The Mac ArthurWilson theory of extinction time

The 1967 book of Robert MacArthur and E. O. Wilson (MacArthur and Wilson 1967) was an absolutely seminal contribution to theoretical ecology and conservation biology. Indeed, in his recent extension of it, Steve Hubbell (2001 ) describes the work of MacArthur and Wilson as a ''radical theory.'' From our perspective, the theory of MacArthur and Wilson has two major contributions. The first, with which we will not deal, is a qualitative theory for the number of species on an island determined by the balance of colonization and extinction rates and the roles of chance and history in determining the composition of species on an island.

The second contribution concerns the fate of a single species arriving at an island, subject to stochastic processes of birth and death. Three questions interest us: (1) given that a propagule (a certain initial number of individuals) of a certain size arrives on the island, what is the frequency distribution of subsequent population size; (2) what is the chance that descendants of the propagule will successfully colonize the island; and (3) given that it has successfully colonized the island, how long will the species persist, given the stochastic processes of birth and death, possible fluctuations in those birth and death rates, and the potential occurrence of large scale catastrophes? These are heady questions, and building the answers to them requires patience.

The general situation

We begin by assuming that the dynamics of the population are characterized by a birth rate !(«) and a death rate ^(w) when the population is size n (and for which there are at least some values of n for which l(n) > ,u(«) because otherwise the population always declines on average and that is not interesting) in the sense that the following holds:

Prfpopulation size changes in the next dt|N(t) = n} = 1 - exp(-(l(n) + ^(n))dt)

PrfN(t + dt)- N(t) = 1|change occurs} = ,. ,1(w) . , (8-3)

Note that Eq. (8.3) allows us to change the population size only by one individual or not at all. Furthermore, since the focus of Eq. (8.3) is an interval of time dt, it behooves us to think about the case in which dt is small. However, also note that there is no term o(dt) in Eq. (8.3) because that equation is precise. For simplicity, we will define dN= N(t + dt) - N(t).

Show that, when dt is small, Eq. (8.3) is equivalent to PrfdN = 1 |N (t) = n} = 2(n)dt + o(dt)

and note that we implicitly acknowledge in Eq. (8.4) that Pr{|dN| > 1|N(t) = n} = o(dt)

All of this should remind you of the Poisson process. We continue by setting p(n, t) = Pr{N(t)=n} (8.5)

and know, from Chapter 3, to derive a differential equation forp(n, t) by considering the changes in a small interval of time:

p(n, t + dt) = p(n - 1, t)1(n - 1)dt + p(n, t)(1 -(1(n) + p(n))dt)

which we then convert to a differential-difference equation by the usual procedure dtp(n, t) = - (1(n) + p(n))p(n, t) +1(n - 1)p(n - 1, t)

This equation requires an initial condition (actually, a whole series for p(n, 0)) and is generally very difficult to solve (note that, at least thus far, there is no upper limit to the value that n can take, although the lower limit n = 0 applies).

One relatively easy thing to do with Eq. (8.7) is to seek the steady state solution by setting the left hand side equal to 0. In that case, the right hand side becomes a balance between probabilitiesp(n), p(n - 1), andp(n + 1) of population size n, n - 1, and n + 1. Let us write out the first few cases. When n = 0, there are only two terms on the right hand side sincep(n - 1) = 0, so we have 0 = - l(0)p(0) + ^(1)p(1) where we have made the sensible assumption that ^(0) = 0 and that 1(0) > 0. How might the latter occur? When we are thinking about colonization from an external source, this condition tells us that even if there are no individuals present now, there can be some later because the population is open to immigration of new individuals. Populations can be open in many ways. For example, if N(t) represents the number of adult flour beetles in a microcosm of flour, then even if N(t) = 0 subsequent values can be greater than 0 because adults emerge from pupae, so that the time lag in the full life history makes the adult population "open" to immigration from another life history stage. For example, Peters et al. (1989) use the explicit form 1(n) = a(n + S)e-cn for which 1(0) = aS.

In general, we conclude that ^(1) = [1(0)/^(1)j¿>(0). When n = 1, the balance becomes 0 = -(1(1) + m(1))p(1) + 1(0)p(0) + ^(2)p(2), from which we determine, after a small amount of algebra, that p(2) = [1(1)1(0)/^(1)^(2)]^(0). You can surely see the pattern that will follow from here.

Figure 8.2. Thinking along sample paths allows us to derive equations for the colonization probability and the mean persistence time. Starting at population size n, in the next interval of time dt, the population will either remain the same, move to n +1, or move to n - 1. The probability of successful colonization from size n is then the average of the probability of successful colonization from the three new sizes. The persistence time is the same kind of average, with the credit of the population having survived dt time units.

Show that the general form foris

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