Energy intensity, residence time, exergy, and ascendency in dynamic ecosystems

Ecological Modelling, 48 (1989) 19-44

19

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND ASCENDENCY IN DYNAMIC ECOSYSTEMS

ROBERT HERENDEEN

Illinois Natural History Survey, 607 East Peabody Drive, Champaign, IL 61820 (U.S.A.) (Accepted 10 April 1989)

ABSTRACT Herendeen, R., 1989. Energy intensity, residence time, exergy, and ascendency in dynamic ecosystems. Ecol. Modelling, 48: 19-44. Several system-wide indicators (energy intensity*net output, exergy, ascendency) have been proposed as optimands in optimizing principles that are useful for predicting how compartment stocks vary over time in perturbed ecosystems. This article investigates how the proposed indicators vary over time in dynamic model ecosystems. The hypothesis is that the indicators will show different sensitivities (i.e., rate of change over time) at times of abrupt changes of availability of energy and rates of cropping/stocking, and that the more sensitive one(s) deserve further study. Application to a four-compartment model ecosystem based on real steady-state data indicates that energy intensity*net output is most sensitive, and ascendency least sensitive. Exergy is or is not sensitive depending on details of the definition; problems of system boundary, disaggregation, and reference levels are still not resolved and are therefore discussed at some length. The dynamic behavior of residence time is also presented.

1. INTRODUCTION

Several quantities (energy intensity, exergy, and ascendency) have been proposed for multicompartment ecosystems to summarize, at a system-wide level, the distribution of material among compartments and/or the pattern of flows between them. Typically these have been calculated and discussed for steady-state ecosystems (Hannon, 1973, 1979, 1985a; Finn, 1976, 1980; Patten and Finn, 1979; Ulanowicz, 1980, 1986; Herendeen, 1981; Patten, 1985). All three of the quantities have been incorporated in proposed optimizing principles which can be used to predict compartment stocks over time in perturbed ecosystems (reviewed in Herendeen, forthcoming; detailed references below). Even though questions of applicable spatial and temporal 0304-3800/89/$03.50

© 1989 Elsevier Science Publishers B.V.

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R. H E R E N D E E N

scale are not yet fully resolved, and given that experimental tests are difficult and expensive, it is instructive to investigate how the quantities change in a dynamically varying ecosystem. The purpose of this article is to compare energy intensity, ascendency and exergy, and to demonstrate their behavior in a dynamic ecosystem, one whose stocks and flows are varying in response to dynamic resource perturbations (DRP), that is, changes in abundance of energy, nutrients, and cropping/stocking. 1 The underlying goal is eventual simulation of stocks vs. time in such a dynamic system. The basis for demonstration is a hypothetical four-compartment dynamic system based on real steady-state data. Residence time will also be discussed, although it is not a part of any optimization principle, nor is it a system-wide concept as used in this article. Many of the conceptual issues addressed here do not apply to a steadystate system; the interest, and difficulties, arise in explicit treatment of dynamic behavior. A fundamental difficulty is that of proper time step for simulation. An ecosystem is usually divided into compartments for reasons of function (trophic considerations) or experimental constraints. 2 In compartmental analysis, the compartment's internal structure is not known and is not desired to be known. The temporal and spatial scale of analysis is constrained in order to maintain that appropriate level of knowledge and ignorance. In particular the temporal scale (time step) must be long enough that-age class effects, seasonal factors, etc., do not affect the compartment's functional relationship between inputs and outputs. 3 This also constrains the time scale of the perturbations that can be modelled (see Appendix 1). The model faster phenomena requires a smaller time step, which necessitates disaggregating one or more compartments, thereby producing new compartments which can be considered internally homogenous on the shorter time scale.

1 In this article 'cropping' and 'exports' will usually be used interchangeably. They do, however, have different connotations: cropping connotes an activity exogenous to the system, while exports connotes an endogenous activity. These two interpretations correspond to recipient and donor control, respectively. 2 The question of the correct number of compartments has been raised but not answered (Herendeen, forthcoming); this article raises other conceptual issues. Especially relevant is whether to include detritus as a compartment when calculating energy intensity, exergy and ascendency. Of the twelve potential intercompartment flows in the system I analyze here (Table 1), six are nonzero, and five of these involve the detritus compartment. Not considering detritus would likely produce quite different results. 3 This relationship may depend explicitly on time, or it may have implicit dependence via quantities that are themselves time-varying, but it must not depend on the length of the time step.

