e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Investigation of thermodynamic properties in an ecological
model developing from ordered to chaotic states
Sandip Mandal a , Santanu Ray b,∗ , Samar Roy a , Sven Erik Jørgensen c
a
b
c
Department of Physics, Visva Bharati University, Santiniketan 731235, India
Department of Zoology, Visva Bharati University, Santiniketan 731235, India
DFH, Institute A, Miljokemi, Universitesparken 2, 2100 Copenhagen O, Denmark
a r t i c l e
i n f o
a b s t r a c t
Article history:
In any ecosystem the equilibrium condition may gradually turn into a chaotic situation
Received 22 August 2005
for different reasons. In this paper a three species (phytoplankton, zooplankton and fish)
Received in revised form
model is proposed. Rate parameters are changed according to the change of the size of the
17 December 2006
organisms. The model is run in different conditions with different sizes of zooplankton by
Accepted 19 December 2006
increasing the grazing rate and consequently decreasing the half saturation constant of this
Published on line 25 January 2007
organisms following allometric principles. The system exhibits different states (equilibrium
point–stable limit cycle–doubling and ultimately chaos) by gradual increment of zooplank-
Keywords:
ton grazing rate and decrease of half saturation constant. This paper also tests the high level
Exergy
of exergy (thermodynamic goal function) of the systems at the edge of oscillation before
Phytoplankton
entering into the chaotic situation. This high level of information supports the hypothesis
Zooplankton
that the system can coordinate the most complex behavior in these situations.
© 2007 Elsevier B.V. All rights reserved.
Fish
Order
Oscillation
Chaos
1.
Introduction
Chaos is mathematical term referring to system dynamics in
which patterns never repeat themselves because there are no
stable equilibrium points and no stable cycles. The characteristics of chaos and its presence in nature are much discussed
in ecology (Godfray and Grenfell, 1993; Hastings et al., 1993;
Perry et al., 1993; Jørgensen, 1995; May, 1987). A number of
mathematical model have been developed to detect chaotic
system dynamics using time–density data (Hastings et al.,
1993). To assess the ecological implications of chaotic dynamics in different natural systems, it is important to explore
changes in the dynamics when structural assumptions of the
system are varied. One approach to the study of the dynam-
∗
Corresponding author. Tel.: +91 3463 261268; fax: +91 3463 261268.
E-mail address: santanu [email protected] (S. Ray).
0304-3800/$ – see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2006.12.014
ics of ecological community is its food web and the coupling of
interacting species with each other. Hastings and Powell (1991)
produced a new example of a chaotic system in a three species
food chain model with type II functional responses. Eisenberg
and Maszle (1995) revisited Hastings and Powell (1991) three
species food chain model and observed, that gradual addition of refugia provide a stabilizing influence for which the
chaotic dynamics collapsed to stable limit cycles. Doveri et al.
(1993) described seasonality and chaos in plankton fish model.
In their model they studied the dynamics of a plankton fish
model comprising phosphorus, algae, zooplankton and young
fish are analyzed for different values of average light intensity, phosphorus concentration in the inflow, and fish biomass.
Large number of bifurcations was studied in the model by
41
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
using varied annual water temperature and light intensity and
also the fixed realistic values of other parameters. The nonlinear system of the model showed multiplicity of attractors,
catastrophic transitions, sub-harmonics of various periods,
cascades of period doublings, and strange attractors arise for
suitable values of the parameters.
Ecosystems are conceived as conceptually open, selfadapting systems, which constantly produce novelty and new
parameters which cannot be severed from their environment
(Jørgensen et al., 1998; Haag and Kaupenjohann, 2001). The
properties of an ecosystem to adapt to changed conditions are
rooted in the interplay between self-organization and selection. Systems at the edge of chaos are adaptable to the most
complex behavior (Kauffman, 1993). For assessing the selforganization, different principles such as exergy, ascendency,
emergy, indirect effect, etc. are proposed by different authors
(Ulanowicz, 1986; Odum, 1988; Jørgensen, 1995; Patten, 1995).
