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Ecological
Indicators
29
(2013)
522–528
Contents
lists
available
at
SciVerse
ScienceDirect
Ecological
Indicators
jo
ur
n
al
homep
ag
e:
www.elsevier.com/locate/ecolind
Mathematical
convergences
of
biodiversity
indices
Benjamin
Bandeira, JeanLouis
Jamet, Dominique
Jamet, JeanMarc
Ginoux∗
Université
du
Sud
ToulonVar,
Equipe
d’Ecologie
et
de
Biologie
des
Milieux
Aquatiques
(EBMA),
France
a
r
t
i
c
l
e
i
n
f
o
Article
history:
Received
30
October
2012
Received
in
revised
form
23
January
2013
Accepted
27
January
2013
Keywords:
Diversity
indices
Biodiversity
Ecological
diversity
Community
structure
a
b
s
t
r
a
c
t
Various
indices
are
used
in
the
scientiﬁc
literature
to
describe
biodiversity
changes.
Nevertheless,
the
appropriateness
of
an
index
rather
than
another
to
transcribe
trends
in
biodiversity
of
plankton
commu
nities
is
not
clearly
established.
So,
starting
from
the
deﬁnitions
of
the
diversity
indices
of
Simpson,
GleasonMargalef,
Menhinick,
Brillouin,
Shannon,
Patten,
Piélou
and
Hurlbert,
the
aim
of
this
work
is
to
state,
under
the
assumption
that
the
total
number
of
individual
is
great,
a
mathematical
convergence
between
the
indices
of
Brillouin,
Shannon,
Simpson’s
reciprocal,
Hurlbert
on
the
one
hand
and
between
the
indices
of
Piélou
and
Patten
on
the
other
hand.
More
particularly,
it
will
be
also
established
that
these
last
two
indices
are
complementary
provided
that
the
total
number
of
individual
is
greater
than
the
number
of
species.
GleasonMargalef’s
and
Menhinick’s
indices
will
be
considered
as
independents.
Thus,
such
a
convergence
will
lead
to
propose
a
classiﬁcation
of
these
indices
into
three
great
groups
reducing
their
number
from
eight
to
four.
This
theoretical
result
will
be
then
applied
on
phytoplankton
and
zooplankton
communities
of
two
neighbouring
bays
differently
affected
by
anthropogenic
inputs
in
NW
Mediterranean
Sea
(Toulon
area,
France)
throughout
three
consecutive
annual
cycles.
A
strong
statistical
correlation
between
the
indices
belonging
to
the
same
group
seems
to
conﬁrm
the
validity
of
our
classiﬁcation.
©
2013
Elsevier
Ltd.
All
rights
reserved.
1.
Introduction
In
each
work
concerning
the
study
of
the
diversity
of
commu
nities,
the
use
of
diversity
indices
is
a
necessary
tool
to
calculate
and
quantify
the
diversity
status
(Van
Strien
et
al.,
2012).
In
addi
tion,
these
indices
estimate
biological
and
ecological
quality
of
an
ecosystem
through
the
structure
of
the
community
(Danilov
and
Ekelund,
1999);
they
are
also
possible
indicators
to
monitor
the
level
of
pollution
in
environment
(Washington,
1984).
In
so
far
as
the
use
of
a
single
numerical
index
to
describe
the
structure
of
a
community
and
the
ecological
quality
of
an
ecosystem
oversimpli
ﬁes
its
real
biodiversity,
the
literature
proposes
different
indices
to
study
the
diversity
of
communities
in
aquatic
ecosystems
(Travers,
1971;
Frontier,
1976;
De
Pauw
et
al.,
1992;
Beaugrand
et
al.,
2002).
According
to
Frontier
(1976),
Washington
(1984)
and
Lévêque
and
Mounolou
(2001)
the
concept
of
diversity
of
a
community
covers
two
fundamental
aspects:
the
number
of
species
S
and
the
regularity,
i.e.,
the
more
or
less
equal
or
unequal
way
with
which
individuals
N,
for
a
number
of
species
given,
are
distributed
between
those.
The
principal
challenge
of
second
half
of
the
∗Corresponding
author
at:
Laboratoire
PROTEE,
EA
3819,
B.P.
20132,
F83957
La
Garde
cedex,
France.
Tel.:
+33
685234362.
Email
addresses:
ginoux@univtln.fr,
jmginoux@orange.fr
(J.M.
Ginoux).
twentieth
century
was,
for
considerable
authors
(Margalef,
1958;
Lloyd
and
Ghelardi,
1964;
Piélou,
1966),
to
ﬁnd
an
index
likely
to
connect
these
two
components
of
diversity.
One
of
the
main
problems
with
diversity
indices
was
that
the
components
are
interrelated
and
often
compete
with
each
other.
