  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 prove that such kind of nonlocal and local diffusion
problems has a unique global solution, and then show that a spreading-vanishing
dichotomy holds. Moreover, criteria of spreading and vanishing, and long time
behavior of solution when spreading happens are established for the classical
Lotka-Volterra competition and prey-predator models. Compared with free
boundary systems with local diffusions as well as with nonlocal diffusions, the
present paper involves some new difficulties, which should be overcome by use
of new techniques. This is part I of a two part series, where we prove the
existence, uniqueness, regularity and estimates of global solution. The
spreading-vanishing dichotomy, criteria of spreading and vanishing, and
long-time behavior of solution when spreading happens will be studied in the
separate part II.