E N E R G Y INTENSITY, R E S I D E N C E TIME, EXERGY, A N D A S C E N D E N C Y IN D Y N A M I C ECOSYSTEMS

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In Section 2 dynamic energy intensity is derived and calculated for a time-varying four -compartment ecosystem. In Section 3 the same is done for residence time. Section 4 contains a parallel discussion for exergy and ascendency. Finally in Section 5, the dynamic behaviors of these quantities, and the implications for their suitability for use in optimizing principles, are compared. 2. DYNAMIC ENERGY INTENSITY

Energy intensity for a compartment is defined as the total energy that must be input to the system in order for that compartment to produce one unit of its output. The starting point for calculating energy intensity is the balance diagram for embodied energy in a steady-state system (Fig. 1) (Bullard and Herendeen, 1975). At steady state there is no reference to stock; energy intensity is totally flow-based. While the flows are often measured in terms of energy (e.g., biomass, as they are in the bog example used in this article, Table 1), they need not be. They can even be expressed in different units for different compartments (kcal for one, grams for another, etc.). To emphasize this point, I use the generic flow unit of 'gloof'/time. Figure 1 represents a balance of embodied energy for each compartment, in which the energy embodied in the output is equal to that embodied in the inputs plus the actual energy input, if there is any (as for autotrophs). Embodied energy associated with a flow is not necessarily actual consumable energy; it is the amount of energy that must be consumed in the entire system for that flow to occur. The balance for compartment j is expressed mathematically as: n

£ ¢iXij + Ej=,jXj

(1)

i=1

where X,/is flow from compartment i to compartment j (units: gloof/time), X/ total output (throughflow) of compartment j (units: gloof/time), E/ energy input to compartment j (units: energy/time), c / e n e r g y intensity of compartment j ' s output (units: energy/gloot'), and ~ net output of compartment j = X / - •7=lXji (units: gloof/time). In Table 1, net output is the sum of basal metabolism, cropping, and stock change. The latter is zero at steady state. The solution to the resulting set of n balance equations, in vector notation, is: ~ = E ( . ! ( - X) -1

(2)

where c is vector of energy intensities, X matrix of intercompartment flows, .I? diagonal matrix of total outputs, and E vector of energy inputs.

0 0 0 0 987.55

38.05 0 0 36.9 0

Animals

0 0 0 584.9 0

Decomposers 337.4 58.21 305 0 0

Detritus 304/304 8.37/8.37 139.95/139.95 0/0

Dissipation (search/basal) 4.1 0 0 78.81

Cropping 0 0 0 0

Stock change

987.55 74.95 584.9 700.61

Output

8490 1.25 35 8836

Stock

Units: flows, kcal m - 2 year-l; stocks, kcal m - 2 . Small corrections have been made to balance the table. In the initial steady state shown here, dissipation is arbitrarily split 1:1 between search energy (dependent on output and on relative scarcity of inputs) and basal metabolism (proportional to stock). Flow is from row to column; for example, animals ingest 38.05 kcal m 2 of plants. Decomposers are defined as 'fungi and bacteria biomass'; detritus as 'the mass of dead trees, dead roots, and litter combined.., excluding peat'.

Plants Animals Decomposers Detritus Energy input

Plants

Energy flows and stocks in a four-compartment model (Russian bog, Logofet and Alexandrov, 1984)

TABLE 1

tO tO

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to other compartments

! i=1

EiX ij "~

-" EjXI Ej

Yj

Fig. 1. Steady-state embodied energy balance.