In this paper we have used the model of Hastings and
Powell (1991). Here the system dynamics moves from ordered
state to chaotic condition for the change of two important
parameters of zooplankton. A thermodynamic goal function,
exergy (Jørgensen, 1995) is tested to examine the behavior of
the system in relation to the dynamics of that system from
order to chaos. The hypothesis to be tested is that a system
at the edge of oscillation, i.e. when system dynamics from
equilibrium condition enter into oscillatory state contains the
highest level of information. Evidence is presented in this
paper which supports the hypothesis when the system moves
towards the edge of chaos. The key parameters chosen for the
analysis are the growth rate and half saturation constant of
zooplankton.
When the system moves from order to chaos the level of
information also changes according to the prevailing conditions and we recorded the highest level in particular moment.
This is of utmost importance if the hypothesis that the edge
of the chaos coincides with the highest level of information is
to be supported.
2.
The model
The basic mathematical model is now represented as a set of
three ordinary differential equations describing the change of
phytoplankton (P), zooplankton (Z) and fish (F) over time:
P
dP
= R0 P 1 −
dT
K0
of equations:
a1 pz
dp
= p(1 − p) −
,
dt
1 + b1 p
a2 zf
df
=
− d2 f
dt
1 + b2 z
C1 A1 PZ
,
B1 + P
dZ
A1 PZ
A2 ZF
=
−
− D1 Y,
dT
B1 + P
B2 + Z
dF
C2 A2 ZF
=
− D2 F
dT
B2 + Z
(1)
To characterize interference between phytoplankton and
zooplankton population, Holling types I, II and III functional
responses are considered to study the behavior of the system. For determination of exact combination of parameters
which are controlling the behavior of the system the number of parameters are reduced by choosing p = P/K0 , z = C1 Z/K0 ,
f = C1 F/C2 K0 , t = R0 T. Making these substitutions in model
(1) and after simplification it yields the following system
(2)
The non-dimensional parameters are defined as a1 = K0 /R0 B1 ,
a2 = A2 C2 K0 /C1 R0 B1 ,
b1 = K0 /B1 ,
b2 = K0 /B2 C1 ,
d1 = D1 /R0 ,
d2 = D2 /R0 .
The observation made in this study are limited to the problem and the system under consideration, which is a general
aquatic food web model, it leads towards chaos from ordered
state. The thermodynamic variable exergy is the appropriate
index to measure the state of the system when it is moving
towards chaos from ordered situation.
During different run of the model the parameter values
(growth rate and half saturation constant) are changed in the
way that zooplankton size in the system are gradually shifted
from larger size to smaller size. According to the change of
the size of the zooplankton both parameters are changed
following allometric principles. We make deductions from
general ecological allometric principles of body size by using
logarithmic scale (log 10) of this size for parameterization of
the growth rate and half saturation constant of zooplankton
(Peters, 1983). Different authors (Sheldon et al., 1972; Kerr,
1974; Radtke and Straškraba, 1980; Tang, 1995; Ray et al.,
2001a,b; Jørgensen et al., 2002) used cell or body volume as a
measure of size for the scaling of allometric relationships with
growth rate and half saturation constant of zooplankton. We
follow the same procedure in our present model that relies on
empirically established allometric relationships between individual cell volume or body volume and these parameters. For
selecting the cell or body volume of zooplankton we surveyed
the literature. We found that normally the zooplankton comprises between the smaller 10 ␮m3 and larger 104 ␮m3 . We use
the notation elsewhere in this paper as Vz for expressing zooplankton body volume Vzi = log(10i ␮m3 ). Blueweiss et al. (1978),
Ahrens and Peters (1991) and Gillooly (2000) noticed that zooplankton growth rate varies according to its size, maximum
recorded in smaller species and minimum in larger species. On
the basis of this observation and on the basis of calculations
by Peters (1983) we propose the growth rate (a1 ) of different
sized zooplankton:
a1 = 0.715 − 0.13 log Vz
−
dz
a1 pz
a2 zf
=
−
− d1 z,
dt
1 + b1 p
1 + b2 z
(3)
Half saturation constant of zooplankton grazing on phytoplankton (b1 ) was studied by many authors (Blueweiss et al.,
1978) and increases logarithmically with body size. Straškraba
and Gnauck (1985), Stickney et al. (2000), Ray et al. (2001a,b)
proposed it varies for nitrate from 0.7 to 4.3 and therefore we
propose the equation:
b1 = 1.2 log Vz − 0.5
(4)
Exergy is defined as the work the system can perform when
brought into thermodynamic equilibrium with a well defined
reference state (for instance the same system at thermodynamic equilibrium at the same temperature and pressure as
42
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
the considered ecosystem). Exergy measures, therefore, the
distance from thermodynamic equilibrium, there is no free
energy available. The amount of exergy stored in the system
has the following advantages as measure of the best organization of the system. Biomass contributes significantly to
the exergy. The contribution is the free energy of biomass,
approximately 18.7 kJ g−1 (Jørgensen et al., 1995). Survival is
measured by the biomass of living organisms. Information has
also exergy accordance. The level of information computed as
the sum of the levels of information for the components that
make up the system. In this context, it is presumed that the
information associated with the ecological network is minor
compared with the information stored in the genetic code.
