129-JBS 00181 EXTREMAL PRINCIPLES

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Journal of Biological Systems, Vol. 14, No. 2 (2006) 255–273 c World Scientific Publishing Company

EXTREMAL PRINCIPLES WITH SPECIAL EMPHASIS ON EXERGY AND ASCENDENCY — THE MODERN APPROACH IN THEORETICAL ECOLOGY

SANTANU RAY Ecological Modelling Laboratory, Department of Zoology Visva-Bharati University, Santiniketan 731 235, India santanu [email protected] Received 3 January 2005 Revised 23 August 2005 Extremal principles or ecological orientors or goal functions are the most modern approach in theoretical ecology. There are many such principles proposed by different theoretical ecologists. In this paper, the most important extremal principles are discussed based on their theoretical backgrounds. Two widely accepted goal functions, i.e. exergy and ascendency are optimized and treated in a quantitative manner in an aquatic ecosystem model of planktonic and fish systems for their appropriateness. In the model varied body sizes of phytoplankton and zooplankton are considered. Parameter values varied according to the allometric principle with the body sizes. For self-organization of the model system two goal functions predict different results, however both are realistic. Keywords: Extremal Principles; Orientors; Goal Function; Model; Exergy; Ascendency; Self-Organization; Aquatic Ecosystem.

1. Introduction Extremal principles are also known as ecological organizing principles or goal functions of the ecosystem to organize itself in the direction of its development. The ecosystem has a self-organizing ability which makes it possible to cope with perturbations by directive reactions which can be described by goal functions.1 Two principles from classical thermodynamics are firmly established for systems near equilibrium.2 The first is the second law which applies to isolated systems: entropy always increases with time and approaches a maximum at equilibrium. The second law is for open systems3 : entropy production always decreases with time and approaches a minimum at steady state. Far from equilibrium, which is where many physical systems and all living systems operate, these principles do not apply.4 The search for organizing principles that do apply has produced a variety of “orientors” dealing with energy, matter flow, biomass and system information.5 The central idea of the orientor approach, which most of the following hypotheses are focused on, originates in modern systems analysis, complexity science and synergetic. It refers to the idea of self-organizing processes that are able to 255

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build up gradients and macroscopic structures from the microscopic “disorder” of non-structured, homogeneous element distributions in open systems, without receiving directing regulations from the outside. In such dissipative structures, the self-organizing process sequences in principle generate comparable series of constellations that can be observed by certain emergent or collective systems features. Thus, similar changes of certain attributes can be observed in different environments. Utilizing these attributes, the development of the systems seems to be oriented specific points or areas in the state space. The respective state variables which are used to elucidate these dynamics are termed “orientors.” Their technical counterparts in modeling are called “goal functions.” Odum6 hypothesized on the trends to be expected in ecosystem development and discussed some of the ecosystem properties as ecological orientors which are biomass, cycling, internal organization, residence time and information. By taking thermodynamic approach recently, Schneider and Kay7 proposed seven ecosystem properties as basic orientors: exergy capture, energy flow, cycling of energy and materials, respiration and transpiration, biomass, average trophic structure and types of organisms. Additional thermodynamic goal functions have been proposed specifically in the context of ecological models. In particular, Bendoricchio and Jorgensen8 made the case that the primary ecosystem goal function is exergy storage. Bastianoni9 and Bastianoni and Marchettini10 suggested minimum empower to exergy ratio as the primary ecosystem goal function, and Jorgensen et al.11 suggested specific dissipation as the primary pattern observed in growth phenomena. Many authors showed a strong correlation between several goal functions. Jorgensen12,13 and Jorgensen and Nielsen14 suggested that perhaps the integration of goal functions could lead to consideration of only one of them. Patten,15 using an earlier development of network analysis, showed that many goal functions have a common basis in the path structure and associated microscopic dynamic of the systems. Most of the widely accepted extremal principles are discussed below. 2. Different Extremal Principles and Their Descriptions 2.1. Maximum power16 Maximum power principle states that systems become organized to maximize their energy throughput. Odum has long championed this principle in ecology, beginning with Odum and Pinkerton17 which argued that maximizing power produced the most energy to perform work and create order (“pump and disorder”). 2.2. Maximum storage18 Energy systems maximize their distance from a thermodynamic reference point by storing usable energy (exergy). The associated accumulation of mass or energy is reflected in structure, function, gradients, order, organization, and information; all of which express in different ways departure from the reference. For entire systems,

