Biodiversity Conservation and Phylogenetic Systematics

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Thissplitsetisvisualizedinaphylogeneticsplitnetwork(Fig.2b). The major
difference to trees is that the interior nodes of a split network cannot be regarded as
representingancestraltaxa.Instead,theweightofasplitA|B indicates the amount
ofdifferencebetweenthetaxonsetA and B. A split is visualized by a single edge or
a set of parallel edges. The former indicates that the split does not conflict any other
splits, while the latter indicates at least one conflict. Therefore, two conflicting splits
arevisualizedbyaparallelogram.Forexample,σ 14 (incyan,Fig. 2 ) and σ 15 (in pink)
contradict each other on the placement of P. emphanum and P. malacense. This
disagreementgeneratesanarrowparallelogramatthebasalPolyplectron.
Ifmorethantwosplitsareindisagreement,thesplitnetworkwillshowmultiple
connectedparallelograms.Forexample,σ 17 (inred,Fig. 2 ) conflicts with σ 19 (in
green) and σ 20 (in yellow). σ 19 also contradicts σ 18 (in blue). Therefore, σ 17 , σ 18 , σ 19
and σ 20 are visualized by three red, two blue, three green, and two yellow parallel
edges,respectively.ThisgeneratesthreeparallelogramswithinGallus(Fig.2b).
Noteverysplitsetcanbevisualizedintwodimensions.Forexample,assuming
that we had a third tree that places G. gallus at the basal Gallus lineage. This would
introduce one split contradicting with both σ 17 and σ 19. These triple-wise conflicting
splits are depicted by a three dimensional parallelepiped. The resulting split network
is not easily visualized anymore. However, for the following it suffices to directly
workonthesplitset(Fig.2a).

The Measure of Split Diversity

Given a split set Σ,theSDofataxonsubsetY is defined as the sum of the weights
λofallsplitsseparatingtaxainY. Here, a split AB| ∈Σseparates Y if YA∩ and
YB∩ are both non-empty. Thus, we get

SDY

Y

()

:

= ∑

σΣ∈ σ

λσ

separates

To illustrate, given ΣinFig. 2 , for Y={P. malacense, P. germaini, P. emphanum,
G. lafayetii} we have SD()Y =+λλ 346 ++λλ 81 ++λλ 3141 +...+λ 9 , where λλi= σi
is defined as the average of the corresponding branch lengths in TCYB and TDCoH 3.
Here, contradicting splits such as σ 17 and σ 19 are considered in the SD
computation.
IfthesplitsetΣcorrespondstoatree(i.e.noconflictingsplitsexistinΣ), then
SDisequivalenttoPD.ThedefinitionofSDthereforegeneralizesPD.Forthis
reason we focus on SD for the remaining of the chapter.

Biodiversity Optimization Problems

Conservationproblemsmainlyfallintotwocategories:taxonselectionandreserve
selection(Fig. 3 ),wheretheconservationtargetsareeithertaxaorgeographical
areas,respectively.UnderPD,thesimplesttaxonselectionproblem(Faith 1992 ) is

Split Diversity: Measuring and Optimizing Biodiversity Using Split Networks