Page 32 - Applied Probability

1. Basic Principles of Population Genetics
(a) Verify the differential equations

r (t)= −r(t)+ u(t)+ v(t)
2
1
s (t)= −s(t)+ v(t)+ w(t)

2 1 15
1

u (t)= −u(t)+ r(t) u(t)+ v(t)
2
1 1

v (t)= −v(t)+ r(t) v(t)+ w(t) + s(t) u(t)+ v(t)

2 2
1

w (t)= −w(t)+ s(t) v(t)+ w(t) .
2
2
1
(b) Show that the frequency r(t) + [u(t)+ v(t)] of the A 1 allele is
3 3 2
constant.
(c) Let p 0 be the frequency of the A 1 allele. Demonstrate that
3
[r(t) − p 0 ] = − [r(t) − p 0 ],
2
and hence
3
− t
r(t) − p 0 =[r(0) − p 0 ]e 2 .
(d) Use parts (a) and (c) to establish
1

lim u(t)+ v(t) = p 0 .
t→∞ 2
(e) Show that
2 t
[(u(t) − p )e ]
0
2 t
t
t
= u (t)e + u(t)e − p e

0
1 t 2 t

= r(t) u(t)+ v(t) e − p e
0
2
3
= p 0 +[r(0) − p 0 ]e − t
2
1 1 − t 3 t 2 t

3
× p 0 − p 0 − [r(0) − p 0 ]e 2 e − p e
0
3 3 2
3
= p 0 +[r(0) − p 0 ]e − t
2
1 − t t 2 t

3
× p 0 − [r(0) − p 0 ]e 2 e − p e
0
2
1
p 0 t 2 −2t
= [r(0) − p 0 ]e − 2 − [r(0) − p 0 ] e .
2 2
Thus,
3
2
u(t) − p 2 0 =[u(0) − p ]e −t + p 0 [r(0) − p 0 ](e −t − e − t
2 )
0
1
2
− [r(0) − p 0 ] [e −t − e −3t ].
4