3. Reaction Diffusion Equations
Consider the following ODE model for population growth
u v ÝtÞ = aÝuÝtÞÞ uÝtÞ,
uÝ0Þ = u 0
where uÝtÞ denotes the population size at time t, and aÝuÞ plays the role of the population
dependent growth rate. The model asserts that the population changes at a rate that is
proportional to the present population size. Moreover, if we suppose
uÝtÞ
aÝuÞ = J 1 ? u K
for constant positive parameters, J, u K
then the population grows when the population is smaller than the ”limiting value” u K and
the population size decreases if u exceeds this value. Thus the model predicts behavior that
is qualitatively consistent with the way we observe populations to behave. This autonomous
differential equation has critical points at u = 0, and u = u K and it is easy to see that the
critical point at zero is unstable while the other is a stable critical point. Then the population
uÝtÞ will tend to the limiting value u K as t tends to infinity, regardless of the initial population
size. This equation is a special case of the more general autonomous equation,
u v ÝtÞ = FÝuÝtÞÞ.
Now the partial differential equation
/ t uÝx, tÞ ? 4 2 uÝx, tÞ = FÝuÝx, tÞÞ
x 5 R n , t > 0,
Ý3.1Þ
can be viewed as an attempt to incorporate the mechanism of diffusion into the population
model. We are going to study equations of this form in the case n = 1 where the equation
can be written more generally as
/ t uÝx, tÞ ? / xx bÝuÝx, tÞÞ = FÝuÝx, tÞÞ,
b v ÝuÞ > 0.
Equations of this form arise in a variety of biological applications and in modelling certain
chemical reactions and are referred to as reaction diffusion equations.
Example 3.1 Fisher’s equation
The reaction diffusion equation with positive constant parameters, D, J, u K
/ t uÝx, tÞ ? D / xx uÝx, tÞ = J uÝx, tÞ 1 ?
uÝx, tÞ
,
uK
Ý3.2Þ
is called Fisher’s equation and it is usually viewed as a population growth model. The
various parameters in the equation have the following dimensions
D = diffusivity ÝL 2 T ?1 Þ
J = growth rate ÝT ?1 Þ
u K = carrying capacity numbers of individuals
1
Reduction to Dimensionless Form
It is often useful to rewrite the partial differential equation in terms of dimensionless
variables. We define
b = Jt
z=x J
D
v = uÝx, tÞ/u K
time scaled to the growth rate
distance scaled to diffusion length
population scaled to carrying capacity
and then the equation becomes
/ b vÝz, bÞ ? / zz vÝz, bÞ = vÝ1 ? vÞ.
Ý3.3Þ
Existence of a Travelling Wave
Now, in order to investigate the existence of travelling wave solutions, we suppose
vÝz, bÞ = VÝz ? cbÞ with VÝsÞ tending to constant values as s tends to plus or minus infinity.
Then
?c V v ÝsÞ ? V”ÝsÞ = VÝsÞÝ1 ? VÝsÞÞ,
? K < s = z ? cb < K.
Ý3.4Þ
This second order equation reduces to the following autonomous dynamical system
V v ÝsÞ = WÝsÞ
W v ÝsÞ = ?cWÝsÞ ? VÝsÞÝ1 ? VÝsÞÞ.
Ý3.5Þ
This system has critical points at Ý0, 0Þ and Ý1, 0Þ. Since
JÝV, WÞ =
0
1
2V ? 1 ?c
we can classify the critical points according to the eigenvalues of this matrix.
at Ý0, 0Þ
V ± = 12 ?c ± c 2 ? 4
a stable node if c > 2 and a stable focus if 0 < c < 2
at Ý1, 0Þ
V± =
1
2
?c ± c 2 + 4
a saddle for all values of c.
In the case 0 < c < 2, the origin is a stable focus and the orbits of the system are curves in
the ÝV, WÞ ? plane
2
V = VÝsÞ,
VÝ?KÞ = V 0
W = WÝsÞ,
WÝ?KÞ = W 0
?K < s < K
with
ÝVÝsÞ, WÝsÞÞ ¸ Ý0, 0Þ as s ¸ K.
