  In this paper we investigate some free boundary problems for the
Lotka-Volterra type prey-predator model in one space dimension. The main
objective is to understand the asymptotic behavior of the two species (prey and
predator) spreading via a free boundary. We prove a spreading-vanishing
dichotomy, namely the two species either successfully spread to the entire
space as time $t$ goes to infinity and survive in the new environment, or they
fail to establish and die out in the long run. The long time behavior of
solution and criteria for spreading and vanishing are also obtained. Finally,
when spreading successfully, we provide an estimate to show that the spreading
speed (if exists) cannot be faster than the minimal speed of traveling
wavefront solutions for the prey-predator model on the whole real line without
a free boundary.