This presumption is justified by the small number of linkages
in the network compared with the vast number of combinations in the genetic code.
The inorganic nutrients have no free energy available as
these are at thermodynamic equilibrium and therefore do
not contain information. Jørgensen et al. (1995) calculated
the probability of a complex protein molecule forming from
straightforward thermodynamics, using the fact that the free
energy of complex organic molecules (e.g. proteins) is round
4.5 kcal g−1 . They found that it would correspond to a concentration of 10−86,000 mg l−1 (Jørgensen et al., 1995). This
represents detritus, and is the probable concentration of complex organic molecules in the inorganic soup. The information
can be calculated from Boltzmann (Boltzmann, 1905; see also
Jørgensen, 1992a,b) as RT ln(1/probability at thermodynamic
equilibrium) = 2.3 × 86,000RT per unit of biomass, where R is
the gas constant.
Besides the contribution from the complex organic
molecules, the level of information for phytoplankton will
contain information in the genetic code. To calculate this
contribution we use the following equation from Boltzmann
(Boltzmann, 1905; Jørgensen, 1992a):
I=
RT ln Weq
W
(5)
The information of any organism can therefore be found if
we know the number of microstates among which the organism has been selected. Amount of DNA per cell could be
used, but the amount of unstructured and nonsense DNA is
different for different organisms, but there is a clear correlation between the number of genes and the complexity of
the organisms. The calculation is based on the number of
genes an organism contains and number of amino acids per
gene. Considering these things the information (also called the
weighting factor) is calculated for all the living state variables
for this model (for detailed calculation and updated ˇi values,
see Jørgensen, 1992a,b; Jørgensen et al., 1995, 2005) and is as
follows:
Exergy = 20P + 232Z + 499F
3.
Results
Equation of model (2) is numerically solved by using modified
fourth order Runge-Kutta method considering the parameter values of Hastings and Powell (1991) except growth rate
and half saturation constant of zooplankton. In the model run
gradually we have decreased the body size of zooplankton by
incorporating the gradual increment and decrement of specific growth rate and half saturation constant of zooplankton
following the equations of (3) and (4), respectively. By considering the different body sizes of zooplankton in the model
we observe equilibrium point, stable limit cycle, doubling and
finally chaos (Figs. 1–4, respectively) in different run. When
model shows equilibrium or stable point we consider the
phase portrait of the system with a1 = 0.291, a2 = 0.1, b2 = 2.0,
d1 = 0.4, d2 = 0.01 and b1 = 3.41 (corresponding zooplankton
body volume Vz3.26 ), phase portrait of the system during limit
cycle with a1 = 0.342, b1 = 2.94 (corresponding zooplankton
body volume Vz2.87 ), and a1 = 0.383, b1 = 2.56 (corresponding
zooplankton body volume Vz2.55 ) in case of doubling period
where W is the number of possible macrostates among which
one has been selected for the focal system. This contribution from information implies that the information carried by
the various species will be included in the amount of exergy
stored in the system. The information is applied by the species
to ensure survival or even growth under the prevailing conditions. Biomass and information are directly linked to the
structure and order of the system in opposition to the random state at thermodynamic equilibrium. The total distance
in energy unit from thermodynamic equilibrium is equal to the
exergy. It can be shown that the exergy can be calculated as
(the system at thermodynamic equilibrium as reference state
as indicated as above):
Exergy =
n
ˇi ci
(6)
i=0
where ˇi is the weighting factor accounting for the information
the species carry, while ci is the concentration in for instance
g m−3 .
(7)
Fig. 1 – Phase portrait of system, with a1 = 0.291, a2 = 0.1,
b2 = 2.0, d1 = 0.4, d2 = 0.01 and b1 = 3.41 (corresponding
zooplankton body volume Vz3.26 ) showing convergent
oscillation and ultimately settles to equilibrium point.