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this principle asserts that total system storage is maximized. For biotic systems this means maximizing biomass. 2.3. Maximum empower and emergy19 The emergy (embodied energy) is the energy of one type required directly and indirectly to make a product. Its unit is the emjoule. Transformity is the energy of one type required to make one joule of another type. To know the emergy flow from sources or storages an energy flow is multiplied by its transformity.19 As solar energy passes through a series of energy transformations, transformity increases at each step of the trophic level. Emergy measures this transformity and empower the associated energy flow.19 2.4. Maximum ascendency20,21 Maximum ascendency principle quantifies network organization as the product of the totals system throughflow (matter or energy) and average mutual information involves the individual flow and a complicated expression of the logarithm of various other flow and organizational components.22 Average mutual information is dimensionless and has a restricted range of values (generally between about 2.0 and 6.0). The total system throughflow, which scales this information quantity, can vary widely over the non-negative real numbers. As a result, throughflow dominates the ascendency measure such that power and ascendency give strongly correlated results.12 2.5. Maximum dissipation7,23–25 Dissipative structures far from equilibrium have been suggested to maximize entropy production. This can be elaborated in exergy forms, the system supplied with an external exergy source will respond by all means available to degrade the received exergy. This amounts to a maximize dissipation principle, and systems and processes satisfying it best gain from the implied work performed. Such gains in work represent a source of selective advantage in physical and biological evolutionary systems. For biological systems dissipation includes respiration plus other usable or unusable exports. Total system export is the sum of dissipative processes over all components. 2.6. Maximum cycling26 This principle states that energy flow caused cycling and this produced organization. The flow of heat from sources to sinks can lead to an internal organization of the systems. The flow of heat can lead to the formation of cyclic flows of material in the intermediate system. The flow of energy causes cyclic flow of matter. The cyclic flow is apart of the organized behavior of the system undergoing energy flux. The

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converse is also true; the cyclic flow of matter such as is encountered in biology requires an energy flow in order to take place. The existence of cycles implies that feedback must be operative in the system. Therefore, the general notions of control theory and the general properties of servo networks must be characteristic of biological systems at the most fundamental level of operation. Control concepts involve negative (deviation-damping) feedback,15 but cycling also opens the possibility for positive (deviation-amplifying) feedback. Ascendency theory20 invokes the latter in “autocatalytic loops” central to system development. Glansdorff and Prigogine1 hypothesized an order-through-fluctuation principle. These authors noted that small deviations in energy flow exist statistically in any thermodynamic system. These are generally damped out by dissipative processes, but as energy gradients (including those reflected in storage) increase, deviation amplification becomes more and more probable. Any damping fluctuations that can better dissipate the gradients became selected for and amplified over time. Odum27 stated that it is probably equivalent to the principle of selection through maximizing power with pulsing. In far-from-equilibrium thermodynamics, pulsing organizations have been referred to as “dissipative structures,”1,28 and the significance of their oscillations is the acceleration of energy flux toward the realization of maximizing power. 2.7. Maximum residence time29 Ecological systems organize to maximize the residence time of energy or matter. The residence time of flow in particular component, τi , is given by the reciprocal of the turn over rate τi−1 . Total system residence time can be found by summing the individual component residence times (τi ) and is the fraction of throughflow that remains as storage. 2.8. Minimum specific dissipation24,30,31 Internal constraints, such as caused in living systems by inefficient energy transfer or limited availability of metabolites, modulate throughflow maximization and divert free energy (exergy) to storage as chemical potential. This tends to minimize dissipation per unit mass or volume, which expresses the least specific dissipation principle. Although this least specific dissipation principle was developed for systems near thermodynamic equilibria, it contends that even if the global system is “far from equilibrium,” sub-systems at finer spatio-temporal-organizational scales may be considered to be in some proximity to a quasi-local steady state — close enough at least for the principle to provide an understanding of “how a system should change.” A question for future research is to determine whether a system is too far or near enough for this to be valid. 2.9. Minimum empower to exergy ratio10 System organization can be measured by a ratio of empower to exergy. Empower to exergy ratio measures the total environmental cost (throughflow) required to

