Math 5110/6830
Homework 5.2
Consider the Moth Wing model:
Assume that the moth’s wing color is determined by only one gene that has
two alleles W and w. Let’s say that W codes for light-colored wings and w codes
for black wings, and W is dominant (moths are diploid).
The wing color is important for moths, because if it does not match the
coloring of the environment, it does not provide good camouflage and the moths
will be eaten by birds. Let us say that α is the fraction of light moths that
survive to reproduce in a given environment, and γ is the fraction of surviving
black moths. Also assume pn is the fraction of W alleles in the population. The
model is:
αpn
pn+1 =
,
2
(γ − α)pn − 2(γ − α)pn + γ
but make sure to explain how you arrive at it.
a) Show that the fixed points are p∗1 = 0 and p∗2 = 1.
b) Perform stability analysis of the fixed points. You will need to consider
cases when either of the phenotypes has selective advantage (α > γ and α < γ).
c) With γ = 0.8 draw a bifurcation diagram with α as the parameter
d) Sketch a solution pn vs. n and describe in word what happens to the moth
population in the your model during the Industrial evolution in England. To
do this start with an environment where light-colored moths have an advantage
because they are effectively camouflaged against the light-coloured trees and
lichens which they rest upon (what does it imply about the relative sizes of
parameters? what does your solution do?). At some point in time the industrial
revolution happens, the lichens die out and the trees become blackened by soot.
How is this represented in your model? What will happen to the solution?
What does it predict about the moth population?
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