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
Fig. 2 – Phase portrait of system with a1 = 0.342, b1 = 2.94
(corresponding zooplankton body volume Vz2.87 ) (all other
parameter values are same as Fig. 1), showing limit cycle.
and finally at the time of chaos the value of a1 = 0.424, b1 = 2.20
(corresponding zooplankton body volume Vz2.24 ). Our aim is to
observe the movement of system dynamics from equilibrium
to chaos and corresponding exergy values of the model.
In all conditions of the model run, exergy values are
recorded and expressed as mg organic matter l−1 . When regular oscillations occur (during limit cycle period), the average
exergy value is used for one oscillation. But during doubling
period and in chaotic conditions we use the average values
of 10 and 100 time units, respectively. Fig. 5a–d shows the
cycle of changes, plotting maximum specific growth rate, half
saturation constant and corresponding body volume of zooplankton and the level of information (exergy). The interesting
observation is that during chaos the recorded value is lowest although the highest specific growth rate of zooplankton
(0.424), lowest half saturation constant (2.20) (smallest sized
Fig. 3 – Phase portrait of system with a1 = 0.383, b1 = 2.56
(corresponding zooplankton body volume Vz2.55 ) (all other
parameter values are same as Fig. 1), showing doubling
period.
43
Fig. 4 – Phase portrait of system with a1 = 0.424, b1 = 2.20
(corresponding zooplankton body volume Vz2.24 ) (all other
parameter values are same as Fig. 1), showing chaos.
zooplankton Vz2.24 ) are considered. During limit cycle and doubling periods the average exergy values are lower than that
in the equilibrium condition but higher than the chaos. As
observed, the highest level of exergy is obtained when the
system moves from equilibrium condition to oscillatory state
(where maximum specific growth rate of zooplankton is 0.328).
Therefore, for this particular model, the highest level of information (exergy value) is obtained at the edge of chaos. This
is a result of adaptation to emerging conditions, and may be
explained by the interplay of self-organization and selection.
4.
Discussion
A biological system evolves towards the edge of chaos, according to a theory proposed by Kauffman (1991, 1992, 1993). By
means of models with realistic parameters, this paper has
examined the relationship between the value of two selected
key parameters (growth rate and half saturation constant
of zooplankton) and the information content of the system.
The model behavior was followed, as the zooplankton growth
rate and half saturation constant are changed. It is found
that the highest level of exergy coincides with the edge of
chaos, with parameter values that are supported by ecological observations. Gradual higher specific growth rate and
lower half saturation constant of zooplankton mean that, the
zooplankton body size gradually decreases from Vz3.26 (during
equilibrium condition) to Vz2.24 (during chaos) following general allometric relationships (Peters, 1983). Ray et al. (2001a,b),
Jørgensen et al. (2002) showed that size combinations between
phytoplankton and zooplankton are very crucial for selforganization of the system. The system adapts by changing
size, but it does not adapt the gradual decrease of zooplankton size and moves towards chaos. In reality it may happen
if non-selective predation by fish or invertebrates is present,
zooplankton yields bigger size (Peters, 1983).
During equilibrium state comparatively minimum specific growth rate and maximum half saturation constant
44
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
Fig. 5 – Body size (Vzi ), half saturation constant (b1 ), specific growth rate (a1 ) and exergy values of the system in different
conditions. In X axis from 1 to 3 when the system moves toward equilibrium point, values at 4 when it is in equilibrium
state, values from 5 to 6 at the junction where oscillation starts (maximum value shown at 6), values at 7, 8 and 9 during
limit cycle condition, doubling period and chaos, respectively.
of zooplankton are considered and when gradual smaller
sized zooplankton are selected the specific growth rate are
increased and half saturation constant decreased the system starts oscillation and at this edge maximum information
occurs. Therefore, the maximum information occurs when
system is occupied by comparatively larger sized zooplankton. Jørgensen (1995) noticed same type of relations between
the size of zooplankton and highest level of information.
