Journal of Modern Physics, 2011, 2, 627-635
doi:10.4236/jmp.2011.226073 Published Online June 2011 (http://www.SciRP.org/journal/jmp)
Predicting Ecosystem Response to Perturbation from
Thermodynamic Criteria
Vasthi Alonso Chávez1, Karo Michaelian2
1School of Mathematics, University of Southampton, Southampton, UK
2Institúto de Física, UNAM, México, Cto. de la Investigación Científica, Cuidad Universitaria, Mexico City, Mexico
Received February 1, 2011; revised April 17, 2011; accepted May 5, 2011
Abstract
The response of ecosystems to perturbations is considered from a thermodynamic perspective by acknowl-
edging that, as for all macroscopic systems and processes, the dynamics and stability of ecosystems is sub-
ject to definite thermodynamic law. For open ecosystems, exchanging energy, work, and mass with the en-
vironment, the thermodynamic criteria come from non-equilibrium or irreversible thermodynamics. For
ecosystems during periods in which the boundary conditions may be considered as being constant, it is
shown that criteria from irreversible thermodynamic theory are sufficient to permit a quantitative prediction
of ecosystem response to perturbation. This framework is shown to provide a new perspective on the popula-
tion dynamics of real ecosystems.
Keywords: Population Dynamics, Ecosystem Perturbation, Non-Equilibrium Thermodynamics, Enrichment
Paradox
1. Introduction
Most ecosystems are under considerable stress, having
been perturbed by human intrusions including, popula-
tion reduction, the introduction of foreign species, habitat
fragmentation, contamination, and general global warm-
ing. Fortunately, ecosystems can often recover from per-
turbations and can even evolve and adapt to new bound-
ary conditions [1]. However, successful recovery de-
pends on the inherent stability of the system, which is a
complex function of the individual interactions among all
the participating species and among species and their
environment. Given that typical ecosystems contain over
3000 species [2], understanding the nature of this stabil-
ity, and thus predicting ecosystem response to perturba-
tion, is far from trivial, but indispensable for developing
a quantitative appro ach to conservation.
Predicting ecosystem response to perturbation is,
therefore, one of the most scientifically taxing yet im-
portant questions o f our time. Most of present ecosyste m
dynamics theory is based on Lotka-Volterra-type equa-
tions which are empirically inspired equations incorpo-
rating one-body parameters, such as inherent birth and
death rates, and two-body effects of one species popula-
tion on another through a “community matrix” of coeffi-
cients representing; competition, predator-prey, symbio-
sis, or neutral interaction. Environment limitations are
incorporated through additional parameters such as the
“carrying capacity”. This two-body “community matrix”
approach, although widely recognized for its usefulness
in revealing the general spectrum of the dynamics of
model ecosystems [3], has had limited application to
predicting real ecosystem response to perturbation.
This is primarily due to the fact that the community
matrix is obtained from fits to time-series population
data, and therefore it can be expected to be representative
of nature only within the limited range of the available
populati on data.
A further problem debilitating the community matrix
approach is that it is a two-body appro ach, while species
interactions are really of a many-body nature. These
many-body effects are usually absorbed within so called
“environmental factors” which are included in the dy-
namical equations as fitted constants. However, these
“constants” are not really constant for perturbed ecosys-
tems and anyhow fail to endow the resultant 2-body equa-
tions with the true dynamics of inherently many-body
natural ecosystems.
A lack of a quantitative ecosystem theory means that
today’s ecosystem health is usually surmised by making

