  Plane Poiseuille flow, the pressure driven flow between parallel plates,
shows a route to turbulence connected with a linear instability to
Tollmien-Schlichting (TS) waves, and another one, the bypass transition, that
is triggered with finite amplitude perturbation. We use direct numerical
simulations to explore the arrangement of the different routes to turbulence
among the set of initial conditions. For plates that are a distance $2H$ apart
and in a domain of width $2\pi H$ and length $2\pi H$ the subcritical
instability to TS waves sets in at $Re_{c}=5815$ that extends down to
$Re_{TS}\approx4884$. The bypass route becomes available above $Re_E=459$ with
the appearance of three-dimensional finite-amplitude traveling waves. The
bypass transition covers a large set of finite amplitude perturbations. Below
$Re_c$, TS appear for a tiny set of initial conditions that grows with
increasing Reynolds number. Above $Re_c$ the previously stable region becomes
unstable via TS waves, but a sharp transition to the bypass route can still be
identified. Both routes lead to the same turbulent in the final stage of the
transition, but on different time scales. Similar phenomena can be expected in
other flows where two or more routes to turbulence compete.
