MATH3070: Natural Resource Mathematics
ASSIGNMENT 4 – SEMESTER 2, 2021
Attempt all questions.

This assignment is due on Wednesday, October 6,
10:00 am. Make sure that you show clearly the reasoning you use to solve the
problems. It is also advisable to keep a photocopy of the assignment you hand
in. You are free to use any reference you wish as long as you cite your source.
However, you must define and explain all notation or concepts used that were not
covered in the lecture or a prerequisite course. Each question is worth 25 points.
Q1. Consider the Wright-Fisher diploid model,
p(t + 1) = wAAp(t)2 + wAap(t)[1 −p(t)]
wAAp(t)2 + 2wAap(t)[1 −p(t)] + waa[1 −p(t)]2 ,
where p(t) is the frequency of allele A at the generation t and wij is fitness of
genotype ij (i,j = A,a).

(a) Reduce the number of parameters.
(b) Derive an allele frequency change in one generation, ∆p = p(t + 1) −p(t).
(c) What are equilibria of p?
(d) Show the parameter condition in which an internal equilibrium (i.e., 0 < ̄p <
1) exists and is stable.
(e) Describe in words how selection can maintain genetic variation i.e., how the
internal equilibrium can exist and be stable.