  New characterizations of the disjoint Dunford-Pettis property of order $p$
(disjoint $DPP_p$) are proved and applied to show that a Banach lattice of
cotype $p$ has the disjoint $DPP_p$ whenever its dual has this property. The
disjoint Dunford-Pettis$^*$ property of order $p$ (disjoint $DP^*P_p$) is
thoroughly investigated. Close connections with the positive Schur property of
order $p$, with the disjoint $DPP_p$, with the $p$-weak $DP^*$ property and
with the positive $DP^*$ property of order $p$ are established. In a final
section we study the polynomial versions of the disjoint $DPP_p$ and of the
disjoint $DP^*P_p$.
