In　this　paper,　a　Leslie-type　predator-prey　system　with　simplified　Holling　type　IV　functional　response　and　strong　Allee　effect　on　prey　is　proposed.　The　dissipativity　of　the　system　and　the　existence　of　all　possible　equilibria　are　investigated.　The　investigation　emphasizes　the　exploring　of　bifurcation.　It　is　shown　that　the　system　exists　several　non-hyperbolic　positive　equilibria,　such　as　a　weak　focus　of　multiplicities　one　and　two,　(degenerate)　saddle-nodes　and　Bogdanov-Takens　singularities　(cusp　case)　of　codimensions　2　and　3.　At　these　equilibria,　it　is　proved　that　the　system　undergoes　various　kinds　of　bifurcations,　such　as　saddle-node　bifurcation,　Hopf　bifurcation,　degenerate　Hopf　bifurcation　and　Bogdanov-Takens　bifurcation　of　codimensions　2　and　3.　With　the　parameters　selected　properly,　there　exhibits　a　limit　cycle,　a　homoclinic　loop,　two　limit　cycles,　a　semistable　limit　cycle,　or　the　simultaneous　occurrence　of　a　homoclinic　loop　and　a　limit　cycle　in　the　system.　Moreover,　it　is　also　proved　that　the　system　has　a　cusp　of　codimension　at　least　4.　Hence,　there　may　exist　three　limit　cycles　generated　from　Hopf　bifurcation　of　codimension　3.　Numerical　simulations　are　done　to　support　the　theoretical　results.　(C)　2021　Elsevier　Ltd.　All　rights　reserved.