Problem SE11

You are required to study the metapopulation of the butterfly Euphydryas casagrandi in the Tuahutu archipelago, which consists of very many small islands. The local extinction probability per unit time in each island is 0.05 year$^{-1}$. Extinction is counterbalanced by immigration, which can take place because of migration from the mainland or from surrounding islands where E. casagrandi is still present. The colonization rate of an empty island from mainland is 0.01 yr$^{-1}$, while the colonization rate from other islands follows the Levins model and is 0.045 yr$^{-1}$. Write down the metapopulation model that governs the dynamics of the fraction of occupied islands. Calculate the fraction of occupied islands at equilibrium.

Suppose then that tourism development plans are devised that aim at urbanizing both the Tuahutu archipelago and the nearby coastline on the mainland. According to plan A the archipelago is heavily urbanized, thus leading to the destruction of one fourth of the insular habitat of E. casagrandi; according to plan B it is the coast of the mainland that is heavily urbanized, thus leading the whole butterfly mainland population to the complete destruction. Evaluate the impacts of both plan A and plan B on the fate of the butterfly metapopulation.