A free boundary problem for some modified predator-prey model in a higher dimensional environment

We focus on the free boundary problems for a Leslie-Gower predator-prey model with radial symmetry in a higher dimensional environment that is initially well populated by the prey. This free boundary problem is used to describe the spreading of a new introduced predator. We first establish that a spreading-vanishing dichotomy holds for this model. Namely, the predator either successfully spreads to the entire space as t goes to infinity and survives in the new environment, or it fails to establish and dies out in the long term. The longterm behavior of the solution and the criteria for spreading and vanishing are also obtained. Moreover, when spreading of the predator happens, we provide some rough estimates of the spreading speed.