The size combination of zooplankton and phytoplankton is
crucial component in any aquatic ecosystem and the combinations are adjusted according to the changing conditions of
the ecosystem. When the system selects smaller size of zooplankton it is obviously beneficial for this group to grow fast
and when this fast growth continues the phytoplankton will be
rapidly exhausted and lead the system into violent oscillatory
state and ultimately chaos. During oscillatory state zooplankton with higher growth rate grow rapidly and which increases
the exergy value, but as their food phytoplankton becomes disappear, the zooplankton and the exergy value decline rapidly
due to lack of food. In the next stage the phytoplankton are
thereby more available, and the zooplankton again grow faster
and so on. Therefore, the system organizes itself by choosing the appropriate growth rate of zooplankton, which itself
adjusts to the availability of phytoplankton resources. This
appropriate higher growth rate corresponds to the edge of
chaos. This situation also coincides with the highest level of
exergy as it corresponds with the best possible grazing of zoo-
plankton on available phytoplankton, i.e. to get on average as
high a concentration of zooplankton and its food (phytoplankton) as possible. In such situation exergy is to be a convenient
measure for the sum of zooplankton and phytoplankton.
When non-selective predation of fish or invertebrates
takes place in the ecosystem, gradually larger size zooplankton become dominant in the system as they are able to
escape predation pressure in comparison to the smaller size.
Jørgensen (1995), Jørgensen et al. (1998) and Ray et al. (2001a,b)
suggested many properties which are associated with selforganizing critical systems. Nicolis and Prigogine (1989) and
more recently Forrest and Jones (1994) elaborately described
these properties for self-organizing critical systems, first of
all the system must be thermodynamically open or in other
word it must be exchanging energy and/or mass with its
environment. The next property is the dynamic behavior of
the system, meaning the system is undergoing continuous
change of some sort. One of the most basic kinds of change
for self-organizing system is to import usable energy from
its environment and export entropy back to it. Inherent local
interaction in the system is an important property for selforganizing system. Non-linearity in the system is another
property, there should be positive and negative feedbacks
in the system, self-organization can occur when feedback
loops exist among component parts and between the parts
and the structures that emerge at higher hierarchical levels. Emergence is one of the most important properties for
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
self-organization. Crutchfield (1994) proposed the theory of
emergence, it says the whole is greater than the sum of the
parts, and the whole exhibits patterns and structures that
arise spontaneously from parts. Emergence indicates there
is no code for a higher-level dynamic in the constituent,
lower-level parts. Emergence also points to the multi-scale
interactions and effects in self-organized systems. All these
properties follow the power law, which are the relationships
between body size and abundance for species in ecosystem, the frequency with which observed changes exceeds a
given change versus the change, the typical frequencies of
‘avalanches’ and the occurrence of ‘avalanches’ according to a
well examined ecosystem model be used to explain the underlying causality. According to power law a distribution of results
such that the larger the effect the less frequently it is seen.
A system subject to power law dynamics exhibits the same
structure over all scales. This self-similarity or scale independent (fractal) behavior is typical of self-organizing systems.
Fig. 5a–d shows plotting of exergy in different conditions
of the systems, gradually the zooplankton size are considered smaller in the model (by increasing specific growth rate
and decreasing half saturation constant following allometric
principle) and system moves from ordered to chaotic state.
Survival of a system can be measured by the information
that the various species are carrying, which is the information
(exergy) used in the model studies considered here. Jørgensen
(1995) proposed it as a tentative hypothesis, linking Darwin’s
theory with Kauffman’s hypothesis (Kauffman, 1991, 1992,
1993) that biological systems evolves towards the edge of
chaos because the exergy level is highest here. The properties
of an ecosystem to adapt to changed conditions are rooted
in the interplay between self-organization and selection. In
the present paper it has been shown that the highest level
of information is obtained at the edge of chaos, where the
organisms are able to coordinate the most complex behavior. The hypothesis that ecosystem evolves toward the highest
level of information which coincides with the edge of chaos,
implies that the ecosystem may show hysteresis phenomena
as a consequence of drastic changes imposed on the system
(Jørgensen, 1999).
The presented methodology can be used to assess the size
of zooplankton (in other similar models or other trophic levels)
that is resulting from adaptation to the prevailing conditions,
based upon the hypothesis that ecosystems attempt to get the
highest possible exergy and information. Several authors like
Ascioti et al. (1993), Caswell and Neubert (1998), Mevinsky et
al. (2001) obtained chaos in different planktonic systems due
to the gradual increment of growth rate of zooplankton. The
size class of zooplankton (Vz2.24 ) in our model during chaotic
conditions corroborate with the size classes of zooplankton of
the model of Mevinsky et al. (2001). They showed that when
the system is dominated by this size class of zooplankton it
moves from regularity to chaos.
