Hurricanes are complex systems that carry large amounts of energy. Their
impact often produces natural disasters involving the loss of human lives and
materials, such as infrastructure, valued at billions of US dollars. However,
not everything about hurricanes is negative, as hurricanes are the main
source of rainwater for the regions where they develop. This study shows a
nonlinear analysis of the time series of the occurrence of hurricanes in the
Gulf of Mexico and the Caribbean Sea obtained from 1749 to 2012. The
construction of the hurricane time series was carried out based on the
hurricane database of the North Atlantic basin hurricane database (HURDAT)
and the published historical information. The hurricane time series provides
a unique historical record on information about ocean–atmosphere
interactions. The Lyapunov exponent indicated that the system presented
chaotic dynamics, and the spectral analysis and nonlinear analyses of the
time series of the hurricanes showed chaotic edge behavior. One possible
explanation for this chaotic edge is the individual chaotic behavior of
hurricanes, either by category or individually regardless of their category
and their behavior on a regular basis.
Introduction
Hurricanes have been studied since ancient times, and their activity is
related to disasters and loss of life. In recent years, there has been
considerable progress in predicting their trajectory and intensity once
tracking has begun, as well as their number and intensity from one year
to the next. However, their long-term and very short-term prediction remains
a challenge (Halsey and Jensen, 2004), and the damage to both materials and
lives remains considerable. Therefore, it is important to make a greater
effort regarding the study of hurricanes in order to reduce the damage they cause. The
periodic behavior of hurricanes and their relationships with other natural
phenomena have usually been performed with linear-type analyzes, which have
provided valuable information. However, we decided to make a different
contribution by carrying out a nonlinear analysis of a time series of
hurricanes that occurred in the Gulf of Mexico and the Caribbean Sea, as
the dynamics of the system are controlled by a set of variables of low
dimensionality (Gratrix and Elgin, 2004; Broomhead and King, 1986).
One of the core sections of this work was the elaborate time series that was
built, especially for the oldest part of the registry, for which it was
possible to compile a substantial and robust collection. This provided our time
series with an amount of data with which it was possible to perform the desired analysis;
otherwise, it would have been impossible to study this natural phenomenon
via nonlinear analysis.
Different methods have been used in the analysis of non-linear,
non-stationary and non-Gaussian processes, including artificial neural
networks (ASCE Task Committee, 2000; Maier and Dandy, 2000; Maier et al.,
2010; Taormina et al., 2015). Chen et al. (2015) use a hybrid neural network
model to forecast the flow of the Altamaha River in Georgia; Gholami et
al. (2015) simulate groundwater levels using dendrochronology and an
artificial neural network model for the southern Caspian coast in Iran. Furthermore, theories of deterministic chaos and fractal structure have
already been applied to atmospheric boundary data (Tsonis and Elsner, 1988;
Zeng et al., 1992), e.g., to the pulse of severe rain time series (Sharifi et
al., 1990; Zeng et al., 1992) and to tropical cyclone trajectory
(Fraedrich and Leslie, 1989; Fraedrich et al., 1990). Natural phenomena occur within different contexts; however, they often exhibit common
characteristics, or may be understood using similar concepts.
Deterministic chaos and fractal structure in dissipative dynamical systems
are among the most important nonlinear paradigms (Zeng et al., 1992). For a
detailed analysis of deterministic chaos, the Lyapunov exponent is utilized
as a key point and several methods have been developed to calculate it. It is
possible to define different Lyapunov exponents for a dynamic system. The
maximal Lyapunov exponent can be determined without the explicit construction
of a time-series model. A reliable characterization requires that the
independence of the embedded parameters and the exponential law for the
growth of distances can be explicitly tested (Rigney et al., 1993; Rosenstein
et al., 1993). This exponent provides a qualitative characterization of the
dynamic behavior and the predictability measurement (Atari et al., 2003). The
algorithms usually employed to obtain the Lyapunov exponent are those
proposed by Wolf (1986), Eckmann and Ruelle (1992), Kantz (1994) and
Rosenstein et al. (1993). The methods of Wolf (1986) and Eckmann and
Ruelle (1992) assume that the data source is a deterministic dynamic
system and that irregular fluctuations in time-series data are due to
deterministic chaos. A blind application of this algorithm to an arbitrary
set of data will always produce numbers, i.e., these methods do not provide a
strong test of whether the calculated numbers can actually be interpreted as
Lyapunov exponents of a deterministic system (Kantz et al., 2013). The
Rosenstein et al. (1993) method follows directly from the definition of the
Lyapunov maximal exponent and is accurate because it takes advantage of all
available data. The algorithm is fast, easy to implement and robust to
changes in the following quantities: embedded dimensions, data set size,
delay reconstruction and noise level. The Kantz (1994) algorithm is similar
to that of Rosenstein et al. (1993).
