x 141• If we
v2t
x + 2bt
write x = x 0 - 8t, where 0 < 8 < 2b, then as long as , x~ is bounded
v2t
e-Atw(x, ~ [~- c 1]Y2te-x 2141 as
X+ 2bt X
If
x
oo,
t)
t-oo.
Roger Lui
16
The coefficient in front of e-x
2
[ exp (A- : )t
complete.
J as t~ oo if x
0
Existence and sta1
goes to zero as t~ oo. Therefore w(x 0 - (Jt, t) =0
2141
is bounded above. The proof of Lemma 4.2 is
0
We now turn to the proof of stability of travelling waves for the case (2.5).
From condition (2.10), we have
~a
oN
[f(p, N)p(1- p )]
~ 0 and~
[g(p, N)N] ;a 0
op
for 0 ;a p ;a 1, N ~ 0. It is well known that comparison theorems are harder to
apply in this case than in the case when both partial derivatives are nonnegative.
The travelling waves p(;), IV(;) with speed (} constructed in Section 3 are
facing in opposite directions. It is more convenient to let r = 1 - p so that (1.2)
becomes
r1 = rxx + F(r, N),
N 1 = Nxx + G(r, N),
(4.3)
where F(r, N) = - f(1- r, N)r(1- r) and G(r, N) = g(1- r, N)N. For each
(} > (}*, let r(;) == 1- p(;) and fii(;) denote a travelling wave solution of (4.3)
that connects (0, 0) to (1, K 3 ). We do not require for N to be monotone.
THEOREM 4.3. Let r, N be solutions to (4.3) with initial data satisfying
0 ;a r(x, 0) ;a 1, 0 ;a N(x, 0) ;aM for all x where M > K 3 . Suppose that
lr(x, 0)- f(x)l ;a Cebx and IN(x, 0)- N(x)l ;a Cebx for x sufficiently small.
Let k~ =max l~(p, N)l, ki =max IFN(p, N)l, h~ =max IGP(p, N)l and hi=
max IGN(p, N)l, where the maximum is taken over o;;ap ;a 1 and o;;aN;;aM. Let
Abe the largest positive eigenvalue of the matrix A= [k~
k!J.
hi hz
1
Suppose (}, b
satisfy the inequalities 2VJ:;. < (} < 2b. Then there exists a positive constant C* such
that
lr(xo- (Jt, t)- f(x 0 )1 ;a C*e<'r 0214 l 1,
IN(x 0 - (Jt, t)- N(x 0 )1 ;a C*e(J.,-
0214 1
)
uniformly for x 0 E [ -L'[i, L] and L > 0, as t~ oo.
Proof. Let r1 (x, t) = r(x, t)- f(x
r1 , N 1 satisfy the equations
r!t
{ Nlt
+ (Jt) and N1(x, t) = N(x,
= r!xx + k1(x,
t)r1 + k 2 (x, t)N1,
= N!xx + h1(x, t)r1 + hz(x, t)N1,
t)- N(x
+ (Jt). Then
(4.4)
where k 1(x, t) = [F(r(x, t), N(x, t))- F(f(x + (Jt), N(x, t))]/r1 (x, t), k 2 (x, t) =
[F(f(x + (Jt), N(x, t))- F(f(x + (Jt), N(x + (Jt))]/N1(x, t), and h 1 , h 2 are similarly
defined with F replaced by G. Since g(p, M) is negative, the theory of invariant
rectangles for parabolic systems implies that 0 ;a r(x, t) ;a 1, 0 ;a N(x, t) ;aM for all
x and t > 0. Therefore lk;(x, t)l ;a kJ, lh;(x, t)l ;a hJ for i = 1, 2. Note that k 2 ;a 0
and h 1 ~ 0 because of conditions (2.1) and (2.8).
Let y, o satisfy the equations
= Yxx + k1(x, t)y- k 2 (x, t)o,
1 = Oxx + h 1(x, t)y + h 2 (x, t)o,
Yt
{0
with initial data
(4.5)
y(x, 0)
= {~ebx if.
if.
