In　this　paper,　we　consider　a　Leslie-Gower　predator-prey　model　with　Allee　effect　on　the　prey　and　a　linear　functional　response.　Here　the　Allee　effect　impacts　the　birth　rate　of　the　prey,　which　is　different　from　the　common　multiplicative　and　additive　Allee　effects.　The　model　is　well-posed,　that　is,　all　solutions　are　bounded.　Applying　the　blow-up　method　indirectly,　we　prove　that　the　origin　which　is　not　an　equilibrium　of　the　system　is　an　attractor.　Then　we　study　the　existence　and　stability　of　equilibria,　which　indicate　that　the　system　undergoes　bifurcations.　With　the　help　of　Sotomayor＇s　theorem,　we　show　the　occurrence　of　saddle-node　bifurcation.　Moreover,　there　is　degenerate　Hopf　bifurcation　of　codimension　at　least　three.　By　choosing　two　(three)　parameters　of　the　system　as　bifurcation　parameters　and　calculating　a　versal　unfolding　near　the　cusp,　we　demonstrate　that　the　system　undergoes　Bogdanov-Takens　bifurcation　of　codimension　two　(three).　These　theoretical　results　are　supported　with　numerical　simulations.　©　2022　World　Scientific　Publishing　Company.