We constructed a database of occurrences of hurricanes in the Gulf of Mexico
and the Caribbean Sea to perform a nonlinear analysis of the time series,
the results from which can aid in the construction of hurricane occurrence
models, which in turn will help to reinforce prevention measures
for this type of hydrometeorological phenomenon.
Materials and methodsData set description
A detailed analysis of historical reports was carried out in order to obtain the annual time series of hurricane
occurrence, from category one to five on the Saffir–Simpson
scale, in the study region from 1749 to 2012. The time series was composed using
the historical ship track of all vessels sailing close to registered
hurricanes, the aerial reconnaissance data for hurricanes since 1944 and the
hurricanes reported by Fernández-Partagas and Díaz (1995a, b,
1996a, b, c, 1997, 1999). All of the abovementioned information in addition to the
database of the HURDAT re-analysis project (HURDAT is the official record of
the United States for tropical storms and hurricanes occurring in the
Atlantic Ocean, the Gulf of Mexico and the Caribbean Sea) was used in a comparative
way in order to build our time series (Fig. 1), which is currently the longest time
series of hurricanes for the Gulf of Mexico and the Caribbean Sea. This
makes our series ideal for performing a nonlinear analysis, which would be
impossible with the records available in other regions.
Hurricanes between 1749 and 2012. The dashed line shows the linear
trend (after Rojo-Garibaldi et al., 2016).
Historical hurricanes were included only if they were reported in two or more
databases and met both of the following criteria: the reported hurricanes
that touched land and those that remained in the ocean; on the other hand,
the followed hurricanes were studied considering their average duration and
their maximum time (9 and 19 days, respectively). This was done in order to
avoid counting more than one specific hurricane reported in different places
within a short period time; to do this, we followed the proposed method by
Rojo-Garibaldi et al. (2016).
Phase diagrams corresponding to the time series of hurricanes that
occurred between 1749 and 2012 in the Gulf of Mexico and the
Caribbean Sea. The x axis in the four plots indicates the time lag (τ).
Data reduction and procedures
Before performing the nonlinear analysis of the time series, we removed the
trend; thus, the series was prepared according to what is required for this
type of analysis. To uncover the properties of the system, however, requires
more than just estimating the dimensions of the attractor (Jensen et al.,
1985); therefore, three methods were applied in this study:
The Hurst exponent is a measure of the independence of the time series
as an element to distinguish a fractal series. It is basically a statistical
method that provides the number of occurrences of rare events and is usually
called re-scaling (RS) rank analysis (Gutiérrez, 2008).
According to Miramontes and Rohani (1998), the Hurst exponent also provides
another approximation that can be used to characterize the color of noise,
and could therefore be applied to any time series. The RS helps to find
the Hurst exponent, which provides the numerical value which makes it possible
to determine the autocorrelation in a data series.
The Lyapunov exponent is invariant under soft transformations, because it
describes long-term behavior, providing an objective characterization of the
corresponding dynamics (Kantz and Schreiber, 2004). The presence of chaos in
dynamic systems can be solved using this exponent, as it quantifies the
exponential convergence or divergence of initially close trajectories in the
state space and estimates the amount of chaos in a system (Rosenstein et al.,
1993; Haken, 1981; Wolf, 1986). The Lyapunov exponent (λ) can take
one of the following four values: λ < 0, which corresponds to a
stable fixed point; λ= 0, which is for a stable limit cycle;
0 < λ < ∞, which indicates chaos; and λ=∞, a Brownian process, which agrees with the fact that the entropy of
a stochastic process is infinite (Kantz and Schreiber, 2004).
