The　dynamics　of　a　nonautonomous　predator-prey　system　with　the　Beddington-DeAngelis　functional　response　is　studied　from　the　perspective　of　extinction　of　the　predator.　With　the　help　of　a　Fluctuation　Lemma,　we　obtain　a　set　of　new　sufficient　conditions　on　the　global　asymptotic　stability　of　the　boundary　solution　(which　means　the　extinction　of　the　predator).　The　result　not　only　improves　but　also　complements　some　existing　ones.　Moreover,　the　result　indicates　that　the　effect　of　the　parameters　is　accumulative　rather　than　pointwise.　©　2008　Elsevier　Inc.　All　rights　reserved.