The
indices
of
diversity
thus
try
to
put
in
adequacy
the
data
on
abundance
with
the
number
of
species
of
a
community
in
only
one
number,
from
which
the
structure
of
the
community
can
the
oretically
be
apprehended.
The
diversity
of
the
species
should
be
a
function
of
the
number
of
species
and
the
distribution
of
the
abun
dance
of
these
species.
It
is
possible
for
diversity
to
grow
while
the
number
of
species
decreases
if
the
distribution
increases.
Diver
sity
under
this
deﬁnition
becomes
an
aspect
of
the
structure
of
the
community
in
which
the
rare
species
are
structurally
of
no
impor
tance.
Diversity
indices
can
be
called
“ecological
indices”
or
“species
diversity”
(for
a
meaningful
difference
between
diversity
and
bio
diversity
see
Marquès,
2001).
But
if
a
diversity
index
is
a
parameter
of
community
structure,
how
does
it
relate
to
community
role
and
function?
After
having
brieﬂy
recalled
deﬁnitions
of
diversity
indices
of
Simpson,
GleasonMargalef,
Menhinick,
Brillouin,
Shannon,
Pat
ten,
Piélou
and
Hurlbert,
a
mathematical
convergence
between
the
indices
of
Brillouin,
Shannon,
Simpson’s
reciprocal
and
Hurlbert
and
between
the
indices
of
Piélou
and
Patten
is
established.
Indices
of
GleasonMargalef
and
Menhinick
are
included
in
the
same
kind
of
group
but
are
considered
as
independent.
Such
a
convergence
1470160X/$
–
see
front
matter
©
2013
Elsevier
Ltd.
All
rights
reserved.
http://dx.doi.org/10.1016/j.ecolind.2013.01.028
B.
Bandeira
et
al.
/
Ecological
Indicators
29
(2013)
522–528
523
leads
us
to
propose
a
classiﬁcation
of
these
indices
into
three
great
groups:
•G1=
{Brillouin,
Shanon,
Hurlbert,
Simpsons
reciprocal},
•G2=
{Piélou,
Patten},
•G3=
{GleasonMargalef,
Menhinick}.
Then,
the
computation
of
diversity
indices
of
phytoplankton
and
zooplankton
communities
of
two
neighbouring
bays
differ
ently
affected
by
anthropogenic
inputs
in
NW
Mediterranean
Sea
(Toulon
area,
France)
throughout
three
consecutive
annual
cycles
enables
to
highlight
a
strong
statistical
correlation
between
indices
belonging
to
the
same
group
which
seems
to
conﬁrm
the
validity
of
our
classiﬁcation.
2.
Methods
2.1.
Phytoplankton
and
zooplankton
samples
Toulon
Bay
is
located
on
the
northwest
Mediterranean
coast
of
France
and
is
divided
by
a
breakwater
in
two
coupled
lit
toral
ecosystems,
Little
Bay
and
Large
Bay,
differently
affected
by
anthropogenic
inputs
and
stresses.
Little
Bay
(sampling
station
S1)
harbours
a
major
commercial
and
military
port
and
is
highly
affected
by
anthropogenic
inputs
(i.e.
organic
compounds,
heavy
metals,
toxic
phytoplankton
species).
Little
Bay
is
greatly
affected
by
raw
sewage
from
the
Toulon
area,
as
well
as
maritime
(military
and
commercial)
trafﬁc.
Little
Bay
contains
high
levels
of
chloro
phyll
a
and
is
characterised
by
a
high
abundance
of
phyto
and
zooplankton,
but
by
a
low
plankton
diversity.
Large
Bay
(sam
pling
station
S2)
is
open
to
the
Mediterranean
Sea.
A
recent
study
(Rossi
and
Jamet,
2008)
conﬁrmed
that
Little
Bay
was
more
polluted
than
Large
Bay,
notably
in
terms
of
metal
con
tamination
in
seawater
and
the
plankton
food
web.
Previous
works
have
also
shown
that
phyto
and
zooplankton
commu
nities
are
less
abundant
but
diversity
is
higher
in
Large
Bay
than
in
Little
Bay.
Twelve
samples
were
collected
from
January
2005
to
December
2007
in
Little
Bay
(S1)
and
Large
Bay
(S2),
respectively.
Phytoplankton
cells
were
sampled
with
a
10
L
Niskin
Bottle
at
3
m
depth.
After
an
inverted
ﬁltration
and
a
sedimentation
column,
phytoplankton
populations
were
identiﬁed
to
species
level
if
pos
sible
and
counted
under
an
inverted
microscope
(400×)
according
to
Lund
method
(Lund
and
Talling,
1957;
Lund
et
al.,
1958)
method.
We
recorded
295
and
420
different
phytoplankton
species
in
Lit
tle
Bay
and
Large
Bay,
respectively.
