Math 4600, Homework 11
1. A weird form of the logistic equation is the tent-map: xn+1 = f (xn ), where
µx
f or 0 ≤ x ≤ 0.5
f (x) =
µ(1 − x) f or 0.5 ≤ x ≤ 1
a. Sketch 2 graphs of f : one with 0 < µ < 1, and the other with 1 < µ. Add in the 1-1 line, and mark
any equilibria. On each graph, use cobwebbing to show the stability of each equilibrium (do this by
hand).
b. Find the steady states by hand, and analyze their stability analytically.
c. Sketch a bifurcation diagram with the steady states on the y-axis, and the parameter µ on the x-axis.
2. Computing: The goal of this problem is to graph cobwebbing diagrams of the 3 phenotype model
derived in class. Let Pn be the frequency of the healthy hemoglobin allele (1 − Pn is the frequency of the
sickle cell allele). Plot cobwebbing diagrams in the case where malaria is not present in the population
(R = 0.9, S = 0.3, T = 0.1) and in the case where malaria is present (R = 0.4, S = 0.8, T = 0.1).
a. Explain why this choice of parameters fits the word description of each society
b. What will happen to the disease in each population in the long run (include cobwebbing diagrams).
3. Computing: Let R be the fitness of genotypes HH, Hh, and S denote the fitness of genotype hh, with
R > S. Let Pn denote the fraction of the H allele in the population on generation n. Suppose that
immediately before reproduction takes place, there is a mutation phase where a fraction of H’s turn into
h’s.
a. Write down the difference equation describing this situation. This can be done by replacing the Pn on
the right hand side of our 2 phenotype model with (1 − )Pn (the proportion of H’s after mutation).
b. Pick R = 0.8, S = 0.5, and = .01. Make a plot of Pn with respect to generation, and make
a cobwebbing diagram. How does this result differ qualitatively from the 2 phenotype model we
covered in class? Explain what the comparison of these models tells us about the role of mutation in
maintaining genetic diversity.
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