Since - kz ~ 0 and h, ~ 0, the '
o(x, t) ~ 0 for all t > 0. Define a
assume _that ( a(x, 0), f3(x, 0)) ~ (l
Let J(x, t, u, v) = k 1(x, t)u + k
Th~n f IS decreasing in v and g is
satisfy the inequalities listed in 1
implies that (a, {3) ;a (rt, Nt) ~ ( y,
and o.
Let y, b be solutions to th
(Y(x, 0), J(x_, 0)) = (y(x, 0), o(~,
( y, 0 ) ;a ( y, o) for all x and t > 0. '
defined as in Theorem 4.3. B·
diagonalisable so that there e~
A= diag (At, Az) and A1 > A2 are th
we have A1 > 0.
J
Let [; =
r[ ~]. Then ,u, v satis
.u(x, O) =
{C1Czebx
if.
if
From Lemma 4.2, there exist posi
2
Cs exp [ (At -
~ )tJ and
v(x 0 -
(}
[-LVI, L] ast~oo. Theorem4.3tl
Remark 4.4. Since all the travc
1~1 ~ ±oo, one cannot expect stal
Simply says that the wave with spe
that decay faster than ebx as x ~ _,
Remark 4.5. With some effort c
gap, (} E ( (}*' 2~], where stabil
established. Note that we have use<
to uncouple y and <5 in the proof 01
Remark 4. 6. The inequalities sat1
that r(x, 0) 1 d N(x, 0)
-( ) ~ an - --~ 1 as
r x
N(x)
0 E [
0,
~]
is the smallest of the
linearising (2.9) at (1, o, o, 0). These
are not enough to imply the results
17
Existence and stability of travelling wave solutions
~ oo. Therefore w(x 0 - Ot, t) =0
with initial data
elling waves for the case (2.5).
a
- p)] ~0 and ap [g(p, N)N] ~0
parison theorems are harder to
tial derivatives are nonnegative.
0 constructed in Section 3 are
ent to let r = 1- p so that (1.2)
(4.3)
),
r, N) = g(1 - r, N)N. For each
travelling wave solution of (4.3)
e i' or N to be monotone.
f.3) with initial data satisfying
where M > K 3 • Suppose that
Cebx for x sufficiently small.
h; =max IGp(p, N)l and h~ =
ver O~p ~ 1 and O~N~M. Let
k* k*]
h~ . Suppose 0, b
'rix A= [ h:
!
exists a positive constant C* such
(x, t) = N(x, t)- N(x + Ot). Then
(4.4)
· Ot), N(x, t))]/r1(x, t), k 2 (x, t) =
)]/N1(x, t), and h 1 , h 2 are similarly
[s negative, the theory of invariant
~ r(x, t) ~ 1, 0 ~ N(x, t) ~ M for all
~h7 fori= 1, 2. Note that k 2 ~0
M
D(x, 0)
= { Cebx
if X> 0,
if x ~ 0.
Since - k 2 ~ 0 and ht ~ 0, the maximum principle implies that y(x, t) ~ 0 and
6(x, t) ~ 0 for all t > 0. Define a= - y, f3 = -D. From our hypotheses, we may
assume that (a(x, 0), f3(x, 0)) ~ (rt(x, 0), Nt(x, 0)) ~ (y(x, 0), f3(x, 0)) for all x.
Let /(x, t, u, v) = kt(x, t)u + kz(x, t)v and g(x, t, u, v) = ht(x, t)u + hz(x, t)v.
Then/is decreasing in v and g is increasing in u. Also, from (4.5), a, {3, y and b
satisfy the inequalities listed in (4.2) with dt = d 2 = 1. Therefore Theorem 4.1
implies that (a, {3) ~ (rt, Nt) ~ ( y, D) for all x and t > 0. We now try to estimate y
and 6.
Let y, b be solutions to the y,=f'xx+krr+k;<5, b,=bxx+hfy+h;<5,
(Y(x, 0), b(x, 0)) = (y(x, 0), D($, 0)). Then the maximum principle implies that
(y, 6) ~ (y, b) for all x and t > 0. To solve the above system for y and <5, let A be
defined as in Theorem 4.3. By increasing A, we may assume that A is
diagonalisable so that there exists a matrix r such that Ar = r A, where
A= diag (At, A2 ) and At > A2 are the eigenvalues of A. Since A is a positive matrix,
we have At > 0.
