  We study by duality methods the extinction and explosion times of
continuous-state branching processes with logistic competition (LCSBPs) and
identify the local time at $\infty$ of the process when it is instantaneously
reflected at $\infty$. The main idea is to introduce a certain "bidual" process
$V$ of the LCSBP $Z$. The latter is the Siegmund dual process of the process
$U$, that was introduced in Foucart (2019), as the Laplace dual of $Z$. By
using both dualities, we shall relate local explosions and the extinction of
$Z$ to local extinctions and the explosion of the process $V$. The process $V$
being a one-dimensional diffusion on $[0,\infty]$, many results on diffusions
can be used and transfered to $Z$. A concise study of Siegmund duality for
regular one-dimensional diffusions is also provided.
