## N 2n 12n20 N

There is one unknown left, p(0). We find it by applying the condition S« -P(n) = 1, which can be done only after we specify the functional forms for the birth and death rates, and we will do that only after we formulate the general answers to questions (2) and (3).

On to the probability of colonization. Let us assume that there is a population size ne at which functional extinction occurs; this could be ne = 0 but it could also be larger than 0 if there are Allee effects, since if there are Allee effects, once the population falls below the Allee threshold the mean dynamics are towards extinction (Greene 2000). Let us also assume that there is a population size K at which we consider the population to have successfully colonized the region of interest. We then define w(n) = Pr{N(t) reaches K before ne|N(0) = ng (8.8)

for which we clearly have the boundary conditions w(ne) = 0 and w(K) = 1. We think along the sample paths (Figure 8.2) to conclude that m(w) = EdN(M(n + dN)}. With dN given by Eq. (8.4), we Taylor expand to obtain

0 0