2.4. Proximity of parcels and species populations
Some authors incorporate clusters of close parcels into the concept
of nature reserve. These clusters must have a total area and
populations of different species sufficiently large. This is the case
of Toth et al. (2009) who propose a model that requires the listing
of possible clusters. We present here a model that incorporates the
concept of clusters in the mathematical program itself. Specifically,
a cluster of parcels is a set of parcels such that the maximum distance
between two parcels of the set is less than or equal to dmax.
We assume that a species will persist in the reserve if the reserve
involves a cluster of parcels with a total population of the considered
species greater than a certain threshold. The objective is to select
an optimal subset of parcels, for example of minimum area, to
protect all species considered. Let pik be the population of species
ek in the parcel si, bk be the threshold for survival of ek, and ai be
the area of the parcel si. Let xik be a Boolean variable that is equal
to 1 iff the parcel si is selected to ensure the survival of the species
ek. Note that a parcel is included in the reserve iff it is used to protect
at least one of the species. The problem can be formulated by
the program