V. A. CHÁVEZ ET AL.
628
painstaking field counts of the populations of particular
species and then, somehow arbitrarily, deciding whether
or not to include those species on an “in danger of ex-
tinction” list. Such one dimensional vigilance of ecosys-
tems is not satisfactory for a number of reasons: first, it
fails to treat the ecosystem as an integrated whole and
could thereby miscalculate the gravity of the situation
about to unfold; second, since many ecosystems have a
natural cyclical, or even chaotic, but stable dynamics, it
may be difficult to distinguish normal stable periodic or
chaotic behavior from a dangerous fall toward extinction;
third, our human p erspective tends to fo cus on species in
which the individuals are physically large, easily ob-
servable, or likable, but no t necessarily those key species
that are most important to the stability of an ecosystem.
Most important, however, is the fact that present ecosys-
tem theory provides little information for designing an
integral approach to arresting an impending catastrophe,
other than, perhaps, suggesting that the endangered spe-
cie be protected by law.
There is clearly a need for attempts at constructing
more quantitative approaches to population modeling
based on fundamental science and measurements that can
lead to prediction of ecosystem dynamics in regions in
population space fo r which no data exist. There has been
a growing realization that such a quantitative theory of
ecosystems will ultimately have a thermodynamic basis
[4-10]. The reasons are compelling: First, thermody-
namic laws derive from symmetry principles inherent in
nature and thus are universal, applicable in suitable form
to all macroscopic systems and processes, irrespective of
the types of interactions involved. Second, thermody-
namics deals with a much reduced set of macroscopic
variables which can be related with measurable ecosys-
tem variables (e.g. populations) involved in the dynami-
cal patterns observed in Nature. Third, the strictly hier-
archal control assumed in traditional ecology is sup-
planted by a more integral approach in which systems are
thermodynamically embedded [8,9,11]. Finally, a num-
ber of problems and paradoxes existing in traditional
ecosystem theory appear to have a simple resolution in
terms of thermodynamic directives [7,10].
The objective of this paper is to demonstrate that, for
particular ecosystems under constant boundary condi-
tions, a non-equilibrium thermodynamic framework for
the population dynamics can lead to explicit predictions
concerning ecosystem response to perturbation. In the
following section we briefly outline the thermodynamic
framework for treating ecosystems which has been pre-
sented elsewhere [10]. In Section 3 we present a simple
model ecosystem and demonstrate how its population
dynamics and stability characteristics are determined by
thermodynamic constraints and criteria relating to energy,
work, and mass flow among the species populations, and
with the external environment. In Section 4, we perturb
this ecosystem and analyze the response as predicated on
the basis of non-equilibrium thermodynamic formalism.
Finally, in Section 5, we discuss how our thermodynamic
framework may have relevance in explaining the par-
ticular population dynamics observed in many real eco-
systems.
2. Thermodynamic Framework
To avoid misinterpretation at the outset, it is prudent to
make a clear distinction between two existing, but fun-
damentally different, thermodynamic frameworks. Equi-
librium thermodynamics deals with isolated systems and
the fundamental state variable governing the evolution of
the isolated system toward the stable equilibrium state is
the total entropy, S. Irreversible thermodynamics deals
with open systems or processes, such as ecosystems,
which exchange energy, work, and mass between com-
ponent parts and with the environment. Here, the vari-
able governing the evo lution toward the stable stationary
state (for constant bound ary conditio ns) is the time varia-
tion of the total entropy of the system, dS/dt. Our
framework is based on the latter, irreversible thermody-
namics, and we employ only that part of this framework,
known as classical, developed by Lars Onsager [12] and
Illya Prigogine [13], which has been extensively v erified
empirically.
As for any open system, the ti me variation of the total
entropy of the ecosystem may be divided into a part due
to the internal entropy production arising from irreversi-
ble processes occurring within the ecosystem itself, and a
second part due to the flow of entropy into, or out of, the
ecosystem from the external environment [13],
dd d
ddd
ie
SS S
ttt
. (2.1)
All macroscopic systems and processes, including
ecosystems, are subject to definite thermodynamic law.
The primary among these is the second law of thermo-
dynamics which states that the internal production of
entropy due to irreversible processes occurring within the
system must be positive definite,
d0
d
iS
t. (2.2)
For the case of ecosystems under the condition of con-
stant external cons traints (see [10] for justification of th is
condition for a large class of ecosystems), classical irre-
versible thermodynamic theory states [13] that the sys-
tem will eventually arrive at a thermodynamic stationary
state in which all macroscopic variables, including the
total entropy, are station ary in time,

V. A. CHÁVEZ ET AL.
629
d0
d
S
t. (2.3)
Therefore, from (2.1), at the stationary state,
dd
,
dd
ie
SS
tt
 (2.4)
implying from Equation (2.2) that
d0
d
eS
t. (2.5)
Maintaining an ecosystem in a stable thermodynamic
stationary state thus requires a continuous negative flow
of entropy into the system. This was emphasized by
Schrödinger [14], but was first recognized by Boltzmann
[15].
The internal entropy production dd
iSt can be writ-
ten as a sum of generalized thermodynamic forces X
multiplied by their corresponding generalized thermo-
dynamic flows J, as stated by Prigogine [13]
d,
d
iS
X
J
t



