Asymptotic behavior and stability of positive solutions to a spatially heterogeneous predator-prey system

In this paper, we continue to study a spatially heterogeneous predator-prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14]. We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m. Finally, we further compare the difference of steady-state solutions between m>0 and m=0. It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model.