5.
Conclusion
In this model it has been shown that the system cannot tolerate overexploitation of phytoplankton by zooplankton. And
this overexploitation leads the system to move from ordered
45
to chaotic situation. The system tries to adapt this situation
by changing the size of the organisms. At the edge of chaos
the system shows highest exergy value. Ecosystems are very
different from physical systems mainly due to their enormous adaptability. It is therefore crucial to develop models
that are able to account for this property, if we want to get
reliable model results. The use of thermodynamic goal function (exergy) offers a new insight in the models which is able
to consider the fitness of the system and adaptability and
also to describe shifts of system of dynamics due to selfcontrolled properties of the system such as different biological
parameters.
Acknowledgements
We are thankful to the Department of Zoology, and Department of Physics, Visva Bharati University for laboratory and
other facilities for performing this work. One of the authors
(Santanu Ray) gratefully acknowledges the support from the
CSIR, New Delhi (Project Ref. No. 37/1185/04/EMR-II).
references
Ahrens, A.M., Peters, R.H., 1991. Plankton community respiration:
relationships with size distribution and lake trophy.
Hydrobiolgia 224, 77–87.
Ascioti, F.A., Beltrani, E., Carrol, T.O., Wirick, C., 1993. Is there
chaos in plankton dynamics. J. Plank. Res. 15, 603–617.
Blueweiss, L., Fox, H., Kudzma, V., Nakashima, D., Peters, R.H.,
Sams, S., 1978. Relationships between some body size and
some life history parameters. Oecologia (Berl.) 37, 257–272.
Boltzmann, L., 1905. The second law of thermodynamics.
Populare Schriften, Essay No. 3 (address to Imperial Academy
of Science in 1886). Reprinted in English: Theoretical Physics
and Philosophical Problems, Selected Writings of Boltzmann,
L. D. Reidel, Dodrecht.
Caswell, H., Neubert, M.G., 1998. Chaos and closure term in
plankton food chain models. J. Plank. Res. 20, 1837–1845.
Crutchfield, J.P., 1994. In anything ever new? In: Gowan, G., Pines,
D., Melsner, D. (Eds.), SFI Studies in the Sciences of
Complexity XIX. Addison-Wesley, Massachusetts, USA.
Doveri, F., Scheffer, M., Rinaldi, S., Muratori, S., Kuznetsov, Y.,
1993. Seasonality and chaos in a plankton-fish model. Theo.
Pop. Biol. 43, 159–183.
Eisenberg, J.N., Maszle, D.R., 1995. The structural stability of a
three-species food chain model. J. Theor. Biol. 176, 501–510.
Forrest, S., Jones, T., 1994. Modeling complex adaptive systems
with echo. In: Stoner, R.J., Yu, X.H. (Eds.), Complex Systems:
Mechanism of Adaptation. IOS Press, Amsterdam.
Gillooly, J.F., 2000. Effect of body size and temperature on
generation time in zooplankton. J. Plankton Res. 22, 241–251.
Godfray, H.C.J., Grenfell, B.T., 1993. The continuing quest for
chaos. Trends Ecol. Evol. 8, 43–44.
Haag, D., Kaupenjohann, M., 2001. Parameters, prediction,
post-normal science and the precautionary principle—a
roadmap for modelling for decision-making. Ecol. Model. 144,
45–60.
Hastings, A., Hom, C.L., Ellner, S., Turchin, P., Godfray, H.C.J., 1993.
Chaos in ecology: is mother nature a strange attractor? Annu.
Rev. Ecol. Syst. 24, 1–33.
Hastings, A., Powell, T., 1991. Chaos in three-species food chain.
Ecology 72, 896–903.
46
e c o l o g i c a l m o d e l l i n g 2 0 4 ( 2 0 0 7 ) 40–46
Jørgensen, S.E., 1992a. Integration of Ecosystem Theories: A
Pattern. Kluwer Academic Publishers, Dordecht.
Jørgensen, S.E., 1992. Parameters, ecological constrains and
energy. In: Jorgensen, S.E. (Ed.), Ecosystem Theory and
Ecological Modeling. Ecol. Model. 62, 163.
