1. Slow selection in population genetics
Consider a population with two alleles, a and b, of a particular gene. Every individual
has two copies of the gene, the options being either aa, ab or bb. As derived in class,
the equations for A, 2H, and B, the proportions of the population having each of the
genotypes aa, ab, and bbrespectively are
where b is the birth rate (assumed to be the same for all three genotypes), di is the
death rate for each genotype, and
F(A;H;B) = b d1A 2d2H d3B: (1)
(a) Assume that the death rates take the form di = d + i where << 1. The quan-
tities = A+H and = B +H are the gene frequencies for a and b respectively
(the fraction of all genes in the population that are a and b respectively). Show
that these quantities change slowly compared to A;H and B and that on the
corresponding slow time scale, A;H and B achieve quasi-steady states A = 2
,H = , and B = 2
(these are known as the Hardy-Weinberg Equilibrium).
(b) Assume that 1 = 3 = d and 2 = 0. Write a sentence or two describing what
this assumption means in terms of the relative tness of each genotype. Derive
the equation 0
= d(1 ) for the slow change of . From the phase line of this
ODE for , determine what happens if there are a few b alleles in a population
with predominantly a alleles. Relate this to the values.
2. Download the three .ode les posted on the Canvas code page for this assignment. For
each of the models/.ode les, generate a bifurcation diagram showing steady states and
periodic solutions as a function of the primary parameters (Iapp or depending on the
model). The les set almost everything up for you. You will just have to modify the
Axes limits in AUTO as described in the rst few lines of each ode le. Submit a
sketch, screenshot or export of the diagram.
For each model, describe in words how the branch of periodic solutions is \born” and
how it \dies”. For some, you can simply name the bifurcation (with as much detail as
possible, e.g. supercritical Hopf). For mechanochemForA6.ode this is optional.
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