  In this paper we investigate a free boundary problem for the diffusive
Leslie-Gower prey-predator model with double free boundaries in one space
dimension. This system models the expanding of an invasive or new predator
species in which the free boundaries represent expanding fronts of the predator
species. We first prove the existence, uniqueness and regularity of global
solution. Then provide a spreading-vanishing dichotomy, namely the predator
species either successfully spreads to infinity as $t\to\infty$ at both fronts
and survives in the new environment, or it spreads within a bounded area and
dies out in the long run. The long time behavior of $(u,v)$ and criteria for
spreading and vanishing are also obtained. Because the term $v/u$ (which
appears in the second equation) may be unbounded when $u$ nears zero, it will
bring some difficulties for our study.
