>>30037313each island has different level up rewards and prerequisites
the first island grows pokebeans and leveling it up makes more beans grow (still takes a long time its like a bean or two every 30mins), also unconfirmed but I'm guessing that it also affects the rate of pokemon that come to visit your island
second island is for growing berries. upgrading lets you plant more berries and increases the amount of berries you get from a harvest
third island is a cave that lets pokemon search for random items (can get evolve stones, big pears etc) upgrading lets you search for different stuff and increases the amount of stuff they find
fourth island for exp or evtraining upgrading lets you place more pokemon, not sure what other benefits you get
five island is for happiness, upgrading lets you place more pokemon and increases the happiness level
upgrading is done just by talking to the dude on the island, however in order to unlock/upgrade islands you need to have a certain amount of box pokemon + beans. the last upgrades cost like 170 beans