By summing all energy into the entire system we arrive at an overall balance:

e=,r

(3)

where n

E = Y'~ E i

(units: energy/time)

i=1

Equation (3) states that, at steady state, energy input to the system is all embodied in the flows to net output. In the dynamic case, stock must appear explicitly in the energy balance equation (Fig. 2). The existing stock is considered an input to the production process and an output of that process, and its energy intensity is updated each time step. This is required for consistency. The stock next time is the sum of the stock now and the change in stock. Stock change is contained in output; it is what is left over after consumption by other compartments, dissipation, and by cropping (see Appendix 2). Therefore it has the energy intensity of that output. If the energy intensity of the existing stock did not change, it would have an intensity different from that of the change in stock. Soon different

n

Eit+A t X ijtAt

Ejt+At xjtA t

i=1

/ e I'tS I't

E;jt+At S jt E.tA t I

Fig. 2. Dynamic embodied energy balance.

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R. H E R E N D E E N

portions of the stock in compartment j would have different energy intensities, which violates the assumption of homogeneity within each compartment. (As stated previously, the compartments are aggregated so as to average out age-class effects. If these effects are important, additional compartments should be used. Each would have its own energy intensity.) The presence of the time step At is necessary to make flows and stocks dimensionally commensurate. 4 Solving the balance equation pictured in Fig. 2 gives for the dynamic energy intensity: c t+AT~- ( e t A t -}- c t S t ) ( i Y t A t -

X t A t -~- ~ , ) - 1

(4)

where, in addition to the quantities already defined, the superscript refers to the time, and S ' is diagonal matrix of stocks at time t (units: gloof). Equation (4) shows the time-dynamic nature of energy intensity: its future value depends explicitly on its present value and on the time step At. At steady state, equation (4) reduces to equation (2) Equation 4 has not been normalized by dividing through by compartment outputs; the stocks and flows appear separately. Equation (4) is therefore completely general and can apply to nonlinear as well as linear models, as long as Fig. 2 applies. This includes the case in which a compartment's output may be less than its possible maximum, as might occur in some optimizing approaches. The overall system energy balance equation is now: e ' a t = , ' + a t V ' At + ( , t + ~ ' - - , ' ) S '

(5)

The energy into the system is allocated to net output plus an adjustment to the energy intensity of the existing stock. At steady state, equation (5) reduces to equation (3). Because net output is dissipation plus cropping (exports) plus growth, equation (5) can be written as: E ' At = ct+a'(DlSSIPATION t +CROPPING')At + (£t+~Tst+At -- ~ ' S t )

(6)

In equation (6) the energy input is allocated to dissipation and cropping (exports) plus the difference between the energy embodied in the new stock and that embodied in the old stock. At steady state in a materially closed system, all energy is allocated to dissipation. If there is no dissipation, then equation (6) is interpreted to mean that there can be no energy input. Conversely, if a closed, dissipationless system has positive energy input, then its stock must be growing. 4 Discrete rather than continuous time analysis is used for two reasons: (a) a non-zero minimum time interval is consistent with the idea of a compartmentalized ecosystem; (b) even if continuous analysis is used, in practice a discrete approach will almost surely be needed to solve the resulting differential equations. Hannon (1985a) uses a continuous approach.

E N E R G Y INTENSITY, R E S ID ENCE TIME, EXERGY, A N D A S C E N D E N C Y IN D Y N A M I C ECOSYSTEMS

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Equation (2) has been applied to several ecosystems (Hannon, 1973, 1979; Finn, 1980, Patten, 1985) and to several national economies (Bullard and Herendeen, 1975; Denton, 1975; Herendeen, 1978). All applications have been to the steady state. Here equation (4) will be applied to a dynamic ecosystem. I use Logofet and Alexandrov's (1984) steady-state four-compartment data for biomass energy flows in a Russian bog (Table 1) as a starting point for an assumed dynamic, nonlinear model incorporating both recipient and donor control. The dynamic model is described briefly in Appendix 2. Nonlinearity is added to produce some stability. Linear recipient control with constant coefficients relating inputs to outputs is inexorably unstable for dynamic simulation, s To impart some degree of stability (not complete stability under all DRP, which is just as undesirable as complete instability), I will use two types of nonlinear relationships which are biologically reasonable. Both incorporate density-dependent response of individual compartments to scarcity of their inputs (Fig. 3); then energy intensity will reflect overall scarcity. The first relationship expresses the assumption that a compartment's stock will not be able to produce as much output, and hence consume as much input, when inputs are relatively scarce. The second expresses the necessity of an organism to dissipate a greater fraction of total output (throughflow) in searching for and obtaining inputs when they are scarcer. Because the flow variable in the Russian bog is energy, the treatment of dissipation strongly affects the energy intensities. There are two possible extremes: assigning either all or none of dissipation to net output. Assigning it all solves the problem mentioned above; at steady state there is a nonzero energy input. However, in this case all energy intensities have the same value (1 kcal/kcal). 6 On the other hand, assigning no dissipation to net output

