  This paper deals with a free boundary problem of the Lotka-Volterra type
prey-predator model with variable intrinsic growth rate for predator over a one
dimensional habitat, in which the free boundary represents the spreading front
and is caused only by the prey. In this problem it is assumed that the species
can only invade further into the new environment from the right end of the
initial region, and the spreading front expands at a speed that is proportional
to the prey's population gradient at the front. Moreover, the variable
intrinsic growth rate $r(t,x)$ is positive in pointwise but may be very small
as $t\to\infty$ or $x\to\infty$. The main objective is to realize the
dynamics/variations of prey, predator and free boundary. We first study the
long time behavior of $(u,v)$ for the vanishing case ($h_\infty<\infty$). Then
we find the criteria for spreading and vanishing. At last, the long time
behavior of $(u,v)$ for the spreading case ($h_\infty=\infty$) is discussed.
Theorems 2.2, 2.3 and 4.1 together establish a spreading-vanishing dichotomy.
Moreover, our model exhibits very different properties from the free boundary
problems of prey-predator model in which the free boundary is caused only by
the predator or by both prey and predator.
