  A subset $S$ of vertices of a digraph $D$ is a double dominating set (total
$2$-dominating set) if every vertex not in $S$ is adjacent from at least two
vertices in $S$, and every vertex in $S$ is adjacent from at least one vertex
in $S$ (the subdigraph induced by $S$ has no isolated vertices). The double
domination number (total $2$-domination number) of a digraph $D$ is the minimum
cardinality of a double dominating set (total $2$-dominating set) in $D$. In
this work, we investigate these concepts which can be considered as two
extensions of double domination in graphs to digraphs, along with the concepts
$2$-limited packing and total $2$-limited packing which have close
relationships with the above-mentioned concepts.