 (2.6)
(for example, the entropy production due to heat flow
can be obtained b y multiplying

1
Q
X
T gradient of
the inverse temperature, and JQ = heat flow).
The separation of the entropy production into its
components of thermodynamic forces and flows is
somewhat arbitrary and can often be chosen for conven-
ience in resolving a specific problem. However, there are
a number of conditions that must be met for any particu-
lar choice. The first condition is that the product of the
force and flow gives units of entropy production, and the
second is that symmetry aspects must be respected, for
example, a scalar force cannot give rise to a vector flow
[16]. We have shown [10] that ecosystem dynamics can
be treated consistently within this irreversible thermody-
namic framework by assigning the generalized thermo-
dynamic forces to the species populations (
X
p



)
(where

represents the species type) and the general-
ized flows to the flows of entropy (dd
J
St

) (due to
flows of energy, work, or mass, to or from species

, see
below).
Within this framework, it was shown [10] that the
ecological steady state, prevalent in nature [2,17], has the
stability characteristics of the thermodynamic stationary
state. In view of this, we have made the formal assertion
that the ecological steady state is just a particular case of
the more general thermodynamic stationary state [10].
A further criterion from classical irreversible thermo-
dynamic theory, considered by Prigogine as the most
general result of irreversible thermodynamic theory, and
valid for constant external constraints, is that the rate of
change of the internal entropy production, due to changes
in the generalized forces X (the populations), is negative
semi-definite; the general evolutionary criterium [13],
d d0where
dd
i XS
tt

. (2.7)
Equation (2.7) implies that, under constant boundary
conditions, all natural chang es in the species populations
must be in such a manner so as to reduce the internal
production of entrop y. This is a powerful auxiliary crite-
rion on ecosystem response to perturbatio n and it will be
shown in §3 and that this, together with the second law
of thermodynamics, and the fact that a system with con-
stant external constraints must arrive at a thermodynamic
stationary state, effectively determines the population
dynamics that the ecosystem can assume. In this manner,
we can predict the dynamical response of the ecosystem
to perturbation, be it either toward recovery, toward a
new dynamics, or toward extinction.
3. Model Ecosystem
We now present our thermodynamic framework for an
over simplified but illustrative 3-species model ecosys-
tem, including up to 3-body interaction terms. Two of the
populations, 1 p and 2 p, are conside red variable, wh ile
the third, 3 p is fixed, and represents the constant
boundary conditions over the ecosystem; such as the
constant supply of nutrients due to a primary producer
species.
The total entropy brought into the ecosystem or car-
ried out of it through one-body transport processes can
be written as,
1
d,
d
ne
eSp
t


 


 (3.1)
where the sum is over all n = 3 species and p

is the
population of species

. e

 represents the average
rate of exchange, or flow, of entropy with the external
environment per individual per unit time of species

as a result of energy (including heat), work, or matter
flow (see below).
Similarly, the internal entropy production, including
production and exchange of entropy between individuals
of the species, may be written as a many-body ex pansion
[10],
''''' '''
1'1',''1 (4)
0.
i
nn n
dS
dt
pp ppO
 
 
 



 





 
(3.2)
The


represent the entropy production per indi-

V. A. CHÁVEZ ET AL.
630
vidual of species

due to one-body irreversible proc-
esses occurring within the individual such as; photosyn-
thesis, transpiration, respiration, metabolism, etc. The
'


 represent the entropy production and exchange due
to 2-body interactions between individuals of species

and '

(e.g. those involved in competition, preda-
tor-prey, symbiosis, etc.); '''


 corresponds to similar
but 3-body interactions, and (4)O represents the en-
tropy produc tion due to 4-body and higher order in terac-
tions (for example, those required for the functioning of
societies). The 4-body and higher order N-body terms
will be neglected in what follows since they would nor-
mally be small as they require increasingly improbable
(except for social species) N-body localization in space
and time.
Equation (2.7), for the time change in the entropy
production due to a change in the generalized forces
X
(the populations) then becomes,
'' ''''''
''''
dd
0.
xpp pp  


 



 (3.3)
The dynamics of the ecosystem can now be deter-
mined from Equations (2.4), (3.1), (3.2) and (3.3) once
the '
s
 are specified.
The '
s
 represent entropy production and flow be-
tween individuals of the species and between individuals
and their environment. A general expression for this en-
tropy flow comes from the Gibb’s equation and results
from the flow of energy, work, and matter [13]. For ex-
ample, the energy per individual per unit time taken in
through photosynthesis de

