In　this　paper,　with　the　methods　of　adaptive　dynamics　and　critical　function　analysis,　we　investigate　the　evolutionary　branching　phenomenon　of　predator　species.　We　assume　that　both　the　prey　and　predators　are　density-dependent　and　the　predator＇s　attack　ability　can　adaptively　evolve,　but　this　has　a　cost　in　terms　of　its　death　rate.　First,　we　identify　the　general　properties　of　trade-off　relationships　that　allow　for　a　continuously　stable　strategy　and　evolutionary　branching　in　the　predator　strategy.　It　is　found　that　if　the　trade-off　curve　is　weakly　concave　near　the　singular　strategy,　then　the　singular　strategy　may　be　an　evolutionary　branching　point.　Second,　we　find　that　after　the　branching　has　occurred　in　the　predator　strategy,　if　the　trade-off　curve　is　convex-concave-convex,　the　predator　species　will　eventually　evolve　into　two　different　types,　which　can　stably　coexist　on　the　much　longer　evolutionary　timescale　and　no　further　branching　is　possible.　Crown　Copyright　(C)　2013　Published　by　Elsevier　Inc.　All　rights　reserved.