Question

6. The California fairy shrimp (Linderiella occidentalis) is known to occupy geographically separated habitat patches. Suppose that there are N patches, n of which are occupied initially and N - n which are not (1 <n<N - 1). A cycle consisting of separate colonization and extinction phases occurs as follows. Each of the N n unoccupied patches is colonized independently with probability c leaving Y (2 n) patches occupied, so that Y-n+C, where C has a binomial B(N -n, c) distribution. Next, the Y patches currently occupied each survive the extinction phase with the same probability s, leaving Z ( Y) patches occupied. Thus, given Y-m, Z has a binomial B(m, s) distribution. (a) Use conditional expectation to evaluate E(Z). Hence show that the expected change in the number of occupied patches after both phases have been completed is given by sc(N-n)-(1-s)n. 13]

Answer 1

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