, or the heat dq

per indi-
vidual per unit time transported to the external environ-
ment, the work done on the environment per unit time
dPV

at constant pressure P, and the matter compo-
nents (e.g. nutrients) of type

taken in or given out by
species

, dn


, of chemical potential



, give for
the rate of entropy exchange per individual with the en-
vironment,
dd d 1
dd
d
1
d
eeq V
P
Tt Tt
n
Tt








 
 (3.4)
where th e temp er atu re T (of the participating individuals)
may be approximated as being constant for the ecosys-
tem [5]. A similar expression can be written for the '



,
representing the entropy production and exchange be-
tween individuals of species

and '

, i.e. in terms of
the energy, work, and matter exchanged due to the
2-body interactions between individuals of the species.
The effect on the entropy flows due to a simultaneous
interaction of a third individual (three-body effects) of
species ''

is cons idered in the parameter '''


 .
Determining the '
s

for a real ecosystem therefore
requires the determination of the flows of energy de,
heat dq, volume dV, and mass dn

of type

be-
tween individuals of the participating species and be-
tween individuals and the external environment. Such a
determination is possible in principle but obviously dif-
ficult in practice. Some of the experimental details for
obtaining these types of flow measurements between the
individuals, and between the individuals and their envi-
ronment, can be found in [5] and references therein, as
well as in [18,19].
The entropy production due to the interaction between
individuals '



is difficult to measure. However,
much work has already been performed on determining
the energy flow in ecosystems [20,21] and, as Equation
(3.4) shows, determining the entropy flow requires
merely extending this program to determining also mass
[21] and work flow between individuals, and between
individuals and the environment.
In the absence of real ecosystem data concerning the
production of entropy and the exchange of entropy, here
we generate these coefficients  for a model ecosys-
tem subject to thermodynamic law with the aid of a ge-
netic algorithm [22]. The algorithm begins by randomly
generating sets of initial values for the entropy produc-
tion and exchanges, the '
s

, within fixed ranges. The
algorithm then evaluates the fitness of each set by
checking to see if the population dynamics, as deter-
mined by criteria (3.2) and (3.3), leads to a viable sta-
tionary state in the long time limit, i.e. one with
limd ddd
ie
tSt St 

 (as required by classical irre-
versible thermodynamics for constant external con-
straints, see Equation (2.4)) and with dd
iSt large and
positive (consistent with what is known about the natural
evolution of biotic systems to ever higher entropy pro-
duction regimes (see [6,13]). The best sets of '
s

are
selected and evolved through mutation and crossover,
optimizing (maximizing), the fitness function,
dd ,
dddd
i
ie
St
St St  (3.5)
which, as required, is large for dd
iSt large, and for
limd ddd
ie
tSt St 

.
An example of a set of '
s
 so obtained is given in
Appendix A. Using this set of s' and a starting
population set of populations (11000p,22000p

)
with 3
p fixed at 2000 (the constant external constraint),
and generating infinitesimal variations of the populations
1
dp and 2 dp at random (3 d0p) while only accept-
ing those sets


dp which satisfy the thermodynamic
criteria of Equations (3.2) and (3.3), leads to the stable
cyclic attractor (oscillating) population dynamics as