Jørgensen, S.E., 1995. The growth rate of zooplankton at the edge
of chaos: ecological models. J. Theor. Biol. 175, 13–21.
Jørgensen, S.E., 1999. Biomanipulation and ecological modeling.
In: Tundisi, J.G., Straškraba, M. (Eds.), Theoretical Reservoir
Ecology and its Applications. Backhuys Publishers, Leiden,
The Netherlands, pp. 547–564.
Jørgensen, S.E., Mejer, H.F., Nors-Nielsen, S., 1995. Emergy,
environ, exergy and ecological modeling. Ecol. Model. 77,
99–110.
Jørgensen, S.E., Mejer, H., Nielsen, S.N., 1998. Ecosystem as
self-organizing critical systems. Ecol. Model. 111, 261–268.
Jørgensen, S.E., Ray, S., Berec, L., Straškraba, M., 2002. Improved
calibration of a eutrophication model by use of the size
variation due to succession. Ecol. Model. 153, 269–277.
Jørgensen, S.E., Ladegaard, N., Debeljack, M., Marques, J.C., 2005.
Calculations of exergy for organisms. Ecol. Model. 185,
165–176.
Kauffman, S.A., 1991. Anti-chaos and adaptation. Sci. Am., 78–84.
Kauffman, S.A., 1992. The sciences of complexity and “origins of
order”. In: Mitten, J.E., Baskin, A.G. (Eds.), Principles of
Organization in Organisms, Proceedings of the Workshop on
Principles of Oranization in Organisms. Santa Fe,
p. 303.
Kauffman, S.A., 1993. Origins of Order: Self Organization and
Selection in Evolution. Oxford University Press, Oxford.
Kerr, S.R., 1974. Structural analysis of aquatic communities. Proc.
Ist. Int. Cong. Ecol., 69–84.
May, R.M., 1987. Chaos and the dynamics of biological
populations. Proc. R. Soc. Lond. 413, 27–44.
Mevinsky, A.B., Petrovskii, S.V., Tikhonov, D.A., Tikhonova, I.A.,
Ivanitsky, G.R., Venturino, E., Malchow, H., 2001. Biological
factors underlying regularity and chaos in aquatic
ecosystems: simple model of complex dynamics. J. Biosci. 26,
77–108.
Nicolis, G., Prigogine, I., 1989. Exploring Complexity. W.H.
Freeman, New York, USA.
Odum, H.T., 1988. Self organization, transformity, and
information. Science 242, 1132–1139.
Patten, B.C., 1995. Network integration of ecological extremal
principles: exergy, emergy, power, ascendency, and indirect
effects. Ecol. Model. 79, 75–84.
Perry, J.N., Woiwod, I.P., Hanski, I., 1993. Using response-surface
methodology to detect chaos in ecological time series. Oikos
68, 329–339.
Peters, R.H., 1983. The Ecological Implications of Body Size.
Cambridge University Press, Cambridge.
Radtke, E., Straškraba, M., 1980. Self-optimization in
phytoplankton model. Ecol. Model. 9, 247–268.
Ray, S., Berec, L., Straškraba, M., Jørgensen, S.E., 2001a.
Optimization of exergy and the implications of body sizes of
phytoplankton and zooplankton in an aquatic ecosystem
model. Ecol. Model. 140, 219–234.
Ray, S., Berec, L., Straškraba, M., Ulanowicz, R.E., 2001b. Evaluation
of system performance through optimizing ascendency in an
aquatic ecosystem model. J. Biol. Syst. 9, 1–22.
Sheldon, R.W., Prakash, A., Sutchiffe Jr., W.H., 1972. The size
distribution of particles in the ocean. Limnol. Oceanogr. 17,
327–340.
Stickney, H.L., Hood, R.R., Stoecker, D.K., 2000. The impact of
mixotrophy on planktonic marine ecosystems. Ecol. Model.
125, 203–230.
Straškraba, M., Gnauck, A.H., 1985. Freshwater Ecosystem. In:
Modelling and Simulation. Elsevier, Amsterdam, pp. 309.
Tang, E.P.Y., 1995. The allometry of algal growth rates. J. Plankton
Res. 17, 1325–1335.
Ulanowicz, R.E., 1986. Growth and Development, Ecosystem
Phenomenology. Springer-Verlag, New York.