In　this　paper,　with　the　method　of　adaptive　dynamics　and　critical　function　analysis,　we　investigate　the　evolutionary　diversification　of　prey　species.　We　assume　that　prey　species　can　evolve　safer　strategies　such　that　it　can　reduce　the　predation　risk,　but　this　has　a　cost　in　terms　of　its　reproduction.　First,　by　using　the　method　of　critical　function　analysis,　we　identify　the　general　properties　of　trade-off　functions　that　allow　for　continuously　stable　strategy　and　evolutionary　branching　in　the　prey　strategy.　It　is　found　that　if　the　trade-off　curve　is　globally　concave,　then　the　evolutionarily　singular　strategy　is　continuously　stable.　However,　if　the　trade-off　curve　is　concave-convex-concave　and　the　prey＇s　sensitivity　to　crowding　is　not　strong,　then　the　evolutionarily　singular　strategy　may　be　an　evolutionary　branching　point,　near　which　the　resident　and　mutant　prey　can　coexist　and　diverge　in　their　strategies.　Second,　we　find　that　after　branching　has　occurred　in　the　prey　strategy,　if　the　trade-off　curve　is　concave-convex-concave,　the　prey　population　will　eventually　evolve　into　two　different　types,　which　can　coexist　on　the　long-term　evolutionary　timescale.　The　algebraical　analysis　reveals　that　an　attractive　dimorphism　will　always　be　evolutionarily　stable　and　that　no　further　branching　is　possible　for　the　concave-convex-concave　trade-off　relationship.　Crown　Copyright　(C)　2012　Published　by　Elsevier　Ireland　Ltd.　All　rights　reserved.