1.
1. An organism has a population density U(X, T),
which is assumed to obey the equation
UT = rU(1- |
U N
|
)+DUXX Î 0 < T, 0 < X < L . |
|
Explain the biological significance of the constants r, N and D.
Rescale the variables to derive the non-dimensional equation for u(x, t)
ut = u(1-u)+uxx Î 0 < t, 0 < x < l , |
|
where l is a constant to be determined.
At time t = 0, the population is at a very low level, u << 1. Discuss what
will happen as t®¥ when the boundary conditions are
(a) ux = 0 at x = 0 and x = l,
and
(b) u = 0 at x = 0 and ux = 0 at x = l,
stating any critical relations between the parameters.
2. An enzyme E reacts with a substrate S to form an intermediate
complex C, which then decomposes to form a product, P and the enzyme.
At t = 0, the concentrations of E, S, C and P are, respectively,
e0, s0, 0 and 0. The reactions are written
where the reaction rates are all equal and scaled so that k = 1.
Obtain the equations for the concentrations of substrate, s(t), and
complex, c(t) º e0v(t) in the form
,
|
ds dt
|
= e0[v-s(1-v)] |
dv dt
|
= s(1-v)-2v . |
|
Explain carefully the Michaelis-Menten steady state hypothesis, and show that it
implies that s(t) is given implicitly by
Deduce that for large times s @ s0exp[1/2(s0-e0t)].
Describe what happens in the very early stages of the reaction.
3. The FitzHugh-Nagumo model of neurone behaviour
can be written in the form
u¢ = e(v-u) and v¢ = f(v)-u , |
|
where
f(v) = -v(v-v1)(v-v2) with v2 > v1 > 0 |
|
and e, v1 and v2 are positive constants.
Determine the condition for these equations to have a unique steady state
and discuss its stability.
Assuming that there is only one steady state and that e << 1,
sketch in the (v, u) plane the two trajectories
starting at u(0) = 0 and v(0) = v1±d,
for some small d. You should include the curve u = f(v) on your figure
and give some indication of the time taken to traverse various portions of
the trajectories.
Sketch also the potential v(t) for each trajectory, and explain the
biological significance of the difference in behaviour.
4.
Cells in a tumour consume a nutrient of concentration u at a constant
rate k. The cells die if the local concentration u < u0 or u > u1. If
at the surface of the tumour u < u1 then the cells divide and the tumour
grows in size. The background concentration is u = u¥, where
u¥ > u1 > u0 > 0.
Assuming the tumour grows to an equilibrium size which is a sphere of radius a
all parts of which are alive, the governing equations are
|
|
= Ñ2u º |
1 r2
|
|
d dr
|
|
æ ç
è
|
r2 |
du dr
|
ö ÷
ø
|
in 0 < r < a |
| |
|
| |
|
Find u(r) and deduce that the equilibrium radius is given by
Show that the assumptions of the model require that
5.
The spread of rabies through the European fox population is modelled by the equations
where S(x, t) and I(x, t) are respectively the susceptible and infected
populations, while r, a and d are given constants.
Discuss, briefly, the assumptions of the model.
Seek travelling wave solutions S = S(z), I = I(z), where z = x-ct for
some positive wave speed c. Show that
|
|
| |
= |
c d
|
|
é ê
ë
|
a r
|
lnS-S-I+A |
ù ú
û
|
, |
|
| |
|
where A is a constant. Show graphically or otherwise that this system can
have two equilibrium points with S = S0 or S = S1, where S0 > S1.
Discuss the stability of these points, and explain why the speed of a
travelling wave such that S® S1 as x®-¥ and S® S0 as
x®¥ must satisfy
6. Explain qualitatively how a shear flow combined with weak
cross-flow diffusion can lead effectively to strong downstream diffusion.
Flow of air or blood in a cleft 0 < y < 1, can be modelled by a
steady Poiseuille velocity in the x-direction,
V(y) = by(1-y) for some constant b. |
|
Defining y-averages over the cleft cross-section for some quantity w by
|
w
|
(x, t) = |
ó õ
|
1
0
|
w(x, y, t) dy , |
|
express b in terms of [`V].
The concentration u(x,y,t) of a chemical satisfies the equation
ut+V(y)ux = Duxx+Duyy with uy = 0 on y = 0,1 . |
|
Writing u = [`u](x, t)+u¢(x, y, t), and justifying carefully each stage of the
argument, show that, approximately,
Deduce that
u¢ = |
D
|
|
æ ç
è
|
y3- |
1 2
|
y4- |
1 2
|
y2+ |
1 60
|
ö ÷
ø
|
|
|
and hence that
|
u
|
t
|
+ |
V
|
|
u
|
x
|
= Deff |
u
|
xx
|
where Deff = D+ |
210D
|
. |
|
Interpret this equation and discuss its biological significance.
File translated from TEX by TTH, version 2.22.
On 4 Oct 1999, 15:26.