  We consider the problem of controlling parabolic semilinear equations arising
in population dynamics, either in finite time or infinite time. These are the
monostable and bistable equations on $(0,L)$ for a density of individuals $0
\leq y(t,x) \leq 1$, with Dirichlet controls taking their values in $[0,1]$. We
prove that the system can never be steered to extinction (steady state $0$) or
invasion (steady state $1$) in finite time, but is asymptotically controllable
to $1$ independently of the size $L$, and to $0$ if the length $L$ of the
interval domain is less than some threshold value $L^\star$, which can be
computed from transcendental integrals. In the bistable case, controlling to
the other homogeneous steady state $0 <\theta< 1$ is much more intricate. We
rely on a staircase control strategy to prove that $\theta$ can be reached in
finite time if and only if $L< L^\star$. The phase plane analysis of those
equations is instrumental in the whole process. It allows us to read obstacles
to controllability, compute the threshold value for domain size as well as
design the path of steady states for the control strategy.
