AM107 AM107  Homework 5
Assigned: Thursday, October 5, 2000
Due:  Thursday, October 12, 2000

Problem 1: Recall the Population Genetics problem:

There are two alleles for a genetic trait A and a.
Individuals carry a pair, one allele from each parent: AA, Aa or aa.
We call the frequencies of these three types of individuals: u (freq. of AA), v (Aa) and w (aa).
And we call the frequency of each allele in the population p = u + 0.5v (freq. of A), q = 0.5v + w (freq. of B).
In this problem we will consider what happens to p and q under the hypotheses of random mating, but now with unequal fitness.
a.  Let a(alpha), b (beta) and g (gamma) be the fitness coefficients of the types  AA, Aa and aa, respectively.  Derive the equations:
pn+1 = (apn2 + bpnqn)/(apn2 + 2bpnqn + gqn2)

qn+1 = (bpnqn + gqn2)/(apn2 + 2bpnqn + gqn2).

b.  Recalling that p + q = 1,  reduce these equations to a single equation for pn+1.

c.  Without loss of generality, assume that a = 1 and that g > 1.  Find the fixed points of this equation and evaluate the stability of each fixed point by any means you please. (There should be three cases depending upon the size of b.)

d.  By changing the time scale of the equations in part a, derive the differential equation:

p' = pq [ j(1-2p) + kq ].

e.  Find the fixed points of p and describe the evolution of p.(A graph may help).

f.  Discuss the biological implications of your findings.  You may also wish to comment on any differences or similarities you see in the results of the discrete and continuous models.

Problem 2: Edelstein-Keshet p.155, #17