Systems with nonlocal vs. local diffusions and free boundaries

We study a class of free boundary problems of ecological models with nonlocal and local diffusions, which are natural extensions of free boundary problems of reaction diffusion systems in there local diffusions are used to describe the population dispersal, with the free boundary representing the spreading front of the species. We first prove the existence, uniqueness and regularity of global solution. For the classical competition, prey-predator and mutualist models, we show that a spreading-vanishing dichotomy holds, and establish the criteria of spreading and vanishing, and long time behavior of the solution.