Island Biogeography Preston (1962) and MacArthur and Wilson (1963) investigated the

Chapter 8, Problem 6

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Island Biogeography Preston (1962) and MacArthur and Wilson (1963) investigated the effect of area on species diversity in oceanic islands. It is assumed that species can immigrate to an island from a species pool of size P and that species on the island can go extinct. We denote the immigration rate by I (S) and the extinction rate by E(S), where S is the number of species on the island. Then the change in species diversity over time is dS dt = I (S) E(S) (8.91) For a fixed island, the simplest functional forms for I (S) and E(S) are I (S) = c _ 1 S P _ (8.92) E(S) = m S P (8.93) where c, m, and P are positive constants. (a) Find the equilibrium species diversity S of (8.91) with I (S) and E(S) given in (8.92) and (8.93). (b) It is reasonable to assume that the extinction rate is a decreasing function of island size. That is, we assume that if A denotes the area of the island, then m is a function of the island size A, with dm/dA < 0. Furthermore, we assume that the immigration rate I does not depend on the size of the island. Use these assumptions to investigate how the equilibrium species diversity changes with island size. (c) Assume that S(0) = S0. Solve (8.91) with I (S) and E(S) as given in (8.92) and (8.93), respectively. (d) Assume that S0 = 0. That is, the island is initially void of species. The time constant T for the system is defined as S(T ) = (1 e1)S Show that, under the assumption S0 = 0, T = P c + m (e) Use the assumptions in (b) and your answer in (d) to investigate the effect of island size on the time constant T ; that is, determine whether T (A) is an increasing or decreasing function of A.