  We analyze a one-dimensional two-scalar fields reaction advection diffusion
model for the globally subcritical transition to turbulence. In this model, the
homogeneous turbulent state is disconnected from the laminar one and disappears
in a tipping catastrophe scenario. The model however exhibits a linear
instability of the turbulent homogeneous state, mimicking the onset of the
laminar-turbulent patterns observed in the transitional regime of wall shear
flows. Numerically continuing the solutions obtained at large Reynolds numbers,
we construct the Busse balloon associated with the multistability of the
nonlinear solutions emerging from the instability. In the core of the balloon,
the turbulent fluctuations, encoded into a multiplicative noise, select the
pattern wavelength. On the lower Reynolds number side of the balloon, the
pattern follows a cascade of destabilizations towards larger and larger,
eventually infinite wavelengths. In that limit, the periodic limit cycle
associated with the spatial pattern hits the laminar fixed point, resulting in
a saddle-loop global bifurcation and the emergence of solitary pulse solutions.
This saddle-loop scenario predicts a logarithmic divergence of the wavelength,
which captures experimental and numerical data in two representative shear
flows.