The
iterated function analysis (IFS) is an easier and simpler way to visualize
the fine structure of the time series because it can reveal correlations in
the data and help to characterize its color, referring color to the type
of noise (Miramontes et al., 2001). Together with the Lyapunov exponent, the
phase diagrams, the false close neighbors method, the space-time separation
plot, the correlation integral plot and the correlation dimension were taken
into account, the latter two to identify whether the system attractor was a
fractal type or not. It is important to compute the Lyapunov exponent, so we used
the algorithms proposed by Kantz (1994) and Rosenstein et al. (1993) to
do so.
The mutual information method (a): the x axis
indicates the time lag against the mutual information index (AMI) and the
arrow indicates the first, most pronounced minimum with a value of
τ= 9. The autocorrelation function (b), the x axis
indicates the time lag versus the value of the autocorrelation function, and
the arrow denotes where the first zero of the function τ= 10 was
obtained.
Results and discussions
Figure 1 shows the evolution of the number of hurricanes from 1749 to 2012
and the linear trend. To have a qualitative idea of the behavior of the
number of hurricanes that occurred in the Gulf of Mexico and the Caribbean
Sea from 1749 to 2012, a phase diagram was created using the ”delay
method” (Fig. 2). This was also used to elucidate the time lag for an
optimal embedding in the data set. The optimal time lag (τ) obtained
visually from Fig. 2 was equal to nine, since it was the time in which the
curves of the system were better divided. We must not forget that this was
only a visual inspection, and the delay time is obtained quantitatively
by other methods. In our case, the hurricane dynamics were not
distinguished through the phase diagram; however, as any hurricane
trajectory starts at a close point location on the attractor data set which
diverges exponentially, the phase diagram is a primary evidence of a chaotic
motion according to Thompson and Stewart (1986).
The most robust method to identify chaos within the system is the Lyapunov
exponent. Prior to obtaining the exponent, it was necessary to calculate the
time lag and the embedding dimension, and for the latter, the Theiler window
was used. The time lag was obtained via three different methods:
The method of constructing delays, which is observed visually in Fig. 2.
The method of mutual information, which yields a more reliable result
as it takes nonlinear dynamic correlations into account; in this study, the delay
time was obtained by taking the first minimum of the function – in this case
τ=9.
The autocorrelation function method, which is based
solely on linear statistics (Fig. 3).
There are two ways to obtain the time lag from the autocorrelation function:
the first zero of the function, and
the moment in which the
autocorrelation function decays as 1/e (Kantz and Schreiber,
2004).
We used the criterion of the first zero because the Hurst exponent (H=0.032) indicated that it was a short memory process; therefore, the
criterion of the first zero is the optimal method in this type of case. Using
this method, the value that was obtained was τ= 10. The value of
this parameter is very important, because if it turns out to be very small,
then each coordinate is almost the same and the reconstructed trajectories
look like a line (the phenomenon is known as redundancy). If the delay time
is quite large, however, then due to the sensitivity of the chaotic movement,
the coordinates appear to be independent and the reconstructed phase space
looks random or complex (a phenomenon known as irrelevance) (Bradley and
Kantz, 2015).
False close neighbors with a time lag of 10, where the embedding
dimension of 5 has a 9.4 % and the embedding dimension of 4 has a
16.66 % false close neighbors (lower line). False close neighbors with a
time lag of 9, where the embedding dimension of 5 has a 20.15 % and the
embedding dimension of 4 has a 20.12 % false close neighbors (upper
line). The values in each line indicate the optimal dimension for each lag.
The Hurst exponent helps us to identify the criteria to find a time lag, and
also describes the system behavior (Quintero and Delgado, 2011). This could
indicate that the system does not have chaotic behavior; however, the
remaining methods have indicated the opposite, and as previously mentioned,
the Lyapunov exponent is considered the most appropriate method for this type
of data set. Therefore, different methods will provide different results, but
the time series will indicate the best method and the result we should use.
Lyapunov exponent with m= 4, τ= 9 and
m= 5, τ= 10, with the Kantz method (a). Lyapunov
exponent with the same values with the Rosenstein method (b).
