4. Additional Examples of Reaction Diffusion Problems
(from Mathematical Biology, by Murray) A generalization of Fisher’s equation, is the
equation
/ t uÝx, tÞ = / x Ýu m / x uÝx, tÞÞ + u p Ý1 ? uÞ q for m, p, q = positive parameters
More general still is the equation
/ t uÝx, tÞ = / x ÝDÝuÞ / x uÝx, tÞÞ + fÝuÞ
but without specific information about the functions DÝuÞ, fÝuÞ analytic solutions for this
equation are not possible. Moreover, by clever choices of the parameters, we can
qualitatively mimic more general D and f. Therefore we will consider the less general
equation and we will begin with the case where m = 1 = p = q.
Density Dependent Diffusion
Consider the equation
/ t uÝx, tÞ = / x Ýu / x uÝx, tÞÞ + u Ý1 ? uÞ.
Ý4.1Þ
The density dependent diffusivity DÝuÞ = u implies that the rate proportionality of flux to
gradient increases as u increases. This means that the flow from crowded regions to less
crowded regions proceeds at a higher rate at the crowding increases.
If we suppose uÝx, tÞ = UÝx ? ctÞ = UÝzÞ, then
L
ÝUÝzÞ U L ÝzÞÞ + c U L ÝzÞ + UÝ1 ? UÞ = 0.
The associated dynamical system is
UvÝzÞ = VÝzÞ,
UÝzÞ V L ÝzÞ = ?cVÝzÞ ? VÝzÞ 2 ? UÝ1 ? UÞ
Ý4.2Þ
where UÝzÞ ¸ 1 as z ¸ ?K, UÝzÞ ¸ 0 as z ¸ K, and Ýwe hopeÞ U L ÝzÞ < 0 all z. We note that
the second equation in the system Ý4.2Þ is singular at U = 0. This singularity can be
removed by defining a new variable, s, given by
d =U d
ds
dz
i.e., z L ÝsÞ = UÝzÞ.
Then (4.2) becomes
U L ÝsÞ = U V
VvÝsÞ = ?cV ? V 2 ? UÝ1 ? UÞ,
Ý4.3Þ
1
with critical points at ÝU, VÞ = Ý0, 0Þ, Ý1, 0Þ, and Ý0, ?cÞ. Then
JÝU, VÞ =
V
U
2U ? 1 ?c ? 2V
and it follows that:
Ý0, 0Þ = a stable node, and Ý1, 0Þ, Ý0, ?cÞ are both saddles.
For c small, say c = .3 we have
we see that there is no orbit joining Ý1, 0Þ to either of the other critical points. For a slightly
larger value of c, say c = .707, we have
i.e., in this case there is an orbit joining the saddle at Ý1, 0Þ to the saddle at Ý0, ?cÞ. Note that
this trajectory tends to U = 0, V = U L = ?c as s ¸ K. This means that the graph of
2
U versus z terminates at a point z 0 on the z-axis and UÝzÞ = 0 for z ³ z 0 . The derivative is
discontinuous at this point as is jumps from VÝz 0 ?Þ = ?c, to VÝz 0 +Þ = 0. The trajectory in
this case is a straight line joining Ý1, 0Þ to Ý0, ?cÞ, i.e., V = ?c , Ý1 ? UÞ where c = c , denotes
the critical value of c where this orbit occurs. It follows from the system (4.3) that on this
orbit
2
dV = ?cV ? V ? UÝ1 ? UÞ
UV
dU
hence if V = ?c , Ý1 ? UÞ then we can solve for c , to obtain
U L ÝzÞ = ? 1 Ý1 ? UÝzÞÞ and
2
c, =
1 . Then
2
UÝ?KÞ = 1, UÝz 0 Þ = 0
implies
UÝzÞ = 1 ? exp
z ? z0
2
=0
if z < z 0
if z > z 0
For c > c , we have an orbit joining Ý1, 0Þ to Ý0, 0Þ and this results in the usual sort of
travelling wave.