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produce a unit of organization (structure). This differs from minimizing specific dissipation in that specific dissipation is only a fraction of total system throughflow. The metric was tested on three lagoon systems and showed that the “natural” system had the lowest empower/exergy value. Bastianoni and Marchettini10 concluded that it was the most efficient of the three at processing throughflow to maintain structure. 3. Application of the Extremal Principles in Ecological Models The extremal principles can be estimated by appropriate mathematical tools. Many authors have optimized (maximized or minimized according to the types of principles) any single extremal principle in their respective ecological system with the help of a model. It is an utmost necessity to optimize all the goal functions in a particular model or set of different models and after doing that a conclusion can be drawn regarding similarities and dissimilarities of different goal functions and ultimately a unification of all goal functions may be reached which will be extremely helpful to define the self-organization of the ecosystem. All of these above-mentioned extremal principles are based on different aspects. But most of the principles are not yet complete for application in the model and also it is very difficult to test all in a particular model. Some of the principles are either based on energetic features or on matter flow or information, etc. Generally, modeler collects data of the system which is being modeled for calibration and validation. The data are available in any particular unit and which is used for model currency. To date, few works are executed regarding the application of the extremal principles in different ecosystem models except exergy, ascendency and emergy. In the following section, only exergy and ascendency are applied in a known model where data are obtained in the unit of phosphorus m−3 . These data will not be possible for application for emergy analysis. This analysis needs data in the form of energetics. For application of different goal functions, different varieties of model with data of different units are necessary. This type of work would be very lengthy and time consuming. First of all a set of models of varying ecosystems has to be selected and these models have to be calibrated and validated properly to make them realistic, and then different goal functions can be applied on these models to study the relationships among different extremal principles. In view of the above constraints, the preceding chapter is only restricted to the application of exergy and ascendency in an aquatic ecosystem model. This model is well tested, calibrated and validated properly and these two goal functions, exergy and ascendency, are also completely formulated for application.12,20,32,33 In this aquatic ecosystem model, exergy and ascendency are applied through optimization (maximization). 3.1. The model The role of body size in planktonic system has significant contribution to the system dynamics.34–36 Commonly planktonic ecosystems are described by compartmental

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models, and each compartment represents a trophic level or taxonomic group. Such models are primarily descriptive, because the most important components of the ecosystem are represented by compartments and their interactions are described by linking the compartments.37 However, the use of these models creates numerous problems.38 The unrealistic lumping of all plankton sizes with widely different rates of physiological parameters is very common in the use of present dynamic models.39 To overcome this problem, Moloney et al.37 suggested an increase in the number of compartments, but this approach adds to the complexity in the model and in most cases the output is far from reality. Therefore an alternative approach has been developed in generic models of planktonic systems: the rate processes vary according to the body sizes of planktons and these are incorporated in the dynamic model. This approach was first used for phytoplankton by Lebedeva3 and later by Radtke and Straˇskraba,40 Plat et al.,41 Moloney and Field39 and Ray et al.,32,33 In the present model the same approach in both phytoplankton and zooplankton are used. The model is constructed on the planktonic food chain of Remove reservoir, Czech Republic. The data of this reservoir are used40,42 in the present model. The model is properly calibrated and validated (for details see Refs. 32 and 33). This reservoir is a dimictic reservoir with a well-defined summer stratification; it has a surface area of 2.06 km2 , maximum depth of 47 m, mean depth of 16.7 m, mean discharge of water 4.1 m3 S−1 and mean water renewal time of 96 days (details of the model description can be obtained from Ray et al.32,33 ). Different rate processes of phytoplankton and zooplankton on the basis of allometric relationships with their body sizes as proposed by Peters43 are determined and applied in the present model. For parameterization of the physiological processes of phytoplankton and zooplankton, deductions are done from general ecological allometric principles of body size by using logarithmic scale (log 10) of this size.44 Different authors25,30,34,41 used cell or body volume as a measure of size for the scaling of allometric relationships with physiological processes. The same procedure is followed in this present model that relies on empirically established allometric relationships between individual cell volume or body volume and metabolic processes. For selecting the cell or body volume of phytoplankton and zooplankton, the relevant literature was surveyed. Normally the phytoplankton cell or body volume ranges between 10 (µm3 ) and 107 (µm3 ) is found and zooplankton comprises between the smaller 10 (µm3 ) and larger 104 (µm3 ). Notation is used elsewhere in this paper as VP and Vz for explaining phytoplankton and zooplankton body volume respectively, Vp i and Vz i = log(10i µm3 ).

