## 3.9 Exercises

These exercises ask you to plot the expected time to extinction using the analytical model for the length of phase two of the mutation buildup. Gnuplot, which allows users to define their own functions and then plot them in two or three dimensions, is a good tool for this exercise.

1. Plot , the expected number of generations in phase two of the mutation buildup, as a function of population size (K). Limit the range of K to . Plot three different curves, using values of R = 2, R = 3, and R = 4. The three curves should show an increasing time to extinction for larger values of R. For the other model parameters, use and ; these are realistic values derived from field studies of a variety of different plants and animals.

2. There is an interesting interaction between s, the average effect per mutation, and the expected time to extinction. If mutations are relatively harmless (values of s smaller than .0001), then organisms can carry quite a few mutations and still survive, and it will take a long time for sufficient mutations to accumulate to start affecting the health of the whole population. On the other hand, if mutations are relatively severe (s larger than 0.25) a single mutation is enough to kill an individual, which in turn implies it is very unlikely that any mutations will be passed to the next generation, and thus the buildup of mutations will take a long time. In this case, for large enough values of s no mutations will accumulate and the population will never go extinct (from genetic causes alone). For the meltdown to occur, the selection coefficient must be in the right range.

Plot as a function of s in the range , using a fixed value of K = 50 and . Again plot three curves, for R = 2, R = 3, and R = 4. At what value of s is the population most vulnerable?