In　this　paper,　a　Leslie-Gower　predator-prey　model　with　simplified　Holling　type　IV　functional　response　and　Allee　effect　on　predator　is　studied.　We　discuss　the　existence　and　stability　of　equilibria,　and　show　that　the　system　has　at　most　three　positive　equilibria.　Under　sufficient　conditions,　the　system　can　have　a　weak　focus　of　order　3,　a　cusp　of　codimension　3,　a　nilpotent　focus　or　an　elliptic　equilibrium　of　codimension　3,　or　a　nilpotent　elliptic　equilibrium　of　codimension　at　least　4.　Also,　we　prove　that　the　degenerate　Hopf　bifurcation　of　codimension　3　occurs　at　the　anti-saddle,　cusp　type　degenerate　Bogdanov-Takens　bifurcation　of　codimension　3　occurs　at　the　double　equilibrium,　and　focus　or　elliptic　type　degenerate　Bogdanov-Takens　bifurcation　of　codimension　3　occurs　at　the　triple　equilibrium.　Finally,　some　numerical　simulations　show　that　the　system　can　exhibit　two　limit　cycles,　a　semi-stable　limit　cycle　or　a　limit　cycle　containing　a　homoclinic　loop.　Our　results　imply　that　the　predator　and　prey　can　arrive　a　stable　state　of　coexistence　if　Allee　effect　is　small,　or　the　predator　will　be　extinct　if　Allee　effect　is　large.