The
total
abundances
were
more
important
in
the
Little
Bay
with
less
diversity
than
in
Large
Bay.
Bacillariophyceae
dominated
largely
phytoplankton
commu
nity
particularly
in
Little
Bay
and
were
responsible
for
bloom
in
fall.
Zooplankton
organisms
were
carried
out
by
vertical
hauls
from
bottom
to
surface
with
a
plankton
net
(mesh
size
90
m)
equipped
by
a
ﬂowmeter.
Organisms
were
identify
to
species
level
if
possible
and
counted
according
to
Hensen
method.
We
recorded
60
different
zooplankton
species
in
Little
Bay
and
Large
Bay,
respectively.
Our
results
showed
that
in
both
bays
Copepods
were
the
most
abun
dant
taxonomic
group
(globally,
more
than
80%).
In
Little
Bay,
we
recorded
high
densities
of
zooplankton,
with
a
dominant
species
Oithona
nana
(Cyclopoida);
the
zooplankton
community
showed
a
low
diversity
in
this
bay.
At
contrary,
in
Large
Bay,
the
abundance
of
this
community
was
much
lower
than
in
Little
Bay,
and
biodiver
sity
was
high.
We
did
not
record
a
dominant
species
in
this
bay.
For
further
and
detailed
information
see
Jamet
et
al.
(2005)
and
Rossi
and
Jamet
(2008,
2009).
2.2.
Mathematical
convergences
and
statistical
correlations
In
order
to
state
a
mathematical
convergence
between
some
diversity
indices
we
use
the
classical
Stirling
formulae
(1730)
and
the
wellknown
exponential
and
logarithmic
functions
properties.
The
statistical
correlation
analysis
between
two
indices
consid
ered
as
independent
random
variables
involves
a
nonparametric
test
of
Spearman
(1907)
and
Wilcoxon
(1945).
Thus,
computations
of
statistical
coefﬁcients
of
correlation
r
between
two
different
indices
belonging
to
the
same
station
(S1or
S2)
during
each
year
(2005,
2006,
2007)
are
carried
out.
According
to
Frontier
(1980),
the
correlation
coefﬁcient
between
two
variables
(two
indices)
are
considered
as
strong
if
r2≥
3/4.
Moreover,
taking
account
that
the
degree
of
freedom
of
our
study
d.f.
=
12
−
2
=
10
since
we
have
twelve
samples
per
year,
we
have
deduced
from
a
statistical
table
the
corresponding
pvalue
for
each
coefﬁcient
of
correlation.
In
the
following
Sections
2.3–2.7,
diversity
indices
are
presented
in
their
original
formulation
and
according
to
a
classiﬁcation
pro
posed
by
Travers
(1971)
and
Washington
(1984)
which
consists
in
dividing
them
in
ﬁve
different
categories.
S
is
the
number
of
species
in
the
community,
niis
the
number
of
individuals
in
the
i
species,
n
is
the
total
number
of
individuals
in
the
sample,
N
is
the
total
number
of
individuals
in
the
community.
2.3.
The
indices
of
Simpson
At
the
end
of
the
40s
Simpson
(1949)
wished
to
deﬁne
a
mea
sure
of
“concentration
of
the
classiﬁcation”
in
terms
of
population
constants,
inverse
of
the
notion
of
diversity.
1.
Simpson’s
index:
The
initial
formula
was
=S
i=1p2
iwhere
pi=
ni/n
with S
i=1pi=
1
which
represents
the
proportion
of
indi
viduals
in
the
different
species
(Simpson,
1949).
The
modiﬁed
formula
l
is
the
probability
for
two
random
individuals
to
belong
to
the
same
species
and
would
correspond
to
an
inﬁnite
popu
lation,
but
one
can
have
an
unbiased
estimate,
corresponding
to
a
limited
sample
l
=S
i=1ni(ni−
1)/n(n
−
1).
If
n
and
niare
rather
large,
l
can
be
approximated
by
l
=S
i=1p2
i
where
pi=
ni/n.
According
to
Williams
(1964),
the
index
of
Simpson
l
is
indepen
dent
from
any
theory
concerning
the
distribution
of
frequencies
but
is
not
totally
independent
of
the
size
of
the
sample.
Moreover,
this
index
depends
on
the
most
abundant
species
and
gives
less
impor
tance
to
the
rare
species.
For
Washington
(1984),
the
index
l
of
Simpson
measures
the
intraspeciﬁc
competition
in
trophic
level.
Simpson’s
reciprocal
index:
Krebs
(1972)
proposed
another
def
inition
of
Simpson’s
index
l
as
to
be
the
probability
for
two
individuals
randomly
and
independently
determined
to
belong
to
different
species
1
−
l
=
1
−S
i=1p2
iwhere
pi=
ni/n.
