In a random sample of an animal population (containing multiple species) the number of individuals, N, in the sample is related to the number of species, S, in the sample, by the equation S = α ln 

1 +

N

α



, for some constant α, which depends on the species. This equation was first derived by Fisher et al (1943), and some of their data are reproduced in Fig. 4.13.

a. Write N as a function of S.

b. For a variety of values of α plot S against ln(N). Note that it’s not immediately obvious how to do this, as the right-hand side of the equation is not a function of ln(N)
unless you rewrite it. How do you rewrite it to make this plot possible? Compare your curves to the ones shown in Fig. 4.13.

c. Are the dashed lines in Fig. 4.13 exactly straight? (Hint:

no, they’re not. Why not?) Why do they look so straight?

d. As α gets larger, the curves in Fig. 4.13 get closer together, and eventually cluster around the solid left curve, which corresponds to N = S. This solid curve looks like an You don’t know why this is, yet, but you will learn how to calculate this in Chapter 12.

exponential (you don’t need to explain why this is). Why does the curve N = S look like an exponential when plotted on a ln-linear plot?