M M[3 || 4]A21 Mathematical Biology:    Example sheet 2 A.J. Mestel


Chemical reaction kinetics and Turing pattern formation.


1.    A chemical reaction involves the four molecular types A, B, X and Y. The concentrations of A and B are maintained at constant levels A0 and B0 by fast external processes, while X and Y are produced by the three reactions

A
k1
®

¬
k_
X ,      B
k2
®

 
Y ,      2X+Y
k3
®

 
3X .
Obtain evolution equations for the concentrations x(t) and y(t) of X and Y.

Show that non-dimensionalisation of the form t = k_ t, u(t) = wx and v(t) = wy, where w2 = k3/k_ , produces the system

u¢ = a-u+u2v ,       v¢ = b-u2v ,     (1)
and state the values of a and b (which are of course positive.)



2.    Demonstrate that (1) has a unique stationary point, and find the condition for it to be stable. What is the largest possible value of a for which this equilibrium can be unstable? If it is unstable, what sort of behaviour in the (u,v)-plane do you expect?



3.    Discuss whether or not the system

ut = (a-u+u2v)+Ñ2u       vt = (b-u2v)+dÑ2v
can exhibit Turing-like spatial instabilities, identifying any critical relationships.



4.    The surface of a zebra's leg is modelled as a cone of angle a, so that in terms of spherical polar coordinates (r, qh) it occupies q = a, 0 < r < L, 0 < h < 2p. In these coordinates the Helmholtz equation for f(r, h) is

1
r2sin2a
  2f
h2
+ 1
r2

r
æ
ç
è
r2 f
r
ö
÷
ø
+k2f = 0      with    f
r
(L,h) = 0 .
Find the h-independent eigenfunctions and show that the eigenvalues satisfy tankL = kL. (Hint: write f = y(r)/r.) Roughly how large must k2 be for (a) striped and (b) non-striped leg patterns to occur?


5.    In lectures we considered the following model for testosterone production in males:

R¢ = f(T)-b1R ,       L¢ = R-b2L ,       T¢ = L-b3T      with   bi > 0 .
Suppose the negative feedback function f(T) = 1/(1+Tm) for some m > 0. Show that there is a unique equilibrium which is stable if m £ 8. Hints:
(i) The roots of x3+ax2+bx+c = 0 have negative real parts iff a > 0, c > 0 and ab > c.
(ii) For any three positive numbers 1/3(p+q+r) ³ (pqr)1/3.
[It follows that for m £ 8 no oscillations in hormone levels are possible. If m > 8, however, limit cycles can occur.)


File translated from TEX by TTH, version 2.22.
On 3 Nov 1999, 13:31.