The correlation dimension D2 corresponding to the occurrence of
hurricanes in the years 1749–2012 in the Gulf of Mexico and the Caribbean
Sea. Curves for different dimensions of the attractor
(y axis) (a). Same information for the D2 with a logarithmic
scale on the x axis (b),
It was possible to observe the difference in the time lag obtained through
the autocorrelation function and the mutual information; however, it is
necessary to use only one result. Through the space-time separation graphic
and the false close neighbors method, we obtained embedding dimensions of m=4 for a τ= 9 and m=5 for τ= 10, and the Theiler
window with a value of W= 16 for τ= 9 and W= 18 for
τ= 10 (Fig. 4). The choice of this window is very important so as
not to obtain subsequent spurious dimensions in the attractor. According to
Bradley and Kantz (2015), the Theiler window ensures that the time spacing
between the potential pairs of points is large enough to represent a
distributed sample identically and independently.
The idea of the false close neighbors algorithm is that at each point in the
time series, St‾ and its neighbor Sj‾ should be
searched in a m-dimensional space. Thus, the distance
St-Sj is calculated iterating both
points, given by the following:
Ri=Si+1-Sj+1Sl‾-SJ‾.
If Ri is greater than the threshold given by Rt, then SJ has
false close neighbors. According to Kennel et al. (1992), a value of
Rt= 10 has proven to be a good choice for most data sets, but
a formal mathematical proof for this conclusion is not known; therefore, if
this value does not give convincing results, it is advisable to repeat the
calculations for several Rt (Perc, 2006). In our case, this value gave
relevant results. It may have some false close neighbors even when working
with the correct embedding dimension. The result of this analysis may depend
on the time lag (Kantz and Schreiber, 2004). In a similar fashion to the delay
time, the value of the embedment dimension is crucial not only for the
reconstruction of the phase space but also to obtain the Lyapunov exponent.
Choosing a large value of m for chaotic data will add redundancy and consequently
affect the development of many algorithms such as the Lyapunov exponent
(Kantz and Schreiber, 2004).
The Lyapunov (λ) exponents were obtained using the Kantz and
Rosenstein methods and took the time lag, the embedding dimension and the
Theiler window as the main values; however, an election of the
neighborhood radius for the exploration of trajectories was also made, as
well as the points of reference and the neighbors near these points. The
modification of these parameters is important to corroborate the invariant
characteristic of the Lyapunov exponent. The Kantz (1994) method using a
value of m=4 and τ= 9 gave us an exponent of
λ= 0.483, while for m=5 and τ= 10 the exponent was
λ= 0.483. As λ is a positive value, it was inferred
that our system is chaotic. In addition, the value of λ obtained for
both imbibing dimensions was the same, suggesting that our result is
accurate. Using the Rosenstein et al. (1993) method, the value obtained for
m=4 and τ=9 was λ= 0.1056, and for m=5 and
τ= 10, the exponent was λ= 0.112 (Fig. 5).
There was a difference between placing the attractor in an embedding
dimension of m=4 and one of m= 5; a better unfolding of the attractor
in the embedding dimension was observed in m=4 and τ= 9. This
value of τ was obtained with the mutual information method, which,
according to Fraser and Swinney (1986) and Krakovská et al. (2015),
provides a better criterion for the choice of delay time than the value
obtained by the autocorrelation function.
It was possible to obtain the correlation dimension D2 (Fig. 6) and the
correlation integral (Fig. 6) using the embedding dimension, the delay time
and the Theiler window, following the method of Grassberger and Procaccia
(1983a, 1983b). This was done in order to obtain the possible dimensions of
the attractor. It should be noted that there is a whole family of fractal
dimensions, which are usually known as Renyi dimensions, but these are based
on the direct application of box-counting methods, which demands significant
memory and processing and the results of which can be very sensitive to the length
of the data (Bradley and Kantz, 2015). That is why we chose to use the dimension and
integral correlation, which according to Bradley and Kantz (2015) is a
more efficient and robust estimator.
The iterated functions system (IFS) test applied to the time
series of the number of hurricanes that occurred in the Gulf of Mexico and
the Caribbean Sea between the years 1749 and 2012.
The right panel on Fig. 7 shows the slope trend of the majority of the slopes
of the correlation integral (ε). In the range of
1 < ε < 10, we are required to have straight
lines as an indicator of the self-similar geometry. The value obtained here
corresponds to D2=2.20 which is the aforementioned slope value. Another
method to see the attractor dimension is the Kaplan–Yorke dimension
(DKY), which is associated with the spectrum of Lyapunov exponents and
is given by the following:
DKY=k+∑i=1kλiλk+1,
where k is the maximal integer, such that the sum of the k major
exponents is not negative. The fractal dimension with this method yielded a
value of DKY= 2.26, which is similar to the one obtained
previously.
