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This work is focused on the mathematical and computational modeling of bioconvection, which describes the mixing of fluid and micro-organisms exhibiting negative geotaxis movement under the force of gravity. The collective population moves towards the surface of the fluid, generating a Rayleigh–Taylor instability, where initial fingers of organisms plummet to the bottom. The inherent drive to swim vertically generates large collective flow patterns that persist in time. We model the flow using the Navier–Stokes equations for an incompressible, viscous fluid, coupled with the transport equation describing the concentration of the micro-organisms. We use a nonlinear semigroup approach to prove the existence of solutions. We propose a partitioned, second-order, time adaptive numerical method based on the Cauchy’s one-legged ‘-like’ scheme. We prove that the method is energy-stable, and for small time steps, the iterative procedure in the partitioned algorithm is linearly convergent. The numerical results confirm the expected second-order of accuracy. We also present a computational study of a chaotic system describing bioconvection of motile flagellates.