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

E+S
k
®

¬
k
C       C
k
®

 
 E+P .
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
s+2lns = s0+2lns0-e0t .
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

k
= Ñ2u º 1
r2
 d
dr
 æ
ç
è 		r2 du
dr
 ö
÷
ø 		      in   0 < r < a
0
= Ñ2u Î r > a .
Find u(r) and deduce that the equilibrium radius is given by
a2 = 3
k
 æ
è 		u¥-u1 ö
ø 		 .
Show that the assumptions of the model require that
2u0+u¥ £ 3u1 .

5.       The spread of rabies through the European fox population is modelled by the equations

St
= -r SI
It
= rSI-aI+dIxx ,
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

S¢
= r
c
 SI
I¢
= 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

c ³ 2[d(r S0-a)]1/2 .

File translated from TEX by TTH, version 2.22.
On 9 Jun 1999, 16:25.