Since the origin is a focus, the orbits are such that V and W assume both positive and
negative values as the curve spirals toward the origin. Negative values for V are not
physically meaningful in the population interpretation of V = u. Therefore, we conclude that
there are no relevant travelling wave solutions for wave speeds between zero and 2.
Spiral orbits near the focus
V has both pos and neg values if 0<c<2
In the case c > 2 the origin is a stable node and the orbits in the fourth quadrant that are
attracted to the origin approach the node with V constantly positive and W constantly
negative. Then these are physically relevant orbits. If there exists an orbit with
V 0 = 1, W 0 = 0 that is attracted to the origin, then this orbit, which is in fact a heteroclinic
3
orbit joining the two critical points, corresponds to a travelling wave solution to the Fisher’s
equation. The component V = VÝsÞ of the heteroclinic orbit is a smooth function such that
V v ÝsÞ = WÝsÞ < 0 for all s. In addition, VÝsÞ tends to 1 as s tends to minus infinity and VÝsÞ
tends to 0 as s tends to plus infinity so that 0 and 1 are the state values ahead of and
behind the wave, respectively. The monotone decreasing function V is the wave form that is
propagated at speed c > 2.
V pos and W neg near stable node
V pos and monotone decreasing if c>2
Proof of Existence of the TW solution
To verify that this heteroclinic orbit actually exists, consider the following picture in the
ÝV, WÞ ? plane.
4
Consider the triangular region, OAB, where the stable node is located at point O, the saddle
point is at A, and OB is the line W + bV = 0 for some b > 0. The parabola W = 1c VÝ1 ? VÞ
is the curve along which W v = 0 and V v = W < 0.
In addition,
W v < 0 and V v = 0 along the line OA,
W v > 0 and V v < 0 along the line AB.
Then trajectories that are inside OAB do not leave this region (in the forward direction)
through OA or AB. Now consider the region in the neighborhood of point A.
If, by letting s tend to ?K, we trace backward along an orbit which cuts the arc PQ near the
point P, we see that such an arc must pass out of OAB above the point A. On the other
hand, tracing backward along an orbit which cuts the arc PQ near the point Q, we see that
such an arc must pass out of OAB below the point A. It follows that there exists an orbit
which passes through the point A as s tends to ?K.
Now consider the line OB, whose equation is W + bV = 0 for some positive b. If a trajectory
crosses OB from right to left at s = s 0 then the expression WÝsÞ + bVÝsÞ decreases from a
positive value when s < s 0 on the right of OB to a negative value when s > s 0 on the left
side of OB. Then in order to have a trajectory exit the triangle OAB through OB at
5
ÝVÝs 0 Þ, WÝs 0 ÞÞ, we must have
W v Ýs 0 Þ + bV v Ýs 0 Þ < 0
and
WÝs 0 Þ + bVÝs 0 Þ = 0.
On the other hand, the equations of the dynamical system Ý3.5Þ imply,
W v ÝsÞ + bV v ÝsÞ = ?cWÝsÞ ? VÝsÞÝ1 ? VÝsÞÞ + b WÝsÞ
= Ýb ? cÞWÝsÞ ? VÝsÞÝ1 ? VÝsÞÞ.
In particular, along OB
W v Ýs 0 Þ + bV v Ýs 0 Þ = Ýb ? cÞÝ ? bVÝs 0 ÞÞ ? VÝs 0 ÞÝ1 ? VÝs 0 ÞÞ
and for b = c/2 > 0,
= VÝs 0 Þ ß?bÝb ? cÞ ? 1 + VÝs 0 Þà,
2
W v Ýs 0 Þ + bV v Ýs 0 Þ = VÝs 0 Þß c ? 1 + VÝs 0 Þà > 0
4
if c 2 > 4.