3.2. Physiological parameters of phytoplankton The size-dependent phytoplankton growth rate was observed and studied experimentally by many authors in different environments.45–50 Maximum gross photosynthetic activity (Pmax ) is estimated on the basis of these observations. Pmax (Vp ) = 3.0 − 0.3 log Vp .

(3.1)

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Photosynthesis is temperature dependent and the values in Eq. (3.1) hold true at the temperature of 20◦ C. The dependence on temperature (T ) is exponential, thus Pmax (Vp ) . (3.1a) Pmax (Vp T ) = 6.05 exp(0.09T ) Yurista51 observed that respiration rate (Resp) of phytoplankton varies in relation to body size. For smaller species of phytoplankton the respiratory rate is four times higher than in the larger species. This results approximately in a logarithmic dependence of the form Resp(Vp ) = 0.02 − 0.002 log Vp .

(3.2)

On the basis of physical laws of particle sizes the self-shading of phytoplankton is size dependent.52 Following the law of Kiefer and Austin,53 Radtke and Straˇskraba40 proposed the following equation for the size-dependent self-shading (Ext). Ext(Vp ) = 0.12VP−0.33 .

(3.3)

The natural mortality rate of phytoplankton (Mort) is also size dependent, the rate being high in smaller-but low in larger-sized phytoplankton.54 The mortality rate is estimated by Mort(Vp ) = 0.043 − 0.006 log Vp .

(3.4)

Eppley et al.55 suggested a logarithmic increase of half-saturation constant (Ks ) of phytoplankton for nitrogen according to its size. Applying the same principle for phosphorus the following equation is proposed: Ks (Vp ) = 10 log Vp − 5.

(3.5)

3.3. Physiological parameters of zooplankton Ahrens and Peters,56 Blueweiss et al.46 and Gillooly57 noticed that zooplankton growth rate varied according to its size, maximum recorded in smaller species and minimum in larger species. On the basis of this observation, Peters43 proposed the following equation for growth of zooplankton Growth(Vz ) = 0.715 − 0.13 log Vz .

(3.6)

Ahrens and Peters,56 Blueweiss et al.46 and Wetzel58 noticed that the respiratory rate of zooplankton was size dependent, achieved maximum in smaller species and minimum in larger species. According to the estimate of Peters43 it is assumed like the following: Resp(Vz ) = 0.033 − 0.08 log Vz .

(3.7)

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Blueweiss et al.46 reported the size-dependent relationship of zooplankton mortality rate and noticed the smallest value in larger species and largest value in smaller species. The rate is calculated as Mort(Vz ) = 0.054 − 0.012 log Vz .

(3.8)

The half-saturation constant of zooplankton grazing on phytoplankton (Ks ) was studied by many authors46,59 and increased logarithmically with body size. Straˇskraba and Gnauck60 proposed that it varied for phosphorus from 10 to 55 and therefore the following equation is obtained: Ks (Vz ) = 15 log Vz − 5.

(3.9)

3.4. Mathematical equations of model All of the above size-dependent parameters are incorporated into a dynamic model of phosphorus (N ), phytoplankton (P ), zooplankton (Z), fish (F ) and detritus (D) (Fig. 1). The model equations are presented below and the parameter description, their values and units are given in Table 1. Q Pmax (Vp T )LeP N dN = . (3.10) (Fin − N ) + dD D − dt V [N + Ksp (Vp )] Le = light effect on photosynthesis I . = 1 (Iopt) exp 1 − Iopt I = Is exp[(−Ec + ExtVp )Ze] Pmax (Vp T )LeP N Growth(Vz )ZP dP = − Resp(Vp )P − . dt [N + Ksp (Vp )] [P + Ks (Vz )]

(3.11)

Growth(Vz )ZP = grazing of Z on P [P + Ks (Vz )] dZ (µzf F szf Z) [Growth(Vz )ZP ] = (arz ) − − Mort(Vz )Z − Resp(Vz )Z. dt [P + Ks (Vz )] (sff F + szf Z + Ksf ) (3.12) (µzf F szf Z) = predation of F on Z (sff F + szf Z + Ksf ) dF (µzf F szf Z) (µff F sff F ) = arf − mrf F − rrf F − (1 − arf ) . dt (sff F + szf Z + Ksf ) (sff F + szf Z + Ksf ) (3.13) (µff F sff F ) = self-predation of F (sff F + szf Z + Ksf )

