Math 5110/6830
Instructor: Alla Borisyuk
Homework 5.2
Due: September 29
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 peppered-colored wings and w
codes for black wings, and W is dominant (moths are diploid).
a) What are all possible genotypes and what are their corresponding wing
colors?
The wing color is important for moths, because if it does not match the
coloring of the environment, it does not provide good camouage and the moths
will be eaten by birds. Let us say that is the fraction of peppered moths that
survive to reproduce in a given environment, and is the fraction of surviving
black moths. Also assume pn is the fraction of W alleles in the population.
b) Fill in the following table, where on top there are alleles contrubted by the
mother, on the left - contributed by the father, with corresponding frequency in
parethesis. Enter the progeny genotype in each empty cell and in parenthesis
the corresponding frequency.
Mother
W (pn ) w (1 pn )
(pn )
Father W
w (1 pn )
c) Build a model by nding the expression for pn+1 . Your end result should
be
n
pn+1 = ( )p2 p
;
n 2( )pn + but make sure to explain how you arrive at it.
d) Show that the xed points are p1 = 0 and p2 = 1.
e) Perform stability analysis of the xed points. You will need to consider
cases when either of the phenotypes has selective advantage ( > and < ).
f) With gamma = 0:8 draw a bifurcation diagram with as the parameter
g) 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 peppered-colored moths have an
advantage because they are eectively camouaged 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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