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We develop a reduced model for the slow unsteady dynamics of an isotropicchemically active particle near the threshold for spontaneous motion. Buildingon the steady theory developed in part I of this series, we match a weaklynonlinear expansion valid on the particle scale with a leading-orderapproximation in a larger-scale unsteady remote region, where the particle actsas a moving point source of diffusing concentration. The resulting amplitudeequation for the velocity of the particle includes a term representing theinteraction of the particle with its own concentration wake in the remoteregion, which can be expressed as a time integral over the history of theparticle motion, allowing efficient simulation and theoretical analysis offully three-dimensional unsteady problems. To illustrate how to use the model,we study the effects of a weak force acting on the particle, including thestability of the steady states and how the velocity vector realigns towards thestable one, neither of which previous axisymmetric and steady models were ableto capture. This unsteady formulation could also be applied to most of theother perturbation scenarios studied in part I as well as the dynamics ofinteracting active particles.

Weakly nonlinear dynamics of a chemically active particle near the threshold for spontaneous motion. II. History-dependent motion