In biology, the behaviour of a bacterial suspension in an incompressible fluid drop is modelled by the chemotaxis-Navier–Stokes equations. This paper introduces an exchange of oxygen between the drop and its environment and an additionally logistic growth of the bacteria population. A prototype system is given by{nt+u⋅∇n=Δn−∇⋅(n∇c)+n−n2,x∈Ω,t>0,ct+u⋅∇c=Δc−nc,x∈Ω,t>0,ut=Δu+u⋅∇u+∇P−n∇φ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 in conjunction with the initial data (n,c,u)(⋅,0)=(n0,c0,u0) and the boundary conditions∂c∂ν=1−c,∂n∂ν=n∂c∂ν,u=0,x∈∂Ω,t>0. Here, the fluid drop is described by Ω⊂RN being a bounded convex domain with smooth boundary. Moreover, φ is a given smooth gravitational potential.Requiring sufficiently smooth initial data, the existence of a unique global classical solution for N=2 is proved, where ‖n‖Lp(Ω) is bounded in time for all p<∞, as well as the existence of a global weak solution for N=3.