Midterm
Question 1: A model of meiotic drive
Meiotic drive is defined as the non-Mendelian production of gametes by heterozygotes.
That is, rather than producing equal numbers of "A" and "a" gametes, an Aa heterozygote produces a fraction, k, of "A" gametes and a fraction, 1-k, of "a" gametes.
Let p[t] equal the frequency of allele A among gametes of generation t. Assuming that these gametes unite at random and that there is no selection, the population of diploids at generation t will be at Hardy-Weinberg frequencies (p[t]^2 : 2p[t]q[t] : q[t]^2).
The following table gives the next generation of gametes produced by these diploid individuals in the presence of meiotic drive:
(a: 10 points) With meiotic drive, what will the frequency of the A gametes be in the next generation?
(b: 15 points)When will the frequency of the A allele in the next generation be greater than the frequency in the current generation (i.e. when will p[t+1] be greater than p[t])?
Question 2: Graphical stability analysis
In the following graphs, p[t+1] has been plotted against p[t] for the diploid selection model in which mutation is also included (both from A to a and from a to A).
(a: 15 points) On these two graphs, place
Question 3: Comparing the spread of dominant versus recessive alleles.
The diploid selection model in continuous time can be solved directly in some cases of particular interest.
In the case of an allele (A) being favored that is dominant (WAA=1+s, WAa=1+s, Waa=1), the frequency of the A allele at time t is given by:
In the case of an allele (A) being favored that is recessive (WAA=1+s, WAa=1, Waa=1), the frequency of the A allele at time t is given by:
(a: 10 points) If the fitness of an AA individual is 10% higher than that of an aa individual (i.e. WAA=1.1 and s=0.1), how many generations does it take for the A allele to rise from a frequency of 0.01 to 0.5 when:
(b: 10 points) In 2-3 sentences, describe why selection is less effective in one of these cases, such that it takes a lot longer to accomplish the same change in allele frequency.
Question 4: Population growth model for Nerviosa complexa
Dr. McMuffin has been studying the species, Nerviosa complexa. She has found that while members of this species have plenty of food and are not resource limited, they get extremely agitated when they are crowded and their rate of reproduction falls off rapidly.
She has studied the number of offspring per parent as a function of the total population size and has observed that it decreases with the square of the population size unlike the normal logistic equation:
She decides, on the basis of this graph, that the population changes in size according to the equation:
Dr. McMuffin comes to you with help in analysing the dynamics of Nerviosa complexa.
(a: 7 points) What are the possible equilibrium population sizes?
(b: 20 points) Analyse the stability of the only equilibrium point at which the populatio is not extinct. Specify when an equilibrium is locally stable or unstable and whether or not you expect to observe oscillatory behavior around the equilibrium.
(c: 5 points) Based on this analysis, would you predict that the dynamics for Nerviosa complexa would become chaotic for lower values of r, the same values of r, or higher values of r than in the standard logistic model? (Circle one)
(d: 8 points) Dr. McMuffin also says that the intrinsic rate of growth of Nerviosa complexa is quite low. You then tell her that the discrete equation may be approximated by a differential equation that can be solved generally. Write down this differential equation.
EXTRA CREDIT QUESTION
(a: 10 points) Solve the differential equation in part (d) of question 4 by the method of separation of variables. [Hint: Mathematica tells you that Integrate[1/(n*(1-n^2/k^2)),n] equals Log[n] - Log[-k^2 + n^2]/2.]
(WARNING: This part is tricky. Attempt only if you have enough time.)