Discrete-time　analogues　of　predator-prey　models　with　monotonic　and　nonmonotonic　functional　responses　are　introduced,　respectively.　The　discrete-time　analogues　are　considered　to　be　numerical　discretizations　of　the　continuous-time　models　and　we　study　their　dynamical　characteristics.　It　is　shown　that　the　discrete-time　analogues　preserve　the　periodicity　of　the　continuous-time　models　with　monotonic　functional　responses.　Moreover,　it　is　the　first　time　that　multiplicity　of　periodic　solutions　are　studied　when　modeled　with　nonmonotonic　functional　responses.　Unlike　other　types　of　functional　responses,　nonmonotone　functional　response　declines　at　high　prey　densities.　Thus　the　existing　arguments　for　obtaining　bounds　of　solutions　to　the　operator　equation　Lx　=　lambda　Nx　are　inapplicable　to　our　case　and　some　new　arguments　are　employed　for　the　first　time.　(c)　2006　Elsevier　Ltd.　All　rights　reserved.