20 Fri Nov 6: Evolutionary ecology

DUE DATE: Fri Nov 13

20.1 Required reading

Otto, Sarah P., and Troy Day. 2007. A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press. Beginning at 3.3 on p76 and ending at p82. Link.

20.2 Questions

  1. What are population-genetic models? [1 mark]

  2. Consider the definition of population dynamics. How are population dynamics related to population-genetic models? You will need to mention inheritance in your answer. [2 marks]

  3. What type of population growth is described by equations 3.6a and 3.6b? [1 mark]

  4. Typically, in this course, we have used \(\lambda\) to be the geometric growth rate. What are \(W_A\) and \(W_a\) in equations 3.6a and b, and for what values of \(W_A\) and \(W_a\) is the number of individuals with the \(A\) and \(a\) alleles increasing? [2 marks]

  5. What are \(n_A(t)\) and \(n_a(t)\) in equations 3.6a and b? [1 mark]

  6. What formula would you use to calculate the frequency of the \(A\) allele at time, \(t\)? [1 mark]

  7. Give two versions of a formula that you would use to calculate the frequency of the allele \(A\) at time, \(t+1\)? [2 marks]

  8. In equation 3.8d, what is \(p(t+1)\) and what is \(V_A\)? [1 mark]

  9. If \(V_A > 1\), what does this imply about the geometric growth rates for individuals that inherit the \(A\) allele relative to those that inherit the \(a\) allele? [1 mark]

  10. Using calculus, the author's derive a continuous time equation (3.11b) from the discrete time equation 3.8d. What is the formula for \(s_c\)? Provide the meaning of all parameters in your equation for \(s_c\). [2 marks]

  11. Although not provided in the book, for the continuous time version of this population-genetic model, the equations for the number of individuals with each allele would be,

\[\begin{eqnarray} \frac{dn_A(t)}{dt} & = & r_A n_A(t) \\ \frac{dn_a(t)}{dt} & = & r_a n_a(t) \end{eqnarray}\]

where \(r_A = b_A - d_A\) and \(r_a = b_a - d_a\), with \(b_i\) the per capita birth rate, and \(d_i\) the per capita mortality rate for each of the \(A\) and \(a\) alleles respectively. If \(r_A > 0\) and \(r_a > 0\), what does this imply about the change in the number of individuals with the \(A\) and \(a\) alleles over time? [1 mark]

  1. What is \(p=p(t)\) in equation 3.11b? Assuming that \(0 \leq p(t) \leq 1\), for what values of \(s_c\) does \(p(t)\) increase? How can this condition on \(s_c\), necessary for increasing \(p(t)\), be understood in terms of \(r_A\) and \(r_a\)? [2 marks]