Other exercises we could try would include examining the effect of increasing m, p and q
above the value 1, testing the stability of such solutions and trying to obtain perturbation
approximations for these solutions.
Having considered some examples of TW solutions to scalar reaction diffusion equations,
we will consider now the existence of TW solutions to systems of such equations.
Rabies Epidemic in Foxes
Consider the following simple model for the spread of rabies in a population of foxes. Let
SÝx, tÞ = the number of susceptible foxes at location x at time t
IÝx, tÞ = the number of infected foxes at location x at time t.
Then the following equations
/ t SÝx, tÞ = ?r SÝx, tÞ IÝx, tÞ
/ t IÝx, tÞ = r SÝx, tÞ IÝx, tÞ ? a IÝx, tÞ + D / xx IÝx, tÞ,
assert that the number of uninfected foxes decreases at a rate that is jointly proportional to
the numbers of infected and uninfected foxes. The proportionality factor is denoted by
r > 0. The second equation asserts that the number of infected foxes increases in the
corresponding way but there is also a loss due to deaths of infected foxes. The death rate is
3
denoted by a > 0. Finally, the number of infected foxes is also affected by movement of
these foxes which is reflected in the diffusive term in the second equation.
If SÝx, 0Þ = S 0 > 0, then we can reduce these equations to dimensionless terms by letting
t=
b ,
rS 0
x=
aÝz, bÞ =
SÝx, tÞ
IÝx, tÞ
, RÝz, bÞ =
.
S0
S0
D ,
rS 0
b=
a ,
rS 0
and
Then
/ > aÝz, bÞ = ?a R
/ > RÝz, bÞ = a R ? b R + / zz R.
We are looking for solutions aÝz, bÞ = fÝz ? cbÞ, RÝz, bÞ = gÝz ? cbÞ such that
a ¸ 1 and R ¸ 0 as z ¸ K
a L ¸ 0 and R ¸ 0 as z ¸ ?K
no infections ahead of the wave
no infections behind the wave and a = constant .
These conditions correspond to an pulse of infection propagating into an initially uninfected
population and leaving behind a (possibly) positive number of survivors. It is assumed that
all infected foxes eventually die.
These assumptions lead to
? c f L ÝzÞ = ?fÝzÞ gÝzÞ
? c g vÝzÞ = fÝzÞ gÝzÞ ? b gÝzÞ + g”ÝzÞ.
Then
gÝzÞ = c
f L ÝzÞ
fÝzÞ
f L ÝzÞ
.
fÝzÞ
and
?c g vÝzÞ ? g”ÝzÞ = c f L ÝzÞ ? bc
Then
?c gÝzÞ ? g L ÝzÞ = c f ÝzÞ ? bc log fÝzÞ + C 0 ,
and the condition as . z ¸ K implies c + C 0 = 0. Then
?c gÝzÞ ? g L ÝzÞ = c f ÝzÞ ? bc log fÝzÞ ? c,
4
and
f L ÝzÞ = 1c fÝzÞ gÝzÞ,
g L ÝzÞ = ? c f ÝzÞ + bc log fÝzÞ + c ? cgÝzÞ.
Note that f L = 0 on the axes, f = 0, g = 0, while g L = 0 on g = b log f ? f + 1. These are
the so called ”null clines” where the derivatives vanish. The critical points of the dynamical
system are located at the intersections of these null clines; i.e., at f = 1, g = 0 and at
f = f 0 , g = 0, for some f 0 , 0 < f 0 < 1. The direction field near the null clines is shown on
the following sketch.