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Fig. 1. Conceptual diagram of five-compartment model, the compartments are phosphorus (N ), phytoplankton (P ), zooplankton (Z), fish (F ) and detritus (D). Inflow in (N ) from outside through water inflow from outside, outflow from (N ) through water outflow, outflow of (N ) to (P ) for phytoplankton intake of nutrient from (N ), outflow from (P ) to (Z) for zooplankton grazing, outflow from (Z) to (F ) for fish predation on zooplankton, flow from (P ), (Z) and (F ) to (D) indicates contribution in detritus pool for grazing, predation and mortality, inflow to (D) for water inflow from outside and outflow from (D) to outside also for water outflow, outflow from (D) to (N ) shows remineralization from detritus to nuturient.

Q dD [Growth(Vz )ZP ] = Din + (1 − arz ) + MortVp P + Resp(Vp )P dt V [P + Ks (Vz )] (µzf F szf Z) + Mort(Vz )Z + Resp(Vz )Z + (1 − arf ) (sff F + szf Z + Ksf ) (µff F sff F ) + (1 − arf ) + mrf F + rff F − ds D − dD D. (sff F + szf Z + Ksf )

(3.14)

3.5. Mathematical formulations of exergy and ascendency Jørgensen12 calculated information of the organic matter as RT ln (l/(probability of being at thermodynamic equilibrium)) = 2.3 ∗ 86000 RT per unit of biomass, where R is the gas constant and T is the temperature of the system. Besides the contribution from the complex organic molecules, the level of information for all the living components will contain information in the genetic code. To calculate this contribution Jørgensen12 used the following equation I (information) = RT ln Weq /W , where W is the total number of ways that a particular macrostate can be constituted microscopically. 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

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Table 1.

Parameter symbols, their description, values and units which are used in the model.

Symbol Iopt

Meaning

Is

Optimum surface radiation for photosynthesis Surface solar radiation

Time Ec T

Extinction coefficient Temperature

Time Ze Q V Q/V F in

Pmax (Vp T ) Resp (Vp ) Mort (Vp ) Ks (Vp ) Growth (Vz ) rrz Resp (Vz ) Mort (Vz ) Ks (Vz ) µzf µf f rrf mrf arf szf sff Ksf Din dS dD

Depth of epilimnion Discharge rate of water Volume of epilimnion Concentration of the inflow of N in mesotrophic/oligotrophic/eutrophic conditions Maximum photosynthetic activity of P Respiration rate of P Mortality rate of P Saturation constant of P for N Maximum growth rate of Z Assimilation rate of Z Respiration rate of Z Mortality rate of Z Saturation constant of Z for P Maximum predation rate of F on Z Maximum self-predation rate of F Respiration rate of F Mortality rate of F Assimilation rate of F Predation preference of F on Z Self-predation preference of F Saturation constant of F for Z and F Concentration of D in inflow Settling rate of D Remineralization rate of D to N

Value 300 280 + 210 SIN (Time 58.1) 0.2 12 + 10 SIN (Time 58.1) 4 10−2 20/5/100

0.6

0.14 0.08 0.0015 0.001 0.5 2/3 1/3 70 20 0.01 0.3

Unit cal cm

−2

day−1

cal cm−2 day−1 day m−1 0C day m m3 day−1 m3 mg N m−3 day−1 day−1 day−1 mg N m−3 day−1 day−1 day−1 day−1 mg N m−3 day−1 day−1 day−1 day−1 day−1 dimensionless dimensionless mg N m−3 mg m−3 day−1 day−1

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 the number of amino acids per gene. Considering these things the information (also called the weighting factor) is calculated for all the living state variables and also for the organic matter detritus (for detailed calculation, see Refs. 12, 28 and 31) and is as follows: Exergy = D + 13.8 ∗ P + 757 ∗ Z + 1513 ∗ F, where D = detritus, P = Phytoplankton, Z = Zooplankton and F = Fish.

(3.15)

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Formally, the ascendency can be computed as follows15,22

n fij T fij log A=T , T Ti Tj j=1

265

(3.16)

where T is the total system throughflow, n is the number of compartments, fij is the flow from the compartment i to the compartment j(0th compartment is the environment), Ti is the total flow leaving compartment i, Ti =

n

fij ,

j = i,

(3.17)

j=0

and Tj the total flow entering compartment j, Tj =

n

fij ,

i = j.

