A Comparative Study of Modified BBO Variants and Other Metaheuristics for Optimal Power Allocation in Wireless Sensor Networks - Advances in Heuristic Signal Processing and Applications - page 89

an island depends critically on the balance between the immigration of new species
onto the island and the emigration of established species.
The BBO algorithm operates upon a population of individuals called islands (or
habitats ). Each island represents a possible solution to the problem in hand. The
fitness of each island is determined by its Habitat Suitability Index ( HSI ), a metric
which determines the goodness of a candidate solution, and each island feature is
called a Suitability Index Variable ( SIV ). Good solutions may have a larger number
of species, which represents an island with a low HSI , compared to poor solutions.
The number of species present on the island is determined by a balance between
the rate at which the new species arrive and the rate at which the old species become
extinct on the island. In BBO, each individual has its own immigration rate ( λ ) and
emigration rate ( μ ). These parameters are affected by the number of species ( S )in
an island and are used to probabilistically share information between islands. Islands
with smaller populations are more vulnerable to extinction (i.e., the immigration rate
is high). But as more species inhabit the island, the immigration rate reduces and the
emigration rate increases. In BBO, good solutions (i.e., islands with many species)
tend to share their features with poor solutions (i.e., islands with few species), and
poor solutions accept a lot of new features from good solutions.
But how might immigration and emigration work on an island? The migration
pattern is determined by the immigration rate ( λ ) at which new species immigrate
to the island. The rate of immigration ( λ ) will decline with the number of species
( S ) present on the island. The maximum immigration rate ( I ) occurs when island
is empty and decreases as more species are added. Once all potential colonists are
on the island, then S S max (maximum number of species the island can support)

and immigration rate must be equal to zero. The immigration rate, when there are S
species in the island, is given by

=

I 1

.

S
S max

λ S =

−

(5.29)

The emigration rate ( μ ), at which populations of established species emigrate,
will increase with the number of species ( S ). The maximum emigration rate ( E )
occurs when all possible species are present on the island (when S S max ), and

must be zero when no species are present. The emigration rate, when there are S
species in the island, is given by

=

μ S = E S
S max

.

(5.30)

Figure 5.2 graphically represents the relationships between the number of species
( S ), emigration rate ( μ ), and immigration rate ( λ ). Over time, the countervailing
forces of emigration and immigration result in an equilibrium level of species rich-
ness. The equilibrium value ( S ∗ ) is the point at which the rate of arrival of species
( λ ) is exactly matched by the rate of emigration ( μ ). We have assumed that μ and λ
are constant linear relationships, but different mathematical models of biogeography
that included more complex variables are presented in [ 15 ].