Even when all the requirements necessary to apply the nonlinear analysis to
our time series are present, one final requirement must be fulfilled to know
whether we can obtain a dimension and whether the complete spectrum of
Lyapunov exponents (another method to visualize chaos) still needs to be
employed.
Eckmann and Ruelle (1992) discuss the size of the data set required to
estimate Lyapunov dimensions and exponents. When these dimensions and
exponents measure the divergence rate with near-initial conditions, they
require a number of neighbors for a given reference point. These neighbors
may be within a sphere of radius (r) and of a given diameter (d) of the
reconstructed attractor.
We then have the requirements for the Eckmann and Ruelle (1992)
condition to obtain the Lyapunov exponents as
logN>Dlog1ρ,
where D is the dimension of the attractor, N is the number of data points
and rd=ρ. For ρ=0.1 in Eq. (3), N may be chosen such
that
N>10D.
Our time series met this requirement; therefore, it supports our previous
results.
The attractor dimension was mainly obtained because this value tells us the
number of parameters or degrees of freedom necessary to control or understand
the temporal evolution of our system in the phase space and helps us to determine
how chaotic our system is. Using the previous methods, a final fractal
dimension of D2= 2.2 was obtained. Following the embedding laws, it
stands to reason that m>D2 (Sauer and Yorke, 1993; Kantz and Schreiber, 2004;
Bradley and Kantz, 2015). The criterion of Ruelle (1990) was used to
corroborate that the obtained dimension of the attractor is reliable, where
N=10D22; once the data fulfill this
requirement, we can say that the dimension of the attractor is reliable.
Finally, the results indicated that at least three parameters are needed to
characterize our system, since the 2.2 dimension indicates that the attractor
dimension falls between two and three.
The spectrum of the Lyapunov exponent gives 0.09983, -0.07443, -0.23387
and -0.73958; therefore, the total sum is λi=-0.9480, and
according to the previous theory, it is enough have at least one positive
exponent in the spectrum of our system in order to have chaotic behavior.
Finally, the total sum of the spectrum of Lyapunov exponents was negative,
indicating that there is a stable attractor, as mentioned previously.
However, since the stable attractor was not easily distinguished, we used a
final method in order to confirm if our system presented some chaotic dynamic
behavior. This method comprised the iterated functions system test (IFS)
(Fig. 7).
Using Fig. 7, it can be observed that the points representing our system
occupy the entire space; according to the IFS test, there are two possible
explanations:
The distribution belongs to a white noise signal and in systems without
experimental noise, the point distribution gives a single curve (Jensen et
al., 1985). However, the previous Hurst exponent obtained was not equal to
zero; therefore, the white noise was also discarded with the autocorrelation
function.
The system is chaotic with high dimensionality. So far, our
results have converged on the occurrence of hurricanes in the Gulf of Mexico
and the Caribbean Sea being a chaotic system, so it is feasible to adopt the
second explanation. Conversely however, our Lyapunov exponent figure was not
flat and it did not seem to flatten as the dimension of embedding increased,
which, according to Rosenstein et al. (1993), would mean that our system is
not chaotic; although the Lyapunov exponent increased with the decrease in
the embedment dimension, which is, again, a characteristic of chaotic
systems. It was then also possible to obtain a dimension of the attractor and
a positive Lyapunov exponent.
Our results were not easy to interpret because the series presented
certain periodic characteristics in an
oscillatory fashion and simultaneously showed chaotic behavior. According to
Rojo-Garibaldi et al. (2016), the series of hurricanes which had spectral
analyzes carried out presented strong periodicities that correspond to
sunspots, which are believed to have caused the periodic behavior mentioned above.