Then for c 2 > 4, no trajectory can exit OAB through OB. Then forward orbits do not exit
through any of the sides of the triangle and we say the triangle OAB is a trapping region
for the dynamical system. It follows that the trajectory that tends to A as s tends to minus
infinity must also tend to O as s tends to plus infinity. This is a consequence of the fact that
any triangle OA’B’ that is similar to OAB is also a trapping region (i.e., it does not permit
orbits to exit through any of the sides) and as A’,B’ tend toward O, the orbits in the triangle
are forced toward O. This proves the existence of the orbit joining the saddle at A to the
stable node at O. Then the component VÝsÞ satisfies (3.4) and the conditions
VÝsÞ ¸ 1 as s ¸ ?K, VÝsÞ ¸ 0 as s ¸ K. and vÝx, tÞ = VÝx ? ctÞ is a TWS for the equation
(3.3) as long as c 2 > 4. i.e., there is a minimum speed but no maximum.
Estimating the Wave Speed
The existence of arbitrarily high wave speeds is not physically reasonable. In order to
resolve this apparent anomaly, consider again the dynamical system (3.5). Note that
WÝsÞ = V v ÝsÞ is negative along trajectories and therefore at a point s = s 1 where
WÝsÞ reaches a negative mininum, we have
W v Ýs 1 Þ = 0
i.e.,
or
? cWÝs 1 Þ = VÝs 1 ÞÝ1 ? VÝs 1 ÞÞ ²
1
4
,
max | V v ÝsÞ| ² | WÝs 1 Þ| ² 1 .
4c
This asserts that the wave speed, c, is inversely proportional to the maximum slope of the
wave form V = VÝsÞ. That is, a steep wave form travels more slowly than a gradually sloped
wave form. Then a bound on the slope of the wave form implies a bound on the speed of
the travelling wave. In particular, max | / x vÝx, 0Þ| = max |V v ÝsÞ| can be estimated from the
6
data and an upper bound on the speed of the travelling wave solution can be obtained.
Stability of the Travelling Wave
We have seen that the equation (3.3) has a travelling wave solution vÝz, bÞ = VÝz ? cbÞ for
c > 2. We wish to consider the stability of this solution to see if it is stable under small
perturbations. In order to do this, we transform to a moving reference frame, one that is
moving with the wave speed, c; i.e., we let wÝz, bÞ = vÝz ? cb, bÞ. Then
/ b wÝz, bÞ = / b vÝz ? cb, bÞ ? c / z vÝz ? cb, bÞ
= / b wÝz, bÞ ? c / z wÝz, bÞ
/ b wÝz, bÞ = / zz wÝz, bÞ + c / z wÝz, bÞ + wÝz, bÞÝ1 ? wÝz, bÞÞ.
We look in particular for a solution of the form
wÝz, bÞ = VÝzÞ + dÝz, bÞ
where
|dÝz, bÞ| << 1
and
dÝz, bÞ = 0 for
| z| > L
Here L denotes a large positive number. Then, in view of equation (3.4), we have
/ b dÝz, bÞ = / zz dÝz, bÞ + c / z dÝz, bÞ + dÝz, bÞÝ1 ? 2VÝzÞÞ ? dÝz, bÞ 2
Since d is assumed to be small, we may neglect the term dÝz, bÞ 2 . Then
/ b dÝz, bÞ = / zz dÝz, bÞ + c / z dÝz, bÞ + dÝz, bÞÝ1 ? 2VÝzÞÞ
and we look for solutions to this equation that have the form dÝz, bÞ = fÝzÞ e ?Vb . For a
solution of this form we must have
f ”ÝzÞ ? c f v ÝzÞ + ÝV + 1 ? 2VÝzÞÞ fÝzÞ = 0,
fÝzÞ = 0 at z = ±L.
