Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion The work of the first author was supported by the NUS ARF Grant R-146-000-034-112. Part of the paper was prepared during the visit of the second author to the National University of Singapore. He expresses his gratitude for the support of the NUS ARF Grant R-146-000-034-112, and PRC NSF Grant NSFC-19831060 and the 333 Project of JiangSu Provice, China.