  We extend our recent phenomenological study of $pN \to NN\pi\pi$ reactions to
the $\bar{p}N \to \bar{N}N\pi\pi$ reactions for anti-proton beam momenta up to
3.0 GeV within an effective Lagrangian approach. The contribution of
$N^*(1440)$ with its $N\sigma$ decay mode is found to be dominant at the
energies close to threshold for $\bar NN\pi^+\pi^-$ and $\bar NN\pi^0\pi^0$
channels. At higher energies or for $p\bar n\pi^-\pi^-$ and $\bar
NN\pi^\pm\pi^0$ channels where $N^*(1440)\to N\sigma$ mode cannot contribute,
large contributions from double-$\Delta$, $\Delta(1600) \to N^*(1440)\pi$,
$\Delta(1600) \to \Delta\pi$ and $\Delta(1620) \to \Delta\pi$ are found. In the
near-threshold region, sizeable contributions from $\Delta \to \Delta\pi$,
$\Delta \to N\pi$, $N \to \Delta\pi$ and nucleon pole are also indicated.
Although these results are similar to those for $pN\to NN\pi\pi$ reactions, the
antinucleon-nucleon collisions are shown to be complementary to the
nucleon-nucleon collisions and may even have advantages in some aspects. The
PANDA/FAIR experiment is suggested to be an excellent place for studying the
properties of relevant $N^*$ and $\Delta^*$ resonances.