This is an eigenvalue problem for áV n , f n ÝzÞâ where positive eigenvalues imply d
decreases with increasing b this is a stable solution and negative eigenvalues imply that
d grows with increasing b an unstable TW solution . Let fÝzÞ = SÝzÞ e a z and note that for
a = c/2, we have
2
S”ÝzÞ + V ? 2VÝzÞ + c ? 1
4
SÝzÞ = 0,
SÝzÞ = 0 at z = ±L.
Note that
2
qÝzÞ := 2VÝzÞ + c ? 1 ³ 2VÝzÞ > 0 if c > 2.
4
Then
SÝzÞ S”ÝzÞ + ßV ? qÝzÞà S 2 ÝzÞ = 0
and
L
L
L
v
2
v
2
X ?L SÝzÞ S”ÝzÞ dz = SÝzÞ S v ÝzÞ| z=L
z=?L ? X ?L S ÝzÞ dz = ? X ?L S ÝzÞ dz
implies
L
V=
X ?L ßS v ÝzÞ 2 + qÝzÞ SÝzÞ 2 àdz
L
X ?L S ÝzÞ 2 dz
> 0.
7
Since the eigenvalues are all positive, the TW solution is stable under small perturbations
that vanish outside some finite interval. This is not stability under all possible perturbations
but only stability under this special class of perturbations. Nevertheless, it shows that the
TW solution has some degree of stability.
Perturbation Approximation for VÝzÞ
The TW solution, VÝzÞ, satisfies, for c > 2,
V”ÝzÞ + c V v ÝzÞ + VÝzÞÝ1 ? VÝzÞÞ = 0,
VÝ?KÞ = 1, VÝKÞ = 0,
z 5 R,
and since V is translationally invariant we can add the condition that VÝ0Þ = 1/2 in order to
fix the wave. This equation is not exactly solvable for general c and hence we will try to
construct an approximate solution. In order to do this, we will make use of the small
parameter, P = 12 which appears in the equation and attempt a so called perturbation
c
expansion in powers of P.
Let c = 1 Ýi.e., P = 12 < .25 Þ and write
P
c
P V”ÝzÞ + V v ÝzÞ + P VÝzÞÝ1 ? VÝzÞÞ = 0,
z 5 R,
which suggests that for small values of P (i.e. large c) the dominant term in the ODE is
V v ÝzÞ; i.e., the equation is approximated by V v ÝzÞ = 0 for large c. The solution VÝzÞ =
constant is valid when z is large negative (there VÝzÞ ¸ 1) and when z is large and positive
(in that region, VÝzÞ ¸ 0) but there is a large interval ”in the middle” where VÝzÞ decreases
from 1 to zero and this approximate equation is not valid there. In order to shrink this
interval to an interval whose length is of magnitude order one, we introduce a scaled
variable
so
VÝzÞ = V
s
P
V v ÝzÞ = U v ÝsÞ s v ÝzÞ =
P U v ÝsÞ,
V”ÝzÞ = Ý P Þ U ”ÝsÞ
s = cz = P z
:= UÝsÞ
Then
2
and
0=
=
P V”ÝzÞ + V v ÝzÞ + P VÝzÞÝ1 ? VÝzÞÞ
P PU”ÝsÞ +
P U v ÝsÞ + P UÝsÞÝ1 ? UÝsÞÞ,
Then UÝsÞ solves,
P U ”ÝsÞ + U v ÝsÞ + UÝsÞÝ1 ? UÝsÞÞ = 0
UÝ?KÞ = 1, UÝ0Þ = 12 , UÝKÞ = 0.
Ý3.6Þ
Note that the parameter P appears in the highest order term of the equation, which suggests
that the equation obtained by deleting this higher order term is really the dominant part of
the problem. To see whether something along these lines is true, we suppose that the
solution U can be expanded in a power series in the small parameter P,
UÝsÞ = U 0 ÝsÞ + P U 1 ÝsÞ + P 2 U 2 ÝsÞ ...