According to Zeng et al. (1990), the spectral power analysis is often used to
distinguish a chaotic or quasi-periodic behavior of periodic structures and
to identify different periods embedded in a chaotic signal. Although, as
Schuster (1988) and Tsonis (1992) mention, the power spectrum is not only
characteristic of a process of deterministic chaos but also of a linear
stochastic process. In our case, this behavior was not observed in the
spectra obtained, which allowed us to detect periodic signals. The spectra
identify two types of behavior in our system. On one hand, there are periodic behaviors
associated with external forcing, such as the sunspot cycle, giving the
system sufficient order to develop; whilst on the other hand, external forcing
presents a chaotic behavior, which gives the system a certain disorder and allows it to be
able to adapt to new changes and evolve. The IFS test showed that the
occurrence of hurricanes in the Gulf of Mexico and the Caribbean Sea is
chaotic with high dimensionality. Fraedrich and Leslie (1989) analyzed the
trajectories of cyclones in the region of Australia and calculated the
dimensionality of this process, obtaining a result of between six and eight, i.e.,
a chaotic process of high dimensionality, which is similar to what we find
with the IFS method. Halsey and Jensen (2004) furthermore postulate
that hurricanes contain a large number of dimensions in phase space.
One possible explanation is localized within a boundary where chaos and
order are separated; this boundary is commonly known as the ”edge of
chaos” (Langton, 1990; Miramontes et al., 2001). Miramontes et al. (2001)
found this type of behavior in ants of the genus Leptothorax, when studying
them individually and in groups. In the former, the behavior was
periodic, whilst in the latter, the behavior was chaotic. In our case, we
believe that the chaotic behavior is due to the individual behavior or the
hurricane category, as the high dimensionality suggested by the IFS test
agrees with the high dimensionality reported by Fraedrich and Leslie (1989)
obtained by studying the trajectories of cyclones – that is, by studying
them individually – while the periodic response is due to the behavior of
hurricanes as a whole.
Finally, an entropy test was performed using non-linear
methods and locally linear prediction (making the prediction at one
step), with both methods showing a predictability value of 2.78 years. The locally
linear prediction method was applied as follows: the last known state of the
system, represented by a vector x= [x(n), x(n+τ),…, x(n+ (m-1) τ)], was determined, where m is the embedment dimension and τ
is the delay time. We then found the p nearby states (usually close neighbors
of x) of the system which represent what has happened in the past, these were
obtained by calculating their distances to x. The idea is then to adjust a
map that extrapolates x and its neighboring p to determine the following
values (Dasan et al., 2002). Based on the above, the value of the embedding
dimension and the delay time were changed. Different values of m were used in
order to elucidate the most accurate result; this was obtained with a
dimension of m= 4 and τ= 9, which are the values for which the
attractor of the system was obtained. Therefore, a good prediction is
possible until t=t0+3 (Fig. 8). Furthermore, if we get away from the
measured data (reported hurricanes) the uncertainty grows in an oscillatory
way. For the first two data (2013 and 2014), the absolute error in the
prediction (observed value – predicted) is less than 0.2, for the third and
fourth value (2015 and 2016) it is between 0.2 and 0.3 and for 2017 the
error is much greater and gives an overestimation of one hurricane (Fig. 8).
An important result of this study is that it allows one to establish the
predictability range of a system.
Prediction of the hurricanes number in the Gulf of Mexico and the
Caribbean Sea by means of non-linear methods, and the entropy test. The
solid black line represents the number of hurricanes observed from 2013 to
2017. The black points are the prediction and the triangles are the error in
the prediction, considered as the observed values minus the predicted values.
Conclusions
The results obtained with the nonlinear analysis suggested a chaotic
behavior in our system, mainly based on the Lyapunov exponents and
correlation dimension, among others. However, the Hurst exponent indicated
that our system did not follow a chaotic behavior. In order to be able
to corroborate our results, we employed the IFS method, which led us to
believe that the hurricane time series in the Gulf of Mexico and the Caribbean
Sea from 1749 to 2012 had a chaotic edge. It is important to emphasize that
this study was prepared as an attempt at understanding the behavior of the
occurrence of hurricanes from a historical perspective, as this type of
phenomenon is part of an ocean–atmosphere interaction that has been changing
over time, hence the value of our contribution. However, we are aware that
from the time the study was conducted to the present date there are new
records, which will make it possible to carry out new studies and apply new
methods.
Data availability
The data are available upon request from the author.
Author contributions
All the authors contributed equally to this work.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was financially supported by the Instituto de Ciencias del Mar y
Limnología de la Universidad Nacional Autónoma de México,
projects 144 and 145. BR-G is grateful for the CONACYT scholarship that
supported her study at the Posgrado en Ciencias del Mar y Limnología,
Universidad Nacional Autónoma de México.
Edited by: Vicente Perez-Munuzuri Reviewed by: three
anonymous referees
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