Chapter 4.1, Problem 11E

Reminder Round all answers to two decimal places unless otherwise indicated.

The MacArthur-Wilson Theory of Biogeography Consider an island that is separated from the mainland, which contains a pool of potential colonizer species. The MacArthur-Wilson theory of biogeography hypothesizes that some species from the mainland will migrate to the island, but that increasing competition on the island will lead to species extinction. It further hypothesizes that both the rate of migration and the rate of extinction of species are exponential functions, and that an equilibrium occurs when the rate of extinction matches the rate of immigration. This equilibrium point is thought to be the point at which immigration and extinction stabilize. Suppose that, for a certain island near the mainland, the rate of immigration of new species is given by

$I=4.2\times {0.93}^t\text{species}\text{per}\text{year}$

and that the rate of species extinction on the island is given by

$E=1.5\times {1.1}^t\text{species}\text{per}\text{year}$.

According, to the MacArthur-Wilson theory, how long will be required for stabilization to occur, and what will be the immigration and extinction rates at that time?