  In the NP-hard Optimizing PD with Dependencies (PDD) problem, the input
consists of a phylogenetic tree $T$ over a set of taxa $X$, a food-web that
describes the prey-predator relationships in $X$, and integers $k$ and $D$. The
task is to find a set $S$ of $k$ species that is viable in the food-web such
that the subtree of $T$ obtained by retaining only the vertices of $S$ has
total edge weight at least $D$. Herein, viable means that for every predator
taxon of $S$, the set $S$ contains at least one prey taxon. We provide the
first systematic analysis of PDD and its special case s-PDD from a
parameterized complexity perspective. For solution-size related parameters, we
show that PDD is FPT with respect to $D$ and with respect to $k$ plus the
height of the phylogenetic tree. Moreover, we consider structural
parameterizations of the food-web. For example, we show an FPT-algorithm for
the parameter that measures the vertex deletion distance to graphs where every
connected component is a complete graph. Finally, we show that s-PDD admits an
FPT-algorithm for the treewidth of the food-web. This disproves a conjecture of
Faller et al. [Annals of Combinatorics, 2011] who conjectured that s-PDD is
NP-hard even when the food-web is a tree.