Now the Jacobian of the system is
JÝf, gÞ =
hence
g
f
c
c
bc ? c ?c
f
at Ý1, 0Þ we have V =
at Ýf 0 , 0Þ we have V =
1
2
1
2
?c ± c 2 ? 4Ý1 ? bÞ
?c ± c 2 + 4Ýb ? f 0 Þ
making Ýf 0 , 0Þ a saddle point for all values of c, while Ý1, 0Þ is a stable focus when
c 2 < 4Ý1 ? bÞ and a stable node when c 2 > 4Ý1 ? bÞ. The stable focus leads to a heteroclinic
orbit with both negative and positive values for g, which is not admissible when g denotes a
population value. Then the physically reasonable solutions are associated with
c 2 > 4Ý1 ? bÞ, which defines a minimal speed for travelling wave solutions. The phase plane
portrait in this case is as shown below
5
The heteroclinic orbit joining the saddle at Ýf 0 , 0Þ to the stable node at Ý1, 0Þ is the travelling
wave
Note that f 0 , as the solution of b log f = f ? 1, is an increasing function of the parameter
b= a ,
rS 0
6
f-1, b log(f) versus f
Then as the death rate of infected foxes, a, increases or as the infection rate for the
uninfected foxes, r, goes down the survival level f 0 goes up. This is exactly as you would
suppose. On the other hand, f 0 , goes down as S 0 , the initial number of uninfected foxes
goes up. This is less to be expected.
Transport of a Chemical Solute
Consider a tube of uniform cross section A containing a porous medium (soil) through
which a fluid containing chemical solute is flowing. As the solute moves through the porous
medium, some of the solute comes out of solution and sorbs to the soil. Conversely,
chemical which is sorbed to the soil can dissolve in the solute and move away from the
sorption site. We will derive a model for this process and examine it for existence of
travelling wave solutions.
Let d denote the porosity, the fraction of total volume that can be occupied by fluid. Then
the state variables for describing the transport of chemical solute through the soil are as
follows
3
qÝx.tÞ = flux of solute at Ýx, tÞ
CÝx, tÞ = solute concentration at Ýx, tÞ
NÝx, tÞ = mass concentration of solute sorbed to the soil
7
Let Ýa, bÞ and Ýt 1 , t 2 Þ denote an arbitrary interval of the tube and an arbitrary interval of time
during the process, respectively. Then the conservation of solute statement for this process
is,
: a d CÝx, t 2 Þ A dx = : a d CÝx, t 1 Þ A dx + : t 2 qÝa, tÞ A dt ? : t 2 qÝb, tÞ A dt ? : t 2 : a / t NÝx, tÞ dx dt.
b
b
t
t
1
b
t
1
1
In the usual way, this is becomes
: t 2 : a Ýd / t C + / x q + / t NÞ dt dx = 0
b
t
1
and, since Ýa, bÞ and Ýt 1 , t 2 Þ are arbitrary,
d / t CÝx, tÞ + / x qÝx, tÞ + / t NÝx, tÞ = 0 at each Ýx, tÞ.
We suppose that solute flux is composed of two parts,
qÝx, tÞ = ?D / x CÝx, tÞ +
diffusion
1
2
d B CÝx, tÞ 2
bulk transport
where the first term is the usual diffusion mechanism while the second part reflects
transport by movement of the chemically contaminated fluid through the soil.
Finally, there are various ways in which, NÝx, tÞ, the mass concentration of chemical that
has been sorbed to the soil can be related to, CÝx, tÞ , the concentration of of chemical in
solution. In order of increasing complexity, three possible choices are
iÞ
NÝx, tÞ = J CÝx, tÞ
iiÞ / t NÝx, tÞ = KÝN 0 ? NÝx, tÞÞ
iiiÞ / t NÝx, tÞ = RÝN, CÞ = K
N 0 C 2 ? NÝx, tÞ
C 20 + C 2
The first choice is mathematically simple but physically not very realistic. The second
choice is more realistic in that it recognizes that the sorption rate must depend on the
current value of N and it assumes there is some maximum value for the amount of solute
that can be sorbed to the soil. Finally, the last choice treats the sorption rate as depending
on the current value of both C and N. The following figure shows the curve where R = 0. To
the left of this curve we have / t NÝx, tÞ < 0 and to the right, / t NÝx, tÞ > 0.
