Some Recent Results on Diffusive Predator-prey Models in Spatially Heterogeneous Environment

We present several recent results obtained in our attempts to understand the influence of spatial heterogeneity in the predator-prey models. Two different approaches are taken. The first approach is based on the observation that the behavior of many diffusive population models is very sensitive to certain coefficient functions becoming small in part of the underlying spatial region. We apply this observation to three predator-prey models to reveal fundamental differences from the classical homogeneous case in each model, and demonstrate the essential differences of these models from each other. In the second approach, we examine the influence of a protection zone in a Holling type II diffusive predator-pre model, which introduces different mathematical problems from those in the first approach, and reveals important impacts of the protection zone.