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
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) |
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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 |
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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, q, h) it occupies q = a,
0 < r < L, 0 < h < 2p. In these coordinates the Helmholtz equation for f(r, h) is
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1 r2sin2a
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¶2f ¶h2
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+ |
1 r2
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¶ ¶r
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æ ç
è
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r2 |
¶f ¶r
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ö ÷
ø
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+k2f = 0 with |
¶f ¶r
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(L,h) = 0 . |
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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 . |
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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.)
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On 3 Nov 1999, 13:31.