1. 1.       A stretch of coastline is inhabited by an animal with (positive) population density N(X, T). Its evolution is modelled by the equation
NT = r N- P N
N+N0
 +D NXX ,
where r, P, N0 and D are positive constants. Explain the biological processes included in this model.

Defining suitable new variables, obtain the non-dimensional equation

ut = u- p u
1+u
 +uxx .
Discuss the large-time behaviour of spatially uniform solutions of this equation.

When p > 1, investigate travelling wave solutions u = f(x-ct) in the (f, f¢) plane. For what values of c does there exist a solution obeying the boundary conditions

u®0      as   x®-¥,       u®(p-1)      as   x®¥ ?

2.       Two pigments, with non-dimensional concentrations u(x, y, t) and v(x, y, t), occupy a rectangular area W. The chemicals react and diffuse according to the equations

ut
= 1+3u-u2-3v+Ñ2u
vt
= 6+4u-9v-v2+20Ñ2v
 ü
ý
þ 		      in    ì
í
î
0
< x < 5p
0
< y < 4p
 ,
together with the no-flux boundary conditions on W
æ
ç
è
u
v
 ö
÷
ø

x 
 = 0      on   x = 0, 5p   and    æ
ç
è
u
v
 ö
÷
ø

y 
 = 0      on   y = 0, 4p .
Investigate the stability of the uniform state u = v = 1, by writing
æ
ç
è
u
v
 ö
÷
ø 		= æ
ç
è
1
1
 ö
÷
ø 		+e æ
ç
è
u0
v0
 ö
÷
ø 		eltf(x,y) ,
where 0 < e << 1 while u0, v0 and the growth rate l are unknown constants. Formulate and solve the problem for the eigenfunctions f(x,y) and associated eigenvalues k2.


Find an equation for l in terms of k2, and deduce that the uniform state is unstable only if 0.2 < k2 < 0.25.

Hence show that only one mode f has a strictly positive growth rate.


Sketch a plausible pattern to emerge from this instability, stating clearly what criterion you are using.

3.       The functions u(t) and v(t) obey the equations

u¢ = e(v-u)      and       v¢ = f(v)-u ,
where the positive constant e and the function f are known. State the conditions for (u, v) = (u0, v0) to be a stable, steady state.


In the FitzHugh-Nagumo model of neurone behaviour,

f(v) = I-v(v-v1)(v-v2)      where   v2 > v1 > 0 .
v(t) is the electric potential, while I, v1 and v2 are known constants.

(a)    When I = 0, determine the condition for there to exist two stable, steady states.


(b)    Assume that only one steady state exists for I = 0 and e << 1. Sketch the two trajectories in the (v, u) plane corresponding to u(0) = 0 and v(0) = v1±d, for some small d. You should include the curve u = f(v) on your figure.

Plot the potential v(t) for each trajectory, and explain the biological significance of the difference in behaviour.


(c)       An experiment is performed with I > 0. Show graphically that it is possible for the system to have a unique equilibrium which is unstable.

Sketch a plausible trajectory in this case.

4.       Blood contains dissolved oxygen, O2, haemoglobin, Hb, and oxygenated haemoglobin, HbO2, with concentrations u, v and h respectively. The total haemoglobin remains constant, v+h = v0, during the reversible reaction

O2+Hb
k+
®

¬
k_
HbO2 .
Find the equilibrium concentration of h as a function of u and sketch it roughly.


Blood flows into the alveoli of the lungs with constant velocity V where, for x > 0, it comes into contact with a high O2 concentration. The oxygen diffuses with coefficient D, much faster than do the larger haemoglobin molecules. The oxygen concentration u(x, y) therefore satisfies

V[u+h(u)]x = Duyy Î x > 0, y > 0 ,      with   
,
u(x,0) = u0,       u(0, y) = u¥ u® u¥      as   y®¥ .
Approximating h¢(u) @ M, a constant, seek a solution of the form
u(x, y) = f(h)      where   h = y
2Ö
nx
 
      and   n = D
(1+M)V
  .
Deduce an ODE with boundary conditions for f(h) and solve it.

Evaluate the oxygen absorption rate, -Duy(x,0).

5.       Some fish are almost spherical when small, but become long and thin as they grow bigger. Suggest a plausible reason for this.

At t = 0, a fish is at rest at X = 0. It swims for a time T, exerting a unit force F and then glides to rest. Its velocity V(t) is modelled by the equation

dV
dt
 = F(t)- 2V
R
 -V2      where   F(t) = ì
í
î
1
      for   0 < t < T
0
      for   T < t .
What might the (constant) parameter R represent?

Writing k = 1/R, u(t) = V(t)+k and c = Ö[(1+k2)], show that

u = ì
í
î
ctanh[ct+A]
      for   0 < t < T
kcoth[kt+B]
      for   T < t ,
where sinhA = k. Give an equation determining B.

Defining X = ò0t V dt, show that

X = ln é
ê
ë 		1
c
 æ
è 		cosh[ct+A] ö
ø 		ù
ú
û 		-kt      for   0 < t < T ,

and that       X® L = ln é
ê
ë 		cosh[cT+A]
2csinh[kT+B]
 ù
ú
û 		+B      as t®¥.
What is the total work done by the fish in travelling this distance?

File translated from TEX by TTH, version 2.22.
On 4 Oct 1999, 15:24.