5 Stability here means that no compartment goes extinct. Scott Overton (personal communication, 1984) brought to my attention the general principle that linear analysis cannot be stable for both positive and negative time. That is, an arbitrary, physically possible (i.e., non-negative) stock vector will ultimately give rise to a physically impossible one (i.e., containing negative entries) in either forecasting or backcasting. Thus dynamic I n p u t - O u t p u t analysis in economics (Leontief, 1970) does well at backcasting the past structure of the economy, but fails at predicting future states, as acknowledged by Leontief and discussed by Dorfman, Samuelson and Solow (1958). Leslie population projection matrices have the opposite quality: an arbitrary, physically possible age distribution leads to physically possible ones in the future, but backcasts to physically impossible past distributions (Pielou, 1977). Additional assumptions can remove these difficulties, but they are nonlinear. 6 At steady state this follows from equation (3). E equals the sum of the net outputs, implying that the average of the energy intensities is 1. Because no intensity can be less than 1, all must equal 1.

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R. HERENDEEN

Input availability

O

"5

Input availability Fig. 3. Assumed nonlinear relationships in dynamic model. A n example of input availability for compartment j :

[ Sx/SJ

S2/Sj ...

S(O)a/S(OL S(O)2/S(OL

L/Sj ]1/n L(O)/S(O)j

where L is light level (kcal/day); '(0)' refers to stock and light levels at an initial steady state, at which availability = 1.0; n is number of inputs, including light if appropriate. The light term is included only if j = autotroph.

(i.e., assuming it is completely embodied in intercompartment flows) leads to different energy intensities, but implies no energy use at steady state by equation (6), which is unacceptable. I take an intermediate approach, dividing dissipation into two parts: basal metabolism, which is considered a part of net output; and search energy, which is considered an inflow dissipation. Basal is assumed proportional to stock, while search depends nonlinearly on scarcity of inputs, as discussed above. Breaking dissipation down this way has some justification; the two types of dissipation are qualitatively different (Costalaza and Hannon, forthcoming). There is, however, a persisting element of arbitrariness. In Table 1, I assume that at the initial steady state, dissipation is split 1 : 1 between the two types, for all compartments. Figure 4 shows the time behavior of the stocks and energy intensities in the four-level bog system in response to a D R P regime that includes abrupt

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND ASCENDENCY IN DYNAMIC ECOSYSTEMS

27

(occurring in one year) changes in light intensity and level of cropping of the plant compartment. Detritus has dynamics different than that of the other compartments, as discussed in Appendix 2, but it is still proper to assign it an energy intensity. Fig. 4a shows the response of the compartment stocks. The plant track the light and cropping monotonically, and the rest of the compartments follow monotonically. The system can tolerate this increased cropping only with the increased light level. If cropped this way without the increased light, the plants are extirpated and the whole system dies. Energy intensities, shown in Fig. 4b, behave somewhat differently. Their steady-state values change little or not at all with a doubling of light level, but they do decrease with increased cropping and increase with decreased cropping of plants. (For part of the assumed regime, cropping is assumed to be negative, corresponding to stocking the system. Details are given in the caption for Fig. 4.) The observed response to cropping is reasonable: for example, cropping maintains the plant stock at a lower level, the available light per plant increases, the plants' search energy decreases, and the energy intensity of plants decreases. This decrease reduces the embodied energy input to the consumers of plants, tending to reduce their energy intensities, and so on. (This argument could have exceptions; for example, scarcity of plants increases search energy of herbivores, thereby tending to increase herbivores' energy intensity.) There is no analogous effect for light because of an asymmetry between light and cropping. When light is increased, stocks tend to increase and to drive the ratio of autotroph stock to fight back towards its original value. However, when autotroph cropping is increased, autotroph numbers are depressed but the light does not follow, and input availability remains greater. What is most interesting is the transient behavior of the energy intensities when the light level is abruptly changed. The energy intensity undergoes a large change in a direction opposed to its final change at the new steady state. These transients occur because of a delay in the tendency of stocks to track the light. The delay is a consequence of the interplay of stock and output for autotrophs, and of all compartments. Suddenly doubling a compartment's output does not immediately result in a doubling of its stock. For example, for plants in Table I the ratio of stock to flow (the steady-state residence time) is 8.6 years, so that a doubling of output would take at least that long to double the stock. Actually it would take much longer because not all of the doubled production would go to stock increase; much would be consumed by animals, whose own stocks would also be increasing. These 'contrary' transients in energy intensity are more likely for light changes then for changes in cropping. The size of the transient can be rather