2.4.
The
indices
of
GleasonMargalef
and
of
Menhinick
The
ﬁrst
indices
of
speciﬁc
diversity
were
elaborated
in
such
a
way
that
the
observed
sample
ﬁts
at
best
with
a
theoretical
law
of
distribution
of
the
individuals
in
species.
Thus,
Gleason
(1922)
indicated
proportionality
between
the
number
of
species
and
the
logarithm
of
the
surface
sampled
in
a
vegetal
community.
The
rep
resentation
of
the
number
of
species
S
in
function
of
the
logarithm
of
the
total
number
of
individuals
N
often
shows
a
convex
pattern
near
the
origin
indicating
that
some
species
are
quickly
inventoried
and
the
other
much
lower.
Margalef
(1951)
noted
that
the
relation
ship
between
the
number
of
species
and
the
number
of
individuals
may
be
reported
in
semilogarithmic
scale
by
linear
application
which
he
named
relation
of
Gleason.
524
B.
Bandeira
et
al.
/
Ecological
Indicators
29
(2013)
522–528
2.
GleasonMargalef’s
index
d
=
(S
−
1)/loge(N)
:
Starting
from
that
of
Gleason
(1922),
Margalef
(1951)
elaborated
this
index
which
is
based
on
the
hypothesis
on
a
linear
relationship
between
the
number
of
species
S
and
the
logarithm
of
the
total
number
of
individuals
N.
Washington
(1984)
considered
that
this
index
which
depends
on
the
size
of
the
samples
has
no
theoretical
fundament
while
Travers
(1971)
and
Whilm
(1967)
showed
that
this
index
was
a
good
indicator
of
the
speciﬁc
diversity
of
the
community.
Then,
numerical
simulations
realized
by
Boyle
et
al.
(1990)
showed
that
the
index
of
Margalef
was
sensitive
to
the
structure
of
the
community,
particularly
to
low
variations
of
the
number
of
species
leading
to
erratic
answer
to
this
index.
3.
Menhinick’s
index
S/√N
:
Menhinick
(1964)
suggested
a
new
index
to
replace
those
of
GleasonMargalef.
He
thought
to
pro
pose
an
index
that
will
be
independent
from
the
size
of
the
sample,
but
Whilm
(1967)
demonstrated
that
this
index
was
more
erroneous
than
those
of
GleasonMargalef.
Nevertheless,
Karydis
and
Tsirtsis
(1996)
showed
that
this
index
was
very
efﬁ
cient
to
evaluate
eutrophication.
2.5.
The
indices
of
the
theory
of
information
In
the
case
of
indices
derived
from
the
information
theory,
diversity
is
measured
by
the
contents
of
the
information,
or
neg
ative
entropy
(“néguentropy”)
which
grows
with
the
increasingly
complex
organisation,
ordered,
improbable
a
priori,
of
the
system.
According
to
Whilm
(1967),
the
contents
of
information
are
a
mea
surement
of
uncertainty
and
thus
a
reasonable
measurement
of
diversity.
In
1948,
Shannon
(1948)
had
compared
this
uncertainty
to
entropy
formally
equivalent
to
the
entropy
in
thermodynamics.
4. Brillouin’s
index:
Brillouin
(1951)
proposed
an
index
of
diver
sity
measuring
the
average
negative
entropy
by
individual
H
=
logeN!/S
i=1Ni!/N.
According
to
Piélou
(1966),
this
index
represents,
the
quantity
of
information
per
symbol
which
is
con
tained
in
a
great
message.
It
depends
on
the
sample
size
and
can
be
used
only
for
a
completely
known
population.
When
N
and
Ni
are
rather
large,
H
can
be
approximated
by
the
formula
of
Stirling
and
leads
to
the
index
of
Shannon.
5.
Shannon
index
H=
−S
i=1piloge(pi)
where
pi=
ni/n
:
Introduced
by
Shannon
(1948),
this
index
which
can
never
be
calculated,
but
only
estimated,
is
used
in
the
information
theory.
It
corresponds,
with
the
sign
and
the
units,
with
the
measurement
of
the
physical
entropy.
Piélou
(1966)
showed
that
both
Hand
H
are
densitydependent.
According
to
Cairns
(1977),
His
insensitive
with
the
rare
species
which
play
a
sig
niﬁcant
role
in
an
ecosystem.
But
it
is
a
functional
role
whereas
His
an
indication
of
structure
of
the
community.
Indeed
the
EPA
(Environmental
Protection
Agency)
estimates
that
Hdoes
not
allow
to
differentiate
a
community
made
up
of
one
or
two
dominant
species
and
of
some
rare
species,
a
community
made
up
of
one
or
two
dominant
species
and
one
or
two
rare
species.
6.
