When a chunk is unloaded, so are the mobs (and all other items) on it, although passive mobs won't be removed.
Running away will usually allow you to stay out of the reach of hostile mobs, since a chunk needs to be loaded a certain amount of time before mobs will start to spawn, but the chunks immediately behind you could have the max allowable mobs, making it dangerous at best to turn around.
Has someone came onto your island and maybe killed your animals
or
I think your island might be set to a biome that doesn't spawn mobs