(3.18)

i=0

The optimization of exergy and ascendency is performed in three conditions of the reservoir ecosystem, i.e. oligotrophic, mesotrophic and eutrophic conditions to know the different combinations of phytoplankton and zooplankton sizes for best system organization in different nutrient loads of the system. The exergy and ascendency to be maximized over phytoplankton and zooplankton sizes were computed when the system reached a steady state. Since this state is always a stable limit cycle, and the exergy and ascendency copied this behavior, the exergy and ascendency are averaged over 365 successive days (duration of one year) in the steady state period of the run. Maximization performed in two steps, for speeding up computations and to avoid being trapped in local maxima. At the first exploratory step, a square lattice is superimposed on the region of admissible phytoplankton and zooplankton sizes, with the edge length 0.2. At each node, the average exergy and ascendency are computed to get a rough picture of the optimized function behavior. Then a small region around the expected maximum is delineated and inserted into the maximization procedure, the fmincon procedure of the MATLAB package (The Math Works Inc., USA). Constrained optimization problems via the technique of sequential quadratic programming are solved by this procedure.61

4. Results and Discussion The size of an organism affects virtually all aspects of its physiology and ecology.3 The work of Schmidt-Nielsen on the physiology of body size,62 Gould on its physiological ramifications,63 and a general theoretical model by McMahon64 have collectively served to bring the importance of body size as a biological variable forcibly to the attention of biologists. The recent explosive growth of both theoretical and empirical studies on the influence of size on all aspects of biology and evolution, and the extension of such studies from a focus on physiology and functional morphology to broader ecological characteristics have resulted in a spate of books on

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the subject.7,29,43 As a result, the field of allometry — the study of relationships between organism size and function — has become rather suddenly a prominent focus in ecology and evolutionary biology. The size distributions of the various organisms comprising an ecosystem provide a holistic description that facilitates comparisons of distinct ecosystems in time and space.65 Gliwicz and Umana proposed that, out of different morphological, behavioral and life history comparisons, body size appears to the factor most responsible for the vulnerability of any system dynamic.66 The allometric relationships of physiological processes of living organisms with their respective body sizes are case as algebraic statements that often include logarithms.41,43 System dynamics are regulated mainly by the physiological rate parameters of all living state variables, and body size contributes significantly to the control of these parameters.34,67 Ecological (and biological) growth and development have very much to do with the evolution of order in organized matter, and work must be done to create this order out of the background (reference state) that is less ordered.68 Purpose is frequently brought into discussions on the origin of order in the form of “goal function,” or “extremal principles” or “orientors”.5 The evolutionary ecology portends that not only will the organisms best fitted to their environment survive but that they will also become dominant under those conditions. The environment of an organism includes both its biotic and abiotic components. The abiotic environment for plankton is characterized mainly by temperature, light and nutrients; and these are determined largely by solar radiation, chemical inputs and the activities of organisms. The biotic environment consists mainly of the other organisms present in the pelagic system. Optimal structure theory informs us that the selection will favor those organism sizes whose physiological rate parameters are best adapted to the system. Evolutionary system ecology strives not simply to take into account the fitness of each individual organism, but the mutual fitness of all organisms living in the community. Detailed optimization results of exergy and ascendency are shown in Figs. 2 and 3, respectively in oligotrophic (2 mg phos. m−3 ), mesotrophic (5 mg phos. m−3 ), and eutrophic (20 mg phos. m−3 ) conditions. Different types of optimization results are obtained for two principles, however both are realistic. For exergy it is found that the best organization of the system is shown by selecting the largest size of zooplankton, but their association with the phytoplankton changes according to different nutrient conditions. In less nutrient conditions, zooplankton are associated with smaller phytoplankton and gradually their association is inclined towards comparatively larger phytoplankton with the increase of nutrient load. By obtaining the optimization result of ascendency, it shows the same system is organized by choosing smaller phytoplankton in all nutrient conditions, however zooplankton size varies according to the changing of nutrient loads. Here the size of zooplankton varies directly with the increase of nutrients input. From thorough literature survey it is found that both the conditions are realistic, any ecological system can be self-organized by different ways, but as to which is the more appropriate one is very

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(a)

(b)

(c) Fig. 2. (a) Plot of average exergy values for different size classes of phytoplankton and zooplankton in a time period of 365 days under oligotrophic condition. (b) Plot of average exergy values for different size classes of phytoplankton and zooplankton in a time period of 365 days under mesotrophic condition. (c) Plot of average exergy values for different size classes of phytoplankton and zooplankton in a time period of 365 days under eutrophic condition.