V. A. CHÁVEZ ET AL.
631
shown in Figure 1.
4. Response to Perturbation
The response to perturbation obtained under the dictates
of the thermodynamic criteria, Equations (3.2) and (3.3),
once the ecosystem has arrived at a stationary state,
Equation (2.4), are shown in Figure 1. This stationary
state, for the particular set of ’s obtained, is a cyclic
attractor. The first perturbation, affected at time k = 5e6
by reducing the population of p1 to 40, can be seen to
have little effect on the ecosystem, a full recovery of the
populations is obtained rapidly. The second perturbation
affected by increasing the population of p1 to 800 at time
k = 8e6 also produces only a small transient effect.
However, if the ecosystem is perturbed in the same
manner but at time k = 1.3e7, population p1 goes nega-
tive (as does the internal production of entropy dd
iSt)
implying the extinction of the species and the thermody-
namic non-viability of the ecosystem. The timing of a
perturbation affected on an ecosystem with cyclic attrac-
tor population dynamics therefore appears to be crucial
in deciding the fate of the ecosystem.
Similar results are obtained if population p2 is per-
turbed. A perturbation of the ecosystem at time k = 1.7e7
by increasing population p2 to 1600 has no long term
consequences. However, when population p2 is reduced
to 600 at time k = 2.0e7, the internal production of en-
tropy and the popu lation p1 both go negative, the ecosys-
tem again becomes untenable.
Figure 2 plots the dynamics of the ecosystem in
population space p1:p2 for 50 different initial popula-
tions. It is apparent from this figure that the recovery or
not of an ecosystem from a particular perturbation de-
pends on the region in population space into which the
ecosystem is perturbed.
Perturbation into the “regions of danger” marked on
the figure leads to either one of the populations going
negative, or to the internal production of entropy going
negative. Both of these results are non physical and
would foretell the collapse of the ecosystem in nature.
Interestingly, these regions of danger do not necessarily
correspond to regions of small population.
We next examine the response of our model ecosys-
tem to permanent changes in the boundary conditions.
Figure 3 shows the ecosystem dynamics under new
boundary conditions of p3 reduced to 200, do wn from its
originally fixed value of p3 = 2000, implying a less nega-
tive flow of entropy into the ecosystem. Without allow-
ing time for the interaction coefficients  to evolve in
response to the new boundary conditions, the cyclic at-
tractor then becomes a point attractor as shown in Figure
3. However, as can be seen from this figure, the point
attractor is not a thermodynamic stationary state since
the internal production of entropy is no longer equal to
the negative of the external flow of entropy (Equation
Figure 1. (a) Populations p1 (solid line) and p2 (dashed line) as a function of time k for the ecosystem given in Appendix A in
response to various (see text) perturbations. (b) diS/dt (solid line) and −deS/dt (dashed line) as a function of time. (c) Trajec-
tory in population space showing the cyclic attractor dynamics and the effects of 4 distinct perturbations (see text).
p2
p1

V. A. CHÁVEZ ET AL.
632
Figure 2. The dynamics in population space for 50 different initial populations showing that there are “regions of danger” in
population space for which a perturbation into these regions would cause the system to become untenable, either because one
of the populations extinguishes (goes negative) or because the internal production of entropy becomes negative (nonphysical).
Figure 3. Ecosystem dynamics for the case where the fixed boundary conditions have been changed from p3 = 2000 to p3 = 200.
The population dynamics becomes that of a point attractor. However, the system is not in a thermodynamic stationary state
since diS/dt ≠ −deS/dt. A subsequent perturbation of p1 to 40 at k = 5e6, which did not affect the stability previously (see Fig-
ure 1), now leads to diS/dt going negative, violating the 2nd law of thermodynamics. The vulnerability of the ecosystem has
thus been increased by reducing the external constraint of p3.
p2
p1

V. A. CHÁVEZ ET AL.
633
(2.4) is no longer satisfied). The ecosystem fitness func-
tion, Equation (3.5), is no longer at a local maximum
value and the system, given time, would evolve its en-
tropy production and exchange coefficients (the set

)
until reaching a new stationary state where the produc-
tion and external flow of entropy are once again equal.
Note that in the perturbed state with the new external
constraint, p3 = 200, the same small perturbation of re-
ducing th e population p1 to 40 at time k = 5e6, which did
not have any lasting affect on the ecosystem previously
(Figure 1), now results in the collapse of the ecosystem
since it moves it into the non-physical, thermodynamic-
cally prohibited, regime of negative internal entropy
production (Figure 3).
Figure 4 shows the opposite effect of increasing the
flow of negative entropy into the ecosystem, obtained by
increasing the value of the fixed external condition to p3
= 2200. Without allowing time for the interaction coeffi-
cients  to evolve, the dynamics remains that of a cy-
clic attractor but the orbit of the attractor increases sig-
nificantly, bringing the population p1 very close to zero
at one point in its orbit. A slight perturbation of popula-
tion p2 at time k = 5e6 is sufficient to cause the popula-
tion p1 to pass through zero and thereby cause the col-
lapse of the ecosystem. We believe that this is a possible
thermodynamic explanation of the “enrichment para-
dox”[23]; contrary to naïve expectation, an increase in
the inflow of nutrients is often observed to make an eco-
system more vulnerable to perturbation. This will be
considered in detail in a forthcoming paper [24].
We have also verified [25] that had the set of