28

R. HERENDEEN

large. Energy intensity thus shows variability that is qualitatively different from that shown by stocks at times of abrupt DRP. Hannon (1979, 1985b) has proposed that ecosystems maximize ~Y, which at steady state is equivalent to maximizing energy input by equation (3). According to equation (5), however, it cannot be equivalent in the dynamic case. This optimand is a product of a term proportional to system size and a term reflecting system structure, i.e., one containing the relative sizes of

1.00.9 0.8 0.7 0.6

i

0.5

plants

0.4 0.3

animals

0.2

decomposers detritus

0"1 1 0.0

2j

,

~

~

,

,

• <-- light

1<-- plants cropping

detritus cropping 4,

.o)

-1 -

plants stocking -->

-2-

0 (o)

40 200 3~0 4~0 5~0 6;0 40 8~0 9~0 1000 year

Fig. 4. Stocks and energy intensities in dynamic model ecosystem. Response of four-compartment bog system to changes in light availability and changes in cropping/stocking of plants. System is initially at steady state as given in Table 1. Light is doubled at year 33 and returned to initial level at year 533. Cropping of plants is increased to 500 kcal m - a day-1 at year 200, returned to original level at year 367, reduced to - 5 0 0 kcal m -2 day - l (i.e., system is stocked) at year 700, and finally returned to original level at year 867. (a) Response of stocks. (b) Response of energy intensities. All graphs are arbitrarily normalized for appearance. Actual initial stocks are given in Table 1. Initial energy intensities: plants, 1.45; animals, 1.94; decomposers, 2.63; detritus, 2.00 (all in kcal/kcal). Time step used in simulation: 0.033 year.

E N E R G Y INTENSITY, RESIDENCE TIME, EXERGY, A N D A S C E N D E N C Y IN D Y N A M I C ECOSYSTEMS

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1.0 0.90.80.7~

0.6-

x E

0.5-

~

0.4-

plants animals decomposers detritus

0.30.20.1 0.0

2j

t -,ight

1-

I plantscropping

&

detritus cropping

$

o 0

.d plants stocking -~

-1-2

100 2(30 300 400 500 600 700 800 900 1000

(b)

year

Fig. 4 (continued). compartments. To see this, define:

Y=LY

(assuming that the sum is justifiable dimensionally)

i=1

yi = g / r

(dimensionless)

Then: t'l

or=

n

E ¢.iYi "~- Y

E ciYi

i=1

i=1

(7)

3. RESIDENCE TIME Residence time can be defined with respect to individual compartments as well as with respect to larger aggregations, including the entire ecosystem. In this article, I confine attention to single-compartment residence time because it is simpler to interpret in a dynamic system, and yet adequate for

