## Nttn 12n nttn nttn 1810

To answer the third question, we define the mean persistence time T(n) by

for which we obviously have the condition T(«e) = 0.

Use the method of thinking along sample paths, with the hint from Figure 8.2, to show that T(n) satisfies the equation

—1 = 2(n)T (n + 1) — (2(n) + p(n))T (n) + p(n)T (n — 1) (8.12) which is also Eq. 4-1 in MacArthur and Wilson (1967, p. 70).

We are unable to make any more progress without specifying the birth and death rates, which we now do.

### The specific case treated by MacArthur and Wilson

Computationally, 1967 was a very long time ago. The leading technology in manuscript preparation was an electric typewriter with a self-correcting ribbon that allowed one to backspace and correct an error. Computer programs were typed on cards, run in batches, and output was printed to hard copy. Students learned how to use slide rules for computations (or - according to one reader of a draft of this chapter -chose another profession).

In other words, numerical solution of equations such as (8.10) or (8.12) was hard to do. Part of the genius of Robert MacArthur was that he found a specific case of the birth and death rates that he was able to solve (see Connections for more details). MacArthur and Wilson introduce a parameter K, about which they write (on p. 69 of their book): "But since all populations are limited in their maximum size by the carrying capacity of the environment (given as K individuals)'' and on p. 70 they describe K as ''... a ceiling, K, beyond which the population cannot normally grow.'' The point of providing these quotations and elaborations is this: in the MacArthur-Wilson model for extinction times (both in their book and in what follows) K is a population ceiling and not a carrying capacity in the sense that we usually understand it in ecology at which birth and death rates balance. In the next section, we will discuss a model in which there is both a carrying capacity in the usual sense and a population ceiling.

For the case of density dependent birth rates, a population ceiling means that where l and p on the right hand sides are now constants. (I know that this is a difficult notation to follow, but it is the one that is used in their book, so I use it in case you choose to read the original, which I strongly

Figure 8.3. Examples of mean persistence times computed by MacArthur and Wilson. The key observations here are that (i) there is a "shoulder" in the mean persistence time in the sense that once a moderate value of K is reached, the mean persistence time increases very rapidly, and (ii) the persistence times are enormous. Reprinted with permission.

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