’s been
chosen such that the thermodynamic stationary state cor-
responded to a point attractor in population space (some-
times referred to as an equilibrium state in the ecological
literature), then increasing the value of the fixed external
condition p3 leads to population oscilla- tions of p1 and p2.
This may have relevance to the sudden outbreak of large
population oscillations of certain spe- cies.
5. Summary and Conclusions
Acknowledging that ecosystems, like all macroscopic
processes, are subject to definite thermodynamic law, we
have demonstrated that under constant external con-
straints, the response of ecosystem s to perturbation can, in
principle, be predicted. The thermodynamic criteria which
direct the dynamics come from non-equilibrium thermo-
dynamic theory. They are; 1) the system must eventually
arrive at a thermodynamic stationary state, Equation (2.4),
2) the internal production of entropy must be positive
definite, in accord with the second law of thermodynam-
ics, Equation (2.2), and 3) any natural change in the
populations must be in such a manner so as to reduce the
internal production of entropy of the entire system,
Prigogine’s general evolution criterion, Equation (2.7 ) .
Figure 4. Ecosystem dynamics for the case where the fixed boundary condition has been increased from p3 = 2000 to p3 = 2200.
In this case, the population dynamics is that of a cyclic attractor with a larger orbit, bringing p1 very close to zero at times. A
subsequent small perturbation of p2 to 800 at k = 5e6 leads to diS/dt going negative and population p1 passing through zero.
We believe that this is the thermodynamic origin of the “enrichment paradox” [23].
p1
p2

V. A. CHÁVEZ ET AL.
634
In the absence of data on the entropy production and
exchange of real ecosystems, we considered a simple
model ecosystem generated by evolving interaction coef-
ficients (representing entropy production and exchange)
through selection with a fitness function favoring the
thermodynamic criteria identified above. We then stud-
ied the response of this model ecosystem to perturbation
of the populations under the same thermodynamic crite-
ria. We found that there ex ist regions in populatio n space
for which perturbation into these regions leads to the
eventual extinction of one or more of the species, or to a
negative internal production of entropy. The latter vio-
lates the second law of thermodynamics and would lead
to ecosystem collapse since physical maintenance proc-
esses require positive production of entropy. An impor-
tant finding is that these regions in population space are a
general feature of the thermodynamic framework and do
not necessarily correspond to regions of small population.
Assigning species to “in danger of extinction” lists, solely
on the basis of the smallness of their populations may
therefore not be an effective conservation policy. Our
proposed approach, based on thermodynamic criteria,
predicts the dynamics over all of population space and
thus leads to quantitative elements for providing more
informed policies for responding to ecosystem perturba-
tion.
Increasing or decreasing the negative flow of entropy
(natural resources) into the ecosystem has the effect of
increasing or decreasing respectively both the amplitude
of the orbit of the attractor in population space and the
internal production of entropy of the system. In either
case, this results in a more vulnerable ecosystem since
the populations pass closer to zero or the internal pro-
duction of entropy may more easily become negative
respectively. We believe that this result gives a thermo-
dynamic explanation of the “enrichment paradox”.
Ecosystems are composed of many thou sands of inter-
acting species and unless the ecosystem can be reduced
to a few key species, the details of the dynamics is, un-
doubtedly, significantly more complicated than the re-
sults obtained here. However, our thermodynamic frame-
work can be straightforwardly applied to a much larger
and more complex ecosystem simply by measuring all
the entropy production and exchange coefficients

for
all the species involved. Work in this direction is under-
way [24].
In conclusion, as for all macroscopic processes, eco-
systems are subject to definite thermodynamic law. For
constant external constraints, these laws are sufficient to
determine ecosystem response to perturbation. Our
analysis of the population dynamics based on thermody-
namic law and the formulation of the interaction coeffi-
cients in terms of physical and measurable quantities (the
production and exchange of entropy) is one step toward a
more quantitative theory of ecosystems.
6. Acknowledgements
K. M. is grateful for the financial support of DGAPA-
UNAM project numbers IN-118206, and IN- 112809,
and for financial assistance while on a 2007 sabbatical
leave. The hospitality afforded during this leave by the
Faculty of Science, Universidad Autonoma del Estado de
Morelos, Mexico, is greatly appreciated.
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