C2

where c2 is a second constant of integration and to which we apply the condition T(0) = 0 to conclude that c2 = (v/2r2)exp((2r/v)K) so that nv T (n) —+ 2r2

Note that this solution involves n, r, K, and v in a nonlinear and relatively complicated fashion.

Foley (1994) uses a different method of obtaining the solution (see his Appendix) and also writes it in a different manner by introducing the parameter s = r/v:

Show that Eqs. (8.37) and (8.38) are the same. Now assume that sK^ 1 and show by Taylor expansion of the exponential to third order in K that

which tells us how the deterministic and stochastic components of the dynamics affect the persistence time. Note, for example, that the mean persistence time now grows as the cube of the population ceiling.

As with the theory of MacArthur and Wilson, this theory is appealing because of its operational simplicity. It tells us to measure the mean and variance of the per capita changes (and, in more advanced form, the autocorrelation of the fluctuations to correct the estimate of variance (Foley 1994, Lande et a/. 2003) and to estimate the ceiling of the population). From these will come the mean persistence time via Eqs. (8.37) or (8.38). It is reasonable to ask, however, how these predictions might depend upon life history characteristics (see Connections), on more general density dependence, or when we ever might see a population ceiling.

The general density dependent case

We now turn to the general density dependent case, so that, instead of Eq. (8.27), the population satisfies the stochastic differential equation dN = b(N)dt + Va(N)dW (8.40)

where b(n) and a(n) are known functions. We will assume that there is a single stable steady state ns for which b(ns) = 0, a population size ne at which we consider the population to be extinct and, although there surely is a true population ceiling, as will be seen we do not need to specify (or use) it.

These ideas are captured schematically in Figure 8.6. We know that T(n) will now satisfy the equation

with one boundary condition T(ne) = 0. For the second boundary condition, as before we require that limn!1 Tn = 0, which by analogy with the previous section, indicates that the population ceiling is infinite. Were it not, we would apply the reflecting condition at K.

We solve this equation using the same method as in the previous section, but now in full generality. To begin, we set W(n) = Tn, so that Eq. (8.41) can be rewritten as h nH

Stochastic and deterministic factors act "in opposition"

Stochastic and deterministic factors "work together"

Figure 8.6. A schematic description of the general case for stochastic extinction. The population dynamics are dN = b(N)dt + Pa(N)dW with a single deterministic stable steady state ns and a population size ne at which we consider the population to be extinct. For starting values of population size smaller than ns, the factors of stochastic fluctuation toward extinction and deterministic increase towards the steady state are acting in opposition, while for values greater than ns they are acting in concert in the sense that the deterministic factors reduce population size.

and we now define

which allows us to write Eq. (8.42) as d [W = e$(n) dn a(n)

and, integrating from n to 1, we conclude that

and we pause momentarily. Note that Eq. (8.45) automatically satisfies the boundary condition lim-!1T- = 0. Also note that the function O(-) defined by Eq. (8.43) involves the ratio of the infinitesimal mean and variance. The bigger the variance - thus the stronger the fluctuations -the smaller the ratio (and thus the integral), all else being equal.

We integrate Eq. (8.45) once more, this time from -e to - (recalling that T(-e) = 0) and end up with the formula for the mean persistence time in the general case

dyds

Equation (8.46) is our desired result. It gives the mean persistence time for a population starting at size n when the dynamics follow the general stochastic differential equation (8.40). This general formulation tells us actually very little about specific situations, but the literature contains many examples of its application once the functional forms for b(n) and a(n) are chosen according to the biological situation at hand (see Connections for some examples).

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