8
Substituting this into the equation and side conditions, then equating like powers of P leads
to
U v0 ÝsÞ = ?U 0 Ý1 ? U 0 Þ,
U 0 Ý?KÞ = 1, U 0 Ý0Þ =
U v1 ÝsÞ = ?U 1 Ý1 ? 2U 0 Þ ? U 0 ”ÝsÞ,
1
2
, U 0 ÝKÞ = 0,
Ý3.7Þ
U 1 ݱKÞ = 0,
etc.
This is a series of first order ODE’s whose solutions are given by
U 0 ÝsÞ =
1
1 + es
U 1 ÝsÞ =
es
4 es
log
s 2
Ý1 + e Þ
Ý1 + e s Þ 2
etc.
Then, finally
VÝzÞ =
z/c
e
1
+ 12
log
z/c
z/c
c
1+e
Ý1 + e Þ 2
z/c
4e
z/c
Ý1 + e Þ 2
+ ...
and this approximation for the wave form VÝzÞ is most accurate for c >> 2 and is least
accurate when c is close to 2. In effect, we are saying that the solution to (3.6) consists of
the solution to (3.7) plus a series of corrections. Or, saying this another way, the basic
behavior of the solution to (3.6) is captured in the solution to (3.7) and the additional terms
can be viewed as refinements on this initial approximation. When this solution is compared
with numerically constructed solutions, taking just the first term of the expansion gives
agreement to within a few percent, even in the worst case when c = 2.
We have seen that the equation
/ t uÝx, tÞ = / xx uÝx, tÞ + uÝ1 ? uÞ,
x 5 R, t > 0,
has travelling wave solutions with a minimum wave speed equal to 2. We have not shown
that, given an arbitrary initial condition, uÝx, 0Þ, the solution will evolve into a travelling wave
solution. It has been shown by Kolmogorov that
1 if x ² x 1
uÝx, 0Þ =
u 0 ÝxÞ if x 1 ² x ² x 2
0 if x ³ x 2
where u 0 ÝxÞ is continuous, nonnegative and u 0 Ýx 1 Þ = 1, u 0 Ýx 2 Þ = 0, then the solution uÝx, tÞ
will evolve into uÝx, tÞ = UÝx ? 2tÞ, a travelling wave with the minimal wave speed. For other
conditions, the solution may depend on the behavior of uÝx, 0Þ for large |x|.
For example, near the leading edge of the wave, u u 0 so u 2 << u and the equation is
approximated by the linear equation
/ t uÝx, tÞ = / xx uÝx, tÞ + u.
If we consider this equation with uÝx, 0Þ u A e ?ax as x ¸ K, where a, A > 0, then
uÝx, tÞ = A e ?aÝx?ctÞ solves the equation if
9
ac = a 2 + 1 i.e.,
c = a + 1a ³ 2.
If 0 < a < 1, then
e ?ax > e ?x and uÝx, 0Þ u A e ?ax as x ¸ K implies c = a + 1a > 2 .
On the other hand, if a > 1, then
e ?ax < e ?x and uÝx, 0Þ u A e ?ax as x ¸ K implies uÝx, 0Þ u A e ?x as x ¸ K .
and uÝx, 0Þ u A e ?x as x ¸ K implies c = 2 .
i.e.,
uÝx, 0Þ u A e ?ax as x ¸ K implies c = a + 1a , if 0 < a < 1, c = 2 if a > 1.
In particular, if uÝx, 0Þ has compact support, then c = 2. These results have been verified
numerically.
The fact that the Fisher equation is invariant under a sign change for x implies that for
each TW solution VÝz ? cbÞ with wave speed c, there is also a TW solution VÝz + cbÞ moving
in the opposite direction. Given an initial state vÝz, 0Þ having compact support, then the
initial state will evolve into a pair of travelling wave fronts, one moving to the left and the
other to the right. If vÝz, 0Þ < 1, then the term vÝ1 ? vÞ > 0 and so acts like a source,
causing the solution to grow until v = 1; i.e., vÝz, bÞ ¸ 1 as t ¸ K for all x.
10