134
that zooplankton organisms exhibit a range of
behaviour including fractional Gaussian motion,
fractional Gaussian noise and pure random noise.
3
2.2.3 The Fractal Nature
of Behavioural States
Self-affine techniques based on the analysis of
frequency distributions of behavioural states
were used to infer the response of zooplankton to
a range of stressors. They include considerations
of the scaling properties of the probability distribution functions (PDFs) of either the time tx
spent in a specific behavioural state x (i.e.
p t
t
x
x
c
( )∝
− ) (Schmitt et al. 2006) or the velocity
v x used to define different behavioural states x
(i.e. p v
v
x
x
c
( )∝
− ) (Michalec et al. 2010). A few
studies investigated the scaling properties of the
cumulative probability distribution functions
(CDFs) of move duration greater than a determined duration t ( P t T
t
≤
(
) ∝
−f 1
) (Seuront and
Leterme 2007) and move lengths L greater than a
determined length l ( N l L
l
≤
( ) ∝
−f 2
) (Seuront
2010a, b, 2011). These studies consistently found
a decrease in the exponents ϕ 1 and ϕ 2 , hence in
behavioural complexity, for a range of copepod
species exposed to sublethal concentrations of
naphthalene (Seuront and Leterme 2007) and the
water-soluble fraction of diesel oil (Seuront
2010a, b; Fig. 3b). In contrast, the exponent c was
shown to increase in Eurytemora affinis following
a short-term exposure to sublethal concentrations
of nonylphenols (Michalec et al. 2013a, b).
Note that the behaviour of an organism alternating between two behavioural states can also
be assessed through the construction of a binary
sequence z t (i) for each behavioural activity taken
from continuous observations. When a specific
activity is observed, z t (i) = 1, and z t (i) = 0 otherwise.
The resulting time series of binary sequences can
further be integrated as w t
z i
i
i
N
t
( ) =
( )
=
∑
1
, where N
3 For instance, the velocity components of Clausocalanus
furcatus were both characterised by β ≈ 0 (Uttieri et al.
2008), indicative of a random process without internal
serial correlation. In contrast, β ranged from 0.30 to
0.75 in Temora longicornis (Moison et al. 2012) and 1.4 to
1.5 in Pseudodiaptomus annandalei (Dur et al. 2010).
is the number of behavioural observations.
The temporal pattern of the integrated variable
w i (t) can then be analysed with self-affine
techniques such as spectral analysis.
3
From Fractals
to Multifractals: A Step
Further in Zooplankton
Stress Assessment
3.1
From Fractals to Multifractals
A measure (i.e. a physical quantity such as mass,
energy, a number of individuals or more
specifically the distance displaced by a copepod;
Fig. 1b) has to be distinguished from its geometric support, which might or might not have a
fractal geometry (Rutherford et al. 2004). Then,
if a measure has different fractal dimensions on
different parts of the support, the measure is a
multifractal. Multifractals are hence a generalisation of fractal geometry initially introduced to
describe the relationship between a given quantity and the scale at which it is measured. Whilst
fractal geometry describes the complexity of a
given pattern with the help of only one parameter
(the fractal dimension), multifractals characterise
its detailed variability by an eventually infinite
number of sets, each with its own fractal
dimensions.
An intuitive interpretation of multifractals is
based on the spatial structure of modern cities
(Rutherford et al. 2004). Consider a city viewed
strictly from above, it can be considered as a
succession of built (buildings) and unbuilt (streets
and parks) areas. The only available information
is hence the distribution of the built and the
unbuilt areas. This is the geometric support of the
city. Now, change the angle of vision by taking a
position not directly above the city, but from the
side. The city initially made of built and unbuilt
areas is now a set of buildings with different
heights. This is the measure we are now interested
in. It is now possible to estimate the distribution
of a wide range of building heights. Each height
will (eventually) be characterised by a fractal
dimension, hence the concept of multifractals.
