Biodiversity Conservation and Phylogenetic Systematics

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xii∈{,^01 },∀taxon (3)

yσ∈{,^01 },∀σsplit (4)

Suppose we are given a total budget B.Letcidenoteconservationcostsfortaxon
i. We can then substitute constraint ( 2 ) by the budget constraint

∑ ≤

i

cxii B

(6)

TogetherwithpreviousconstraintswehavetheIPformulationofProblem 2 by:

Wenowmodelviabilityconstraintsthatoperateontaxonvariablesasfollows.
G. sonneratii depends on P.malacense and P.germaini(Fig. 4 ). Therefore, the viability
constraint for G. sonneratii is simply

xxPM+≥PG xGS

This ensures that xGS is 1 (i.e., G. sonneratii is selected for conservation) only if at
least one of xPM and xPGisalso1.Viabilityconstraintsforalltheothertaxaarelisted
in Table 3 .Now,theIPformulationforviabletaxonselectioncanbeobtainedby
simplyincludingviabilityconstraintstoProblem1:

IP for Reserve Selection Problems

ForreserveselectionweencodeasubsetW of m areas by a binary vector (z 1 ,z 2 ,...,zm),
where zr is 1 if area r is present in W, and 0 otherwise. We call zrareavariables.For
the pheasant habitat (Table 1 ) we have eight area variables zID, zLK, zBT, zIN, zPH, zMY,

IP Formulation of Problem 1

Maximizeobjectivefunction(1),subjecttosubsetsizeconstraint(2),binary

constraints (3, 4), split constraints (5) (see Table 3 ).

IP Formulation for Problem 2

Maximizeobjectivefunction(1),subjecttobudgetconstraint(6),binarycon-

straints (3, 4), and split constraints (5) (Table 3 ).

IP Formulation of Problem 3

Maximizeobjectivefunction(1),subjecttosubsetsizeconstraint(2),binary

constraints (3, 4), split constraints (5), and viability constraints (7) (Table 3 ).

Split Diversity: Measuring and Optimizing Biodiversity Using Split Networks