30

R HERENDEEN

discussing some aspects of the extent to which energy cycles in ecosystems (Patten, 1985; Higashi, 1986; Herendeen, 1988). Whole-system cycling time has been developed and discussed by Finn (1976). At steady state, single-compartment residence time may be defined in two equivalent ways: - the average time a 'particle' of energy, biomass, nutrient, etc. placed in a compartment remains there before leaving for the first time; the average time that a particle now in the compartment has been there continuously. A dynamic definition will combine both approaches: residence time at time t + At depends on its value at time t and on what happens to the compartment during At. The value of residence time is strongly dependent on its definition and the assumed dynamics of the system. For example, at steady state residence time could be infinite if a compartment's stock never interacts with its throughflow. If stock and flow interact to the fullest possible extent, the steady state residence time is given by S / X , where S is the stock (units: gloof), and X the output (throughflow) (units: gloof/time). Values between S / X and infinity are possible depending on the assumed degree of interaction of stock and throughflow (Herendeen, 1988). The maximum interaction of stock and flow is the full mixing assumption, which states that stock (S) and throughflow for the period ( = X At, where At is the time step) are used collectively to satisfy all output from the compartment (ingestion by other compartments, mortality flows to detritus, dissipation, cropping (exports), and growth either somatically or via reproduction). For this assumption to apply requires that the time step be small compared with the lifetime of the organisms in the compartment. It is also assumed that the destination of the flow out of a compartment is independent of the source of the flows into the compartment. This is the Markov assumption used by Patten (1985) and Barber, Patten and Finn (1979). On the other hand, in a dynamic simulation the time step must be large enough for the internal temporal details may be averaged out (Appendix 1): there can be no age-class effects. The goal is then to obtain an expression for residence time that incorporates both a non-zero time step and the full mixing assumption. As stated, residence time derived here is a property of a compartment, not of the whole ecosystem. We can therefore leave off the subscript to identify the compartment: all quantities apply to the same compartment. Given a compartment with S t is stock at time t (units: gloof), X t output (throughflow) at time t (units: gloof/time), At time step, "rt residence time at time t, and "1" t + a t residence time at time t + At. We define residence time under the full mixing assumption by: •

=

xt)s'+ x'

X'

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND ASCENDENCY IN DYNAMIC ECOSYSTEMS

31

Equation (8) states that the stock ages from residence time r ' to r t+~xt in the period At, and that the throughflow has an average residence time of At/2, the flow being uniform across At. Both of these assumptions are only valid if At << organism lifetime. At steady state: ~.,+a, = r ' = r = S / X + At~2

(9)

Equation (9) reduces to r = S / X when At = 0, but for At = 0, equation (8) states that r is unchanged. Thus r = S / X is a possible steady state value, depending on initial conditions. It was argued above that At must > 0 in a proper simulation model, and it is disconcerting to note that equation (9) indicates that r increases without limit as At increases. This is unreasonable: we would expect that residence time should not exceed approximately the average lifetime of the organisms in the compartment. This indicates a flaw in equation (8), which embodies the assumption that the stock S ages continuously during a long At, which it cannot. If is is a living compartment (rather than detritus or a nutrient), death intervenes. Equation (8) applies only to infinitesimal At, and we still seek an expression for finite time step. We do this by using equation (8) m a n y times in sequence, each for a time u. We then take the limit as the n u m b e r of steps, N, goes to infinity while u goes to zero, such that their product, Nu = At, is finite; At is the time step used in simulation, long enough so that internal temporal details of intracompartment processes can be averaged out. In the summing and limit-taking, stock and throughflow are assumed constant over the period At. Then:

,rt+u = [(g t --~ u)gt-ot - X t u ( u / 2 ) ] / [ S ' + X'u] ,rt+2u_--[(,r

t + u -+-

u)St -+-Xttt( u/2)]/[ St + Xt u]

r '+3u= [ ( r '+2u + u ) S ' + X ' u ( u / 2 ) ] / [ S ' + X'u]

,rt+Nu= [(,rt+(N-1)u -+-u)St + Xtu(u/2)]/[ S t q- Xtu]

This series of equations can be solved to yield:

=pN , + u ( p + p2 + p, + ...

