Two kinds of free boundary problems for the diffusive prey–predator model

In this paper we study two kinds of free boundary problems for the diffusive prey–predator model over a one dimensional habitat, in which the free boundary represents the spreading front and is only caused by the prey. In these two problems, it is assumed that the species can only invade further into the new environment from the right end of the habitat, and the spreading front expands at a speed that is proportional to the prey’s population gradient at the front. Asymptotic behaviors of solution as t→∞t→∞ are discussed firstly. The acquired results can help us to comprehend whether the expanding is successful or not. Then we establish the criteria for spreading and vanishing.