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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. 10 100 1000 10 000 100 000 Equilibria! population size(K)

recommend.) For the case of density dependent death rates, MacArthur and Wilson assume that l(n) = In

y whatever needed to go from n > K to K otherwise

From these equations it is clear that in neither case is K a carrying capacity (at which birth rates and death rates are equal); rather it is a population ceiling in the sense that ''the population grows exponentially to level K, at which point it stops abruptly'' (MacArthur and Wilson

1967, p. 70). This point will become important in the next section, when we use modern computational methods to address persistence time. However, the point of Eqs. (8.13) and (8.14) is that they allow one to find the mean time to extinction, which is exactly what MacArthur and Wilson did (see Figure 8.3). The dynamics determined by Eqs. (8.13) or (8.14) will be interesting only if 1 > ^ (preferably strictly greater). Figures such as 8.3 led to the concept of a ''minimum viable population'' size (Soule 1987), in the sense that once K was sufficiently large (and the number K = 500 kind of became the apocryphal value) the persistence time would be very large and the population would be okay.

It is hard to overestimate the contribution that this theory made. In addition to starting an industry concerned with extinction time calculations (see Connections), the method is highly operational. It tells people to measure the density independent birth and death rates and estimate (for example from historical population size) carrying capacity and then provides an estimate of the persistence time. In other words, the developers of the theory also made clear how to operationalize it, and that always makes a theory more popular.

We shall now explore how modern computational methods can be used to extend and improve this theory.

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