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(a)

(b)

(c) Fig. 3. (a) Plot of average ascendency values for different size classes of phytoplankton and zooplankton in a time period of 365 days under oligotrophic condition. (b) Plot of average ascendency values for different size classes of phytoplankton and zooplankton in a time period of 365 days under mesotrophic condition. (c) Plot of average ascendency values for different size classes of phytoplankton and zooplankton in a time period of 365 days under eutrophic condition.

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difficult to predict.40 In the present study the exergy and ascendency show different results and here also it is very difficult to interpret. Straskrabova42 obtained these types of different size combinations which coincided with the higher possibility of fish production. A theoretical explanation of why such different peaks of spectra might occur when the underlying relationships of the determining parameters are smooth was presented by Han and Straskraba.69,70 They demonstrated that the peaks arose because, the slope of the size relationships for individual organism groups are less than the slope across different groups. Other studies presented examples of flatter spectra (on a logarithmic scale) for marine plankton,49 and have indicated how the spectral shape changes with geographical differences. The re-analysis of size spectra models distinguish carefully the units of measurement and to exercise great care in how one group of classes of organisms.70 The size structure of pelagic ecosystems is not constant, but shifts dynamically in response to growth and mortality of individual organisms.69,71 What is missing from these empirical investigations is any demonstration of the degree to which the presence of various organisms with different dependencies on size might affect the dynamics of the ecosystem, particularly if representatives of different groups should mutually interact and modify the abiotic environment. Jorgensen12 suggested that the use of single goal function for assessing the self-organization of a particular ecological system was not appropriate and that even further research was necessary for the application of types of goal function in varied ecological systems. According to Jorgensen12 for a particular model some goal functions are appropriate while others are not, but at the same time it is very difficult to choose proper goal function(s) to know the self-organization of the specific system(s). Much debate and confusion have centered on the appropriateness of these various goal functions, because at a glance the simultaneous realization seems contradictory. Further inspection, however, shows that all these goal functions are in fact mutually consistent. They are all generated by more or less common ecological processes and give complementary perspectives on the spontaneous directions of ecological growth and development. All extremal principles are consistent with their properties. Not only are all these orientors mutually consistent, but they are inter-dependent for fulfillment. Maximizing boundary dissipation and maximizing cycling both contribute to maximizing throughflow. Maximizing throughflow contributes to maximizing storage, subject to turnover considerations.72 According to Jorgensen et al.,11 specific dissipation rather than storage per se is the primary goal function. Minimizing specific dissipation is most encompassing because it captures all three properties above. It is dependent on maximizing storage faster than maximizing dissipation, which is empirically observed.4 In conclusion, it can be supported by the use of a plurality of goal functions because each organizing principle reflects a slightly different aspect of overall system function. In fact, it is probably their complementarity and inter-dependency that has made the identification of a single universal extremal principle difficult.

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Acknowledgments The author is grateful to the late Prof. E. P. Odum, the late Prof. H. T. Odum, the late Prof. M. Straskraba, the late Prof. J. J. Kay, Prof. B. C. Patten, Prof. S. E. Jorgensen, Prof. R. E. Ulanowicz, Dr. S. N. Neilson and Dr. B. Fath, for the ideas of extremal principles received from them when he either worked in their laboratories or met them in scientific conferences. The author is thankful to Dr. Ludec Berec for his help relating to the optimization of exergy and ascendency. The author gratefully acknowledges the support of the CSIR, New Delhi (Project Ref. No. 37/1185/04/EMR-II). He is also thankful to the Department of Zoology, Visva Bharati University for laboratory and computer facilities. Thanks are also due to the Prof. Goutam Ghosal, Department of English, Visva-Bharati University, an interpreter of Sri Aurobindo’s theory of evolution, and who has gone through the language of the present work. Lastly, the author is thankful to the anonymous reviewers for their constructive comments to improve the present work.

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