A　Lotka-Volterra　type　predator-prey　system　with　Allee　effect　on　the　predator　species　and　density　dependent　birth　rate　on　the　prey　species　is　proposed　and　studied.　For　non-delay　case,　such　topics　as　the　persistent　of　the　system,　the　local　stability　property　of　the　equilibria,　the　global　stability　of　the　positive　equilibrium　are　investigated.　For　the　system　with　infinite　delay,　by　using　the　iterative　method,　a　set　of　sufficient　conditions　which　ensure　the　global　attractivity　of　the　positive　equilibrium　is　obtained.　By　introducing　the　density　dependent　birth　rate,　the　dynamic　behaviors　of　the　system　becomes　complicated.　The　system　maybe　collapse　in　the　sense　that　both　the　species　will　be　driven　to　extinction,　or　the　two　species　could　be　coexist　in　a　stable　state.　Numeric　simulations　are　carried　out　to　show　the　feasibility　of　the　main　results.　©　2019　Chen　et　al.,　published　by　De　Gruyter　2019.