Dynamics for a diffusive prey-predator model with different free boundaries

To understand the spreading and interaction of prey and predator, in this paper we study the dynamics of the diffusive Lotka-Volterra type prey-predator model with different free boundaries. These two free boundaries, which may intersect each other as time evolves, are used to describe the spreading of prey and predator. We investigate the existence and uniqueness, regularity and uniform estimates, and long time behaviors of global solution. Some sufficient conditions for spreading and vanishing are established. When spreading occurs, we provide the more accurate limits of (u,v) as t→∞, and give some estimates of asymptotic spreading speeds of u,v and asymptotic speeds of g,h. Some realistic and significant spreading phenomena are found.