Redundancy
R
=
(H
max −
H)/(H
max −
H
min)
:
Patten
(1962)
pro
posed
an
assessment
of
predominance
called
Redundancy.
It
represents
the
way
of
which
the
individuals
are
distributed
among
the
species
and
gets
a
measurement
of
the
predominance
of
one
or
some
species.
Whilm
(1967)
considers
that
the
index
H
of
Shannon
and
the
Redundancy
R
of
Patten
should
be
used
on
the
same
community
so
that
R
can
estimate
the
rate
of
abundance
of
the
species
and
R
is
independent
of
the
base
of
the
logarithms.
7.
Evenness
J=
H/H
max =
H/log2(S)
:
This
index
introduced
by
Piélou
(1966)
represents
the
“relative
diversity”,
i.e.
the
ratio
of
the
diversity
observed
with
observable
maximum
diversity
with
the
same
number
of
species.
It
expresses
the
degree
of
equality
in
species
abundance
in
the
sample.
Liljelund
(1977)
regards
it
as
best
measurement
to
detect
the
appearance
of
new
species
during
the
succession.
2.6.
Probability
of
interspeciﬁc
encounters
(P.I.E.)
8.
Hurlbert’s
index:
Hurlbert
(1971)
with
some
ideas
inspired
of
the
article
of
Simpson
proposed
a
new
index
P.I.E.
=
(N/N
−
1) 1
−S
i=1p2
i.
This
index
represents
the
probability
(P)
of
encounter
(E)
interspeciﬁc
(I).
If
an
individual
enters
a
commu
nity
and
meets
in
a
random
way
two
individuals,
P.I.E.
is
the
probability
so
that
they
belong
to
different
species
and
meas
ures
the
importance
of
the
interspeciﬁc
competition
relating
to
the
total
competition
by
supposing
that
the
encounters
are
random
and
that
each
meeting
represents
a
unit
of
competi
tion.
1
−P.I.E.
is
the
intraspeciﬁc
proportion
of
competition.
This
implies
that
for
nonfatal
meetings
in
a
community,
the
index
l
of
Simpson
measures
the
intraspeciﬁc
competition.
When
the
ﬁrst
individual
encounter
risk
becoming
the
subject
of
the
sec
ond
also
meets,
as
in
a
nonlethal
meeting
the
probability
is
what
Hurlbert
regards
as
a
complement
of
the
index
of
Simp
son.
Thus,
Hurlbert
(1971)
afﬁrms
that
its
P.I.E.
index
and
the
reciprocal
index
of
Simpson
proposed
by
Krebs
(1972)
have
the
same
components
of
speciﬁc
richness
and
regularity.
3.
Results
The
aim
of
this
section
is
to
propose,
on
the
basis
of
mathemat
ical
convergences,
a
new
classiﬁcation
of
the
diversity
indices
into
three
great
groups.
3.1.
Indices
of
Brillouin,
Shannon,
Simpson’s
reciprocal
and
Hurlbert:
group
1
3.1.1.
Indices
of
Brillouin–Shannon
It
is
easy
to
state
that
these
two
indices
converge
for
great
values
of
N.
This
result
is
based
on
the
wellknown
Stirling
formulae
(1730)
recalled
below:
loge(N!)
≈
Nloge(N)
−
N
pour
N
1
By
taking
into
account
the
logarithms
properties,
Brillouin’s
index
reads:
H=1
Nloge(N!)−
logeS
i=1
Ni!
=1
Nloge(N!)−
S
i=1
loge(Ni!)
By
replacing
each
term
by
the
Stirling
formulae,
we
have:
H
=1
Nloge(N!)−
S
i=1
loge(Ni!)
=1
NNloge(N)−
N
−
S
i=1
(Niloge(Ni)−
Ni)
B.
Bandeira
et
al.
/
Ecological
Indicators
29
(2013)
522–528
525
Thus,
after
developing,
we
obtain:
H
=1
NNloge(N)−
N
−
S
i=1
(Niloge(Ni)−
Ni)
=
loge(N)−
1
−1
N
S
i=1
(Niloge(Ni)−
Ni)
=
loge(N)−
1
−
S
i=1
Ni
Nloge(Ni)+
S
i=1
Ni
N
By
taking
into
account
that
N
=
S
i=1
Ni,
it
follows:
H
=
loge(N)−
S
i=1
Ni
Nloge(Ni)=
−
S
i=1
Ni
NlogeNi
N
Provided
that
Ni≈
niwhen
N
→
n
we
establish
the
mathemat
ical
convergence
between
Brillouin
and
Shannon
indices.
H
=
loge(N)−
S
i=1
Ni
Nloge(Ni)
=
−
S
i=1
Ni
NlogeNi
N≈
−
S
i=1
ni
nlogeni
n
3.1.2.