L. Seuront
that zooplankton organisms exhibit a range of
behaviour including fractional Gaussian motion,
fractional Gaussian noise and pure random noise.
3
2.2.3 The Fractal Nature
of Behavioural States
Self-affine techniques based on the analysis of
frequency distributions of behavioural states
were used to infer the response of zooplankton to
a range of stressors. They include considerations
of the scaling properties of the probability distribution functions (PDFs) of either the time tx
spent in a specific behavioural state x (i.e.
p t
t
x
x
c
( )∝
− ) (Schmitt et al. 2006) or the velocity
v x used to define different behavioural states x
(i.e. p v
v
x
x
c
( )∝
− ) (Michalec et al. 2010). A few
studies investigated the scaling properties of the
cumulative probability distribution functions
(CDFs) of move duration greater than a determined duration t ( P t T
t
≤
(
) ∝
−f 1
) (Seuront and
Leterme 2007) and move lengths L greater than a
determined length l ( N l L
l
≤
( ) ∝
−f 2
) (Seuront
2010a, b, 2011). These studies consistently found
a decrease in the exponents ϕ 1 and ϕ 2 , hence in
behavioural complexity, for a range of copepod
species exposed to sublethal concentrations of
naphthalene (Seuront and Leterme 2007) and the
water-soluble fraction of diesel oil (Seuront
2010a, b; Fig. 3b). In contrast, the exponent c was
shown to increase in Eurytemora affinis following
a short-term exposure to sublethal concentrations
of nonylphenols (Michalec et al. 2013a, b).
Note that the behaviour of an organism alternating between two behavioural states can also
be assessed through the construction of a binary
sequence z t (i) for each behavioural activity taken
from continuous observations. When a specific
activity is observed, z t (i) = 1, and z t (i) = 0 otherwise.
The resulting time series of binary sequences can
further be integrated as w t
z i
i
i
N
t
( ) =
( )
=
∑
1
, where N
3 For instance, the velocity components of Clausocalanus
furcatus were both characterised by β ≈ 0 (Uttieri et al.
2008), indicative of a random process without internal
serial correlation. In contrast, β ranged from 0.30 to
0.75 in Temora longicornis (Moison et al. 2012) and 1.4 to
1.5 in Pseudodiaptomus annandalei (Dur et al. 2010).
is the number of behavioural observations.
The temporal pattern of the integrated variable
w i (t) can then be analysed with self-affine
techniques such as spectral analysis.
3
From Fractals
to Multifractals: A Step
Further in Zooplankton
Stress Assessment
3.1
From Fractals to Multifractals
A measure (i.e. a physical quantity such as mass,
energy, a number of individuals or more
specifically the distance displaced by a copepod;
Fig. 1b) has to be distinguished from its geometric support, which might or might not have a
fractal geometry (Rutherford et al. 2004). Then,
if a measure has different fractal dimensions on
different parts of the support, the measure is a
multifractal. Multifractals are hence a generalisation of fractal geometry initially introduced to
describe the relationship between a given quantity and the scale at which it is measured. Whilst
fractal geometry describes the complexity of a
given pattern with the help of only one parameter
(the fractal dimension), multifractals characterise
its detailed variability by an eventually infinite
number of sets, each with its own fractal
dimensions.
An intuitive interpretation of multifractals is
based on the spatial structure of modern cities
(Rutherford et al. 2004). Consider a city viewed
strictly from above, it can be considered as a
succession of built (buildings) and unbuilt (streets
and parks) areas. The only available information
is hence the distribution of the built and the
unbuilt areas. This is the geometric support of the
city. Now, change the angle of vision by taking a
position not directly above the city, but from the
side. The city initially made of built and unbuilt
areas is now a set of buildings with different
heights. This is the measure we are now interested
in. It is now possible to estimate the distribution
of a wide range of building heights. Each height
will (eventually) be characterised by a fractal
dimension, hence the concept of multifractals.
L. Seuront