Predator-prey　interaction　is　considered　a　natural　phenomenon　in　the　ecological　system.　Empirical　research　conducted　on　vertebrates　has　demonstrated　that　the　presence　of　predators　can　have　a　significant　impact　on　the　survival　rates　and　reproductive　capabilities　of　prey　populations.　Recently,　there　has　been　research　on　mathematical　models　of　predator-prey　systems　that　include　different　predator　functional　responses　and　fear　effects.　These　studies　have　overlooked　the　influence　of　fear　on　the　death　rates　of　species　that　are　hunted.　From　the　given　findings,　we　present　a　mathematical　model　of　predator-prey　systems　that　includes　fear　costs　impacting　the　rates　of　reproduction　and　death　in　prey　population.　By　reducing　the　discrete　model　into　different　normal　forms,　we　prove　that　there　exists　a　set　of　codimension-1　and　codimension-2　bifurcations,　which　include　transcritical,　flip,　Neimark-Sacker　bifurcations,　1:2　and　1:4　strong　resonance　bifurcations.　These　findings　indicate　that,　compared　with　the　system　without　fear　effect,　the　increase　of　the　fear　effect　parameter　k1　that　affects　the　birth　rate　of　prey　and　the　fear　effect　parameter　k2　that　affects　the　death　rate　of　prey　will　strengthen　the　oscillation　of　prey　population　and　reduce　the　oscillation　of　predator　population.　In　addition,　the　increases　of　k1　and　k2　have　no　effect　on　the　density　of　the　prey　population　but　reduce　the　density　of　the　predator　population.　When　the　fear　effect　k1　and　other　parameter　values　remain　the　same,　the　system　generates　an　expanding　limit　circle　as　k2　increases,　indicating　that　the　effect　of　fear　effects　on　the　death　rate　enhances　the　stability　of　the　system.