In　this　paper,　we　prpose　a　single-species　stage　structure　model　with　Michaelis–Menten-type　harvesting　for　mature　population.　We　investigate　the　existence　of　all　possible　equilibria　of　the　system　and　discuss　the　stability　of　equilibria.　We　use　Sotomayor’s　theorem　to　derive　the　conditions　for　the　existence　of　saddle-node　and　transcritical　bifurcations.　From　the　ecological　point　of　view,　we　analyze　the　effect　of　harvesting　on　the　model　of　mature　population　and　consider　it　as　a　bifurcation　parameter,　giving　the　maximum　threshold　of　continuous　harvesting.　By　constructing　a　Lyapunov　function　and　Bendixson–Dulac　discriminant,　we　give　sufficient　conditions　for　the　global　stability　of　boundary　equilibrium　and　positive　equilibrium,　respectively.　Our　study　shows　that　nonlinear　harvesting　may　lead　to　a　complex　dynamic　behavior　of　the　system,　which　is　quite　different　from　linear　harvesting.　We　carry　out　numeric　simulations　to　verify　the　feasibility　of　the　main　results.　©　2020,　The　Author(s).