+

lux/s)

or

r'+N"=pUr t + u [ p ( 1 - - p N ) / ( 1 - - p ) ] ( 1 + ½uX/S)

(10)

32

R. HERENDEEN

where p • s's =

St/(St--] - Xtu). Equation (10) has the steady-state solution:

=

s'/x'+ u/2

This expression for steady-state residence time resembles equation (9). A difference is in the superscript 't ', which indicates that this is the steady-state residence time that would result if the stock and throughflow were held at their present values for a long time. However, steady-state residence time still increases with increasing u. Now let u = A t / N and take the limit of equation (10) as N approaches infinity, recalling that: lim(N approaches infinity) of (1 + k / N ) N = exp(k) Then: " / " + A t = T/

e x p ( - At X ' / S t)

+ (St/Xt)[l - e x p ( - A t X t / S t ) ]

(11)

This expression is bounded as At increases, in contrast to equation (8). Equation (11) yields at steady state, for At ~ 0:

rts=St/X '

(12)

Rewriting equation (11):

Tt+At=~sts + ('/'t-- q'sts) exp(-- At/'r•,)

(13)

Equation (13) shows that for a vanishingly small time step At, residence time does not change, and that it can be different from the steady-state value corresponding to the instantaneous stock and flow. For large enough At, if stock and flow hold constant, the residence time at time t + At becomes independent of the initial value and instead depends only on present stock and flow. The characteristic time of the relaxation to the steady-state value is itself the steady-state residence time. This is also evident in the continuous form of equation (13): 0"r/0(A/) = - (~" - %s)/'rss where 0 denotes partial differentiation. Use of a non-zero time step delays and smooths out changes in residence time. It is no surprise that phenomena faster than the time step cannot be simulated. Figure 5 shows the dynamic behavior of residence times calculated using equation (13). Plants' residence times behave as expected. Plants respond to increased light by increasing their output. During a growth period, the ratio of stock to output decreases and residence time drops; the average age of a rapidly growing population is low. Conversely, residence time increases during a period of decline caused by diminished light. Plants' response to changed cropping is a consequence of the assumed dependence of o u t p u t /

E N E R G Y INTENSITY, RESIDENCE TIME, EXERGY, A N D A S C E N D E N C Y I N D Y N A M I C ECOSYSTEMS

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1.0

0"9 0.8t 0.7

plants

_ _

~

g

0.5-

"~

0.4-

S

animals decomposers detritus

0.30.2. 0.10.0 2~

9.o b

light ~-

plantscropping

detrituscropping ,1,

0

E: plantsstocking- *

-1,

-2

0

t;o 2;0 3;0 4;0 ~;0 6;0 7;0 8;0 900 1000 year

Fig. 5. Residence times in dynamic model ecosystem. Regime and normalization as in Fig. 4. Initial values of residence times ( = stock/output from Table 1): 8.60, 0.0167, 0.0598, 12.6 year. stock, which at steady state equals the residence time, on input availability (Fig. 3). Increased cropping reduces stock, there is more light per plant, and o u t p u t / s t o c k increases. The opposite is true when plants are stocked rather than removed. Residence times for animals and decomposers show negligible variation in this example because: - their steady-state residence times are less than 1% of residence times for the compartments from which they receive material, plants and detritus; - they are not directly affected by DRP, having no light dependence and being uncropped. Residence time for detritus also is as anticipated. Detritus is donor-controlled: its inputs are determined by mortality in the contributing compartments (here assumed proportional to their stock). When their stocks in-

34

R. HERENDEEN

crease, their inputs to detritus increase, transiently increasing detritus' output and reducing residence time; this occurs whether the increase results from increased light or decreased cropping. Similarly, detritus residence time increases transiently when input compartments' stock decrease, whether caused by decreased light or increased cropping. 4. EXERGY AND ASCENDENCY Two system-wide indicators of orderedness have been proposed: - exergy, which is based on the orderedness of the distribution of stocks among ecosystem compartments (Jorgensen and Mejer, 1981, 1983; Jorgensen, 1982, 1986); - a s c e n d e n c y , based on the orderedness of intercompartment flows (Ulanowicz, 1980, 1981, 1986). TABLE 2 Expressions for energy intensity (c), residence time (r), exergy (EX), and ascendency (A) Et+AI = (E' r,'+a'=

At +,'gt)(X' At-- X' At + S') r,t exp(- At X[/S[) + (St/X[)[1 - e x p ( - At X[/S[)]

Ex= RTC°[ ln(C°/C°'ref)-(1-C°'ref/C°)+

i=lkXiln(xi/Xi're')]

A=flk=Oj=O y" ~ (XkJ/fl)ln[(Xkj/Xk