8
Then we have the following model for this process
d / t CÝx, tÞ ? D / xx CÝx, tÞ + d B CÝx, tÞ / x CÝx, tÞ + / t NÝx, tÞ = 0
N 0 C 2 ? NÝx, tÞ
C 20 + C 2
/ t NÝx, tÞ ? K
= 0.
This is equivalent to
/t
N
N0
N ? ÝC/C 0 Þ 2
N0
1 + ÝC/C 0 Þ 2
+K
= 0,
1 / ÝC/C 0 Þ ? D / xx ÝC/C 0 Þ + ÝC/C 0 Þ / x ÝC/C 0 Þ + N 0 / t ÝN/N 0 Þ = 0
dBC 0
BC 0 t
dBC 20
Reducing this to dimensionless form by letting
a= C ,
C0
R= N ,
N0
z=x
dBC 0
,
D
b=
dB 2 C 20
t,
D
we obtain
/ > a + a / z a + K / > R = / zz a
/>R + k R ?
a2
1 + a2
K = N0 ,
C0d
k=
= 0,
where
Now we suppose
where
KD .
dBC 20
aÝz, bÞ = fÝz ? vbÞ, and RÝz, bÞ = gÝz ? vbÞ
f, g ¸ 0 as z ¸ K
f L , g L ¸ 0 as z ¸ ?K
9
This amounts to assuming the concentrations of solute and sorbed chemical ahead of the
wave are zero and are equal to unknown constants (to be determined) behind the wave.
Then our system of equations reduces to
?v f L + f f L ? K v g L = f”
?v g L + k g ?
f2
1 + f2
= 0.
It follows from integrating the first equation that
f L ÝYÞ =
1 2
f
2 ÝYÞ
? v fÝYÞ ? K v gÝYÞ
2
g vÝYÞ = kv g ? f 2
1+f
.
Evidently this system has the following null isoclines
fL = 0
on
g=
fÝf ? 2vÞ
2Kv
gv = 0
on
g=
f2
1 + f2
a parabola
ÝR = 0Þ
The null isoclines are shown in the figure:
10
The isoclines meet at the origin, which is one of the critical points for the system. This
critical point can be shown to be a saddle point. The other critical point is located where the
null clines meet in the first quadrant; i.e.,
2Kv f = Ýf ? 2vÞÝ1 + f 2 Þ
or
f 3 ? 2v f 2 + Ý1 ? 2KvÞ f ? 2v = 0.
For each v > 0, this cubic has a positive real root f 0 and then
Ýf 0 , g 0 Þ with g 0 =
f 20
1 + f 20
is a critical point for the system. .
It would appear that there is a heteroclinic orbit joining Ýf 0 , g 0 Þ to Ý0, 0Þ and such an orbit
would be associated with a TW joining the state Ýf 0 , g 0 Þ at Y = ?K to Ý0, 0Þ at Y = K. Since
we cannot solve for Ýf 0 , g 0 Þ without explicit values for K, v we cannot determine the type of
critical point Ýf 0 , g 0 Þ is. (For the values used to generate the figure, it is an unstable node).
However, it is evident from the figure that the direction field of flow points out of the region
W that is bounded on the sides by the two null clines and on the bottom by g = 0. This
means that if we proceed backward in time along the heteroclinic orbit that arrives at
Ý0, 0Þ at Y = +K, the direction of flow is into the region W and that the heteroclinic orbit can
therefore not leave W through any of the three boundaries. In fact, since the side
boundaries are null isoclines, the orbit cannot cross either of these without violating
uniqueness. Then the orbit must tend to the critical point Ýf 0 , g 0 Þ as Y ¸ ?K, and the
corresponding travelling waves then appear as follows
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