In　this　paper,　we　propose　a　Leslie-Gower　predator-prey　model　with　strong　Allee　effect　on　prey　and　fear　effect　on　predator.　We　discuss　the　existence　and　local　stability　of　equilibria　by　making　full　use　of　qualitative　analytical　theory.　It　is　shown　that　the　above　system　exhibits　at　most　two　positive　equilibria　and　it　can　undergo　a　series　of　bifurcation　phenomena.　We　indicate　that　the　dynamical　behavior　of　the　model　is　closely　related　to　the　fear　effect　on　predator.　In　detail,　when　the　fear　effect　parameter　p　=　p*,　the　system　will　undergo　degenerate　Hopf　bifurcation.　There　exist　two　limit　cycles　(the　inner　is　stable　and　the　outer　is　unstable).　However,　when　p　=　p(*),　the　system　will　undergo　degenerate　Bogdanov-Takens　bifurcation.　Also,　by　numerical　simulation,　we　conclude　that　the　stronger　the　fear　effect,　the　bigger　the　density　of　prey　species.　The　above　shows　that　fear　effect　on　predator　is　beneficial　to　the　persistence　of　the　prey　species.　Our　results　can　be　seen　as　a　complement　to　previous　works　[Gonzalez-Olivares　et　al.,　2011;　Pal　&　Mandal,　2014].