  The transition to turbulence in many shear flows proceeds along two competing
routes, one linked with finite-amplitude disturbances and the other one
originating from a linear instability, as in e.g. boundary layer flows. The
dynamical systems concept of edge manifold has been suggested in the
subcritical case to explain the partition of the state space of the system.
This investigation is devoted to the evolution of the edge manifold when a
linear stability is added in such subcritical systems, a situation poorly
studied despite its prevalence in realistic fluid flows. In particular the fate
of the edge state as a mediator of transition is unclear. A deterministic
three-dimensional model is suggested, parametrised by the linear instability
growth rate. The edge manifold evolves topologically, via a global saddle-loop
bifurcation, from the separatrix between two attraction basins to the mediator
between two transition routes. For larger instability rates, the stable
manifold of the saddle point increases in codimension from 1 to 2 after an
additional local saddle node bifurcation, causing the collapse of the edge
manifold. As the growth rate is increased, three different regimes of this
model are identified, each one associated with a flow case from the recent
hydrodynamic literature. A simple nonautonomous generalisation of the model is
also suggested in order to capture the complexity of spatially developing
flows.