Let[~]= r[~]. Then /l,
Ct
!l(x, 0)
v satisfy !lt = !lxx
= { C e bx
2
ifx>O,
"f <o·,
1 x =
+ At/l,
v(x, 0)
v, = Vxx
{C
+ AzV, and
3
= C e bx
4
ifx>O,
"f <o .
1 x =
From Lemma 4.2, there exist positive constants C 5 , C 6 such that !l(x 0
~ )t]
2
Cs exp [ (At -
and v(xo- Ot,
t) ~
~ )t]
-
Ot, t) ~
2
c6
exp [ ( Az-
uniformly for Xo
multiplying[~] by r.
E
0
lsi~
±oo, one cannot expect stability in the supremum norm. Theorem 4.3
simply says that the wave with speed 0 is stable under perturbation by functions
that decay faster than ebx as x - -oo if 2v'f; < 0 < 2Vb.
Remark 4.5. With some effort, one can show that 2v'f; > 0* so that there is a
gap, 0 E ( 0*, 2v'f;], where stability of waves with speed 0 has not been
established. Note that we have used the fact that p and N diffuse at the same rate
to uncouple y and bin the proof of Theorem 4.3.
Remark 4.6. The inequalities satisfied by the initial data in Theorem 4.3 imply
r(x, 0)
N(x, 0)
that i'(x) -1 and N(x) -1 as x--oo, since i'(x), N(x)=O(e=) near -oo;
~).
· k 2(x, t)b,
- h2(x, t)b,
= { Cebx
Remark 4.4. Since all the travelling wave solutions have the same limit as
oo.
lc2(x, t)Nt,
h2(x, t)N1 ,
y(x, O)
[-Lv't, L] as t-HXJ. Theorem 4.3 then follows by
2
- Ot, t)- N(xo)l ~ C*ep.,-11 i4}t
1
if X> 0
if X~ 0'
1
~. The proof of Lemma 4.2 is
a E [ 0,
(4.5)
~]
is the smallest of the four eigenvalues of the matrix obtained by
linearising (2.9) at (1, 0, 0, 0). These two conditions on the initial data apparently
are not enough to imply the results of Theorem 4.3. This is because Theorem 4.1
Roger Lui
18
requires four sets of inequalities involving both the upper and lower comparison
functions to hold simultaneously.
Acknowledgment
p roceed'mgs of the Royal Society of Edinburgh, 115A,
On blow-up and
heat e1
This research was partially supported by NSF grant DMS-8601585 and NSF
grant DMS-8801968.
v
Keldysh Institute of Applied M
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(Issued 28 June 1990)
(MS
Th)e 2asymptotic behaviour of the solut
u In (1+u) fort>O, xe[-.n- .n-J u (t
blows. up a t a fi mte
· time
·
. ' mvestigat
. ' x •
To, IS
functions Uo the behaviour of u(t, x) near
~a(t, x) = exp2{(To- t)-'. cos2 (x/2)} -1
t~on v, = (vx) /(1 + v) + (1 + v) ln2(1 + ;)
tipme fo) r other semilinear or quasilinear p.
( >! 'u,=Au+ufl(/3>1), u =Au+ I
are dtscussed.
'
e
1. Introd1
I~ this paper, the initial-boundary
With source
u, = Uxx + (1 + U) ln 2 ~
ux(t, -.n) =
u(O, x) = u
u0 E C
1
([
-.n, .n]),
1
is considered. Local existence and
are proved in [3]. The source (';;
condition
f+oo
I
:md then a nontrivial nonnegative
mterval and blows up: u(t, x) exists
lim
s
t->Ti) xe[·
Here ToE (0, +oo) is a finite blow-1
global in time follows from the
E(t) = (2~)-1 f':'" u(t, x) dx, which
u(t, x)) In (1 + u(t, x)) dx fort> 0.