Indices
of
Simpson’s
reciprocal
– Shannon
It
is
also
possible
to
state
that
these
two
indices
converge
for
all
values
of
n.
This
result
is
based
on
the
following
formulae:
an=
enloge(a)for
a
>
0,
from
which
we
deduce:
ni
n1
=
eloge(ni/n)
By
using
a
Taylor
series
expansion
of
the
exponential
at
the
ﬁrst
order
we
obtain:
ni
n1
=
eloge(ni/n)≈
1
+
logeni
nif
ni
n
Simpson’
reciprocal
reads:
l
≈
S
i=1ni
n2=
S
i=1ni
n
ni
n
By
replacing
one
of
the
two
ratios
ni/n
by
the
previous
approx
imation,
we
have:
l
≈
S
i=1ni
n2
=
S
i=1ni
n
ni
n≈
S
i=1ni
n
1
+
logeni
n
=
S
i=1
ni
n+
S
i=1
ni
nlogeni
n
But,
since
n
=S
i=1ni,
it
follows:
l
≈
1
+
S
i=1
ni
nlogeni
n
Then,
we
deduce
that
the
index
of
Simpson’s
reciprocal
reads:
1
−
l
1
−
l
≈
1
−1
+
S
i=1
ni
nlogeni
n=
−
S
i=1
ni
nlogeni
n
Another
mathematical
convergence
is
thus
established.
3.1.3.
Indices
of
Simpson’s
reciprocal
–
Hurlbert
It
is
also
possible
to
state
that
these
two
indices
converge
for
great
values
of
N.
In
this
case
Hurlbert’s
index
reads:
P.I.E.
=N
N
−
11
−
S
i=1ni
n2≈
1
−
S
i=1ni
n2
=
1
−
l
Thus,
a
mathematical
convergence
is
established
between:
 the
indices
of
Brillouin
and
Shannon,

the
indices
of
Simpson’s
reciprocal
and
Shannon,

the
indices
of
Simpson’s
reciprocal
and
Hurlbert.
Then,
by
a
simple
transitivity
relationship
we
deduce
that
there
exists
a
mathematical
convergence
between
these
four
indices.
So,
we
propose
to
place
them
in
the
same
group
G1inside
which
we
can
obviously
add
the
index
of
Simpson
which
is
complementary
to
the
index
of
Simpson’s
reciprocal.
So,
it
won’t
be
surprising
to
observe
some
strong
correlations
between
these
different
indices
(see
Section
4).
G1=
{Brillouin,
Shannon,
Hurlbert,
Simpson’s
reciprocal}
3.2.
Indices
of
Piélou’s
Evenness–Patten’s
Redundancy:
group
2
According
to
their
deﬁnition,
indices
of
Evenness
and
of
Redun
dancy
seem
to
be
directly
related
to
that
of
Shannon.
Nevertheless,
the
variability
of
the
number
of
species
S
enables
either
to
include
these
indices
in
the
group
G1or
to
exclude
them.
On
the
other
hand,
when
the
theoretical
value
minimum
of
Htends
to
zero
(H
min ≈
0),
i.e.,
when
N
S,
it
is
easy
to
prove
that
indices
of
Piélou
and
Patten
are
complementary.
If
H
min ≈
0,
we
have:
R
=H
max −
H
H
max −
H
min ≈H
max −
H
H
max =
1
−H
H
max =
1
−
J
Thus,
as
we
will
see
in
the
Section
4
some
strong
negative
corre
lations
will
be
observed
between
these
two
indices.
So,
we
propose
to
place
these
indices
in
the
same
group
G2
G2=Pié
lou,
Patten
3.3.
Indices
of
GleasonMargalef,
Menhinick:
group
3
Concerning
the
indices
of
GleasonMargalef
and
Menhinick,
it
seems
more
difﬁcult
to
state
a
mathematical
convergence
accord
ing
to
their
deﬁnition.
These
indices
would
be
independent
of
the
number
of
individuals
in
the
sample
only
if
the
relationship
between
S
(or
S
−
1)
and
loge(N)
or √N
was
linear.
Unfortunately,
this
is
rarely
the
case.
Thus,
we
propose
to
deﬁne
a
third
group
G3comprising
the
indices
of
GleasonMargalef
and
Menhinick
considered
as
indepen
dents.
G3=
{GleasonMargalef,
Menhinick}
So,
it
appears
that
the
assessment
of
diversity
could
be
done
on
the
basis
of
three
indices.
Each
of
them
could
be
chosen
in
each
group
G1,
G2and
G3according
to
its
ecological
relevance.
Thus,
this
classiﬁcation
enables
to
reduce
from
eight
to
three
the
number
of
diversity
indices.
4.
Discussion
After
having
classiﬁed
the
indices
into
three
great
groups:
G1containing
the
indices
of
Brillouin,
Shannon,
Hurlbert
and
Simpson,
G2containing
the
indices
of
Piélou
and
Patten
and
526
B.
Bandeira
et
al.
/
Ecological
Indicators
29
(2013)
522–528
Table
1
Correlation
coefﬁcient
between
diversity
indices
of
group
1
for
phytoplankton,
zooplankton.
(a)
Phytoplankton
2005
2006
2007
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Brillouin–Shannon
1
1
1
1
1
1
Hurlbert–Simpson−1−1
−1
−1
−1
−1
−1
Hurlbert–Brillouin
0.96
0.92
0.95
0.96
0.95
0.98
Simpson−1–Brillouin −0.96
−0.92
−0.95
−0.96
−0.95
−0.98
Hurlbert–Shannon
0.96
0.93
0.95
0.96
0.95
0.98
Simpson−1–Shannon
−0.96
−0.93
−0.95
−0.96
−0.95
−0.98
(b)
Zooplankton
2005
2006
2007
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Brillouin–Shannon
1
1
1
1
1
1
Hurlbert–Simpson−1−1
−1
−1
−1
−1
−1
Hurlbert–Brillouin
0.92
0.95
0.94
0.94
0.86
0.86
Simpson−1–Brillouin
−0.92
−0.95
−0.94
−0.94
−0.86
−0.86
Hurlbert–Shannon
0.92
0.95
0.94
0.94
0.86
0.86
Simpson−1–Shannon
−0.92
−0.95
−0.94
−0.94
−0.86
−0.86
Table
2
Correlation
coefﬁcient
between
diversity
indices
of
group
2
for
phytoplankton,
zooplankton.
(a)
Phytoplankton 2005 2006
2007
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Piélou–Patten
−1
−1
−1
−1
−1
−1
(b)
Zooplankton
2005
2006
2007
Little
Bay Large
Bay Little
Bay Large
Bay Little
Bay Large
Bay
Piélou–Patten
−1
−1
−1
−1
−1
−1
G3containing
the
indices
of
GleasonMargalef
and
Menhinick
we
have
computed
(with
an
applet
specially
created
for
Math
ematica
and
available
at
http://ginoux.univtln.fr)
the
diversity
indices
of
phytoplankton
and
zooplankton
communities
of
two
neighbouring
bays
differently
affected
by
anthropogenic
inputs
in
NW
Mediterranean
Sea
(Toulon
area,
France)
throughout
three
annual
cycles
(20052006–2007).
These
results
are
presented
in
Tables
1–3.
In
order
to
exemplify
the
mathematical
convergence
established
in
Section
3
and
to
show
that
only
one
index
of
each
group
G1,
G2and
G3is
necessary
to
provide
ecological
signiﬁcance
statistical
correlations
between
indices
belonging
to
each
group
have
been
made
for
each
community,
each
bay
and
each
year.
Let
us
notice
that
in
Tables
1,
2
and
3b,
the
correlation
coefﬁcients
r
>
0.7079.
So,
the
corresponding
pvalue
for
a
degree
of
freedom
equal
to
10
is
p
<
0.01.
In
Table
3a,
the
pvalue
is
indicated
in
parenthesis
below
each
value
of
the
correlation
coefﬁcient.
4.1.
Indices
of
group
1:
Brillouin,
Shannon,
Hurlbert
and
Simpson’s
reciprocal
Concerning
this
ﬁrst
group,
the
correlation
coefﬁcient
between
the
indices
of
Brillouin
and
Shannon
and
between
the
indices
of
Hurlbert
and
Simpson’s
reciprocal
computed
during
this
annual
cycles
(20052007)
for
each
bay
and
each
plankton
community
is
equal
to
∼
100%
(see
Table
1a
and
b).
Accord
ing
to
Washington
(1984),
Simpson’s
reciprocal
and
Hurlbert’s
indices
measure
the
interspeciﬁc
competition
and
“are
thus
closely
related”.
Let’s
notice
that
the
correlations
(respectively
anticorrelations)
between
the
indices
of
Hurlbert–Brillouin,
Table
3
Correlation
coefﬁcient
between
diversity
indices
of
group
3
for
phytoplankton,
zooplankton.
(a)
Phytoplankton 2005
2006
2007
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Margalef–Menhinick
0.23
p
=
0.47
0.54
p
=
0.07
0.36
p
=
0.25
0.39
p
=
0.21
0.57
p
=
0.05
0.86
p
=
0.0003
(b)
Zooplankton
2005
2006
2007
Little
Bay Large
Bay
Little
Bay
Large
Bay
Little
Bay
Large
Bay
Margalef–Menhinick
0.93
0.94
0.94
0.70p
=
0.011
0.92
0.75
p
=
0.005
B.
Bandeira
et
al.
/
Ecological
Indicators
29
(2013)
522–528
527
Simpson’s
reciprocal–Brillouin,
and
between
the
indices
of
Hurlbert–Shannon,
Simpson’s
reciprocal–Shannon
may
be
sim
ply
deduced
from
the
strong
correlations
(respectively
anti
correlations)
between
the
indices
of
Brillouin
and
Shannon
and
between
the
indices
of
Hurlbert
and
Simpson’s
reciprocal.
Thus,
the
mathematical
convergence
is
transcribed
through
the
strong
correlation
that
is
observed.
So,
the
belonging
of
these
indices
to
this
group
seems
to
be
justiﬁed
both
the
mathematical
point
of
view,
from
the
statistical
point
of
view
and
from
the
point
of
view
of
ecological
interpretation.
4.2.
Indices
of
group
2:
Piélou’s
Evenness–Patten’s
Redundancy
For
each
plankton
community,
each
bay
and
during
the
period
(20052007)
a
very
strong
anticorrelation
coefﬁcient
(100%)
between
the
Evenness
and
Redundancy
(see
Table
2a
and
b)
has
been
obtained.
This
statistical
result
is
the
consequence
of
the
mathe
matical
convergence
established
in
the
previous
section.
According
to
Washington
(1984),
Piélou’s
Evenness
is
an
indicator
of
the
struc
ture
of
the
community
while
Patten’s
Redundancy
measures
the
predominance
of
one
or
more
species.
Here
again,
we
can
con
sider
that
the
belonging
of
these
indices
to
the
same
group
is
fully
justiﬁed.
4.3.
Indices
of
group
3:
GleasonMargalef
and
Menhinick
Concerning
this
group,
the
correlation
coefﬁcient
between
the
indices
of
GleasonMargalef
and
Menhinick
computed
during
this
period
(20052007)
for
each
bay
is
very
low
for
phytoplankton
com
munity:
between
23%
to
86%
(see
Table
3a)
while
it
is
very
high
for
zooplankton
community:
between
70%
to
94%
(see
Table
3b).
According
Boyle
et
al.
(1990)
the
index
of
GleasonMargalef
is
sensitive
to
the
structure
of
the
community.
So,
these
statisti
cal
results
suggest
that
the
“resilience”
(a
concept
introduced
by
Holling,
1973)
of
the
phytoplankton
community
is
lower
than
that
of
zooplankton
community.
Especially
in
the
little
bay
which
is
highly
affected
by
anthropogenic
inputs
(i.e.
organic
compounds,
heavy
metals,
toxic
phytoplankton
species).
Moreover,
Danilov
and
Ekelund
(2001)
claimed
that
the
use
of
a
single
index
in
coastal
water
eutrophication
studies
can
lead
to
erratic
conclusions.
That’s
why,
they
suggested
combining
application
of
Menhinick’s
index
with
other
ecological
indices.
Thus,
this
conﬁrms
the
idea
to
keep
these
indices
as
independents
in
the
same
group.
So,
under
the
assumption
that
the
total
number
of
individ
ual
is
great
(N
1
and
N
S
respectively),
we
conclude
that
a
mathematical
convergence
between
the
indices
of
Brillouin,
Shan
non,
Simpson’s
reciprocal,
Hurlbert
and
between
the
indices
of
Piélou
and
Patten
has
been
established
in
this
work.
Although
they
have
been
included
in
the
same
kind
of
group,
indices
of
Gleason
Margalef
and
Menhinick
have
been
considered
as
independents.
Such
a
convergence
led
us
to
propose
a
classiﬁcation
of
these
indices
into
three
great
groups:
•G1=
{Brillouin,
Shannon,
Hurlbert,
Simpsons
reciprocal},
•G2=
{Piélou,
Patten},
•G2=
{GleasonMargalef,
Menhinick}.
The
computation
of
diversity
indices
of
phytoplankton
and
zooplankton
communities
of
two
neighbouring
bays
differently
affected
by
anthropogenic
inputs
in
NW
Mediterranean
Sea
(Toulon
area,
France)
throughout
three
consecutive
annual
cycles
has
enabled
to
highlight
a
strong
statistical
correlation
between
indices
belonging
to
the
same
group
(G1,
G2or
G3)
conﬁrming
thus
the
mathematical
convergence
analytically
established
and
the
valid
ity
of
our
classiﬁcation.
These
mathematical
results
show
that
the
indices
belonging
to
the
same
group
are
completely
equivalent
(except
for
the
group
3)
under
the
assumption
considered.
So,
we
propose
to
choose,
for
the
assessment
of
biodiversity
in
marine
ecosystems,
only
one
index
among
those
of
each
group
G1,
G2and
G3according
to
its
ecological
relevance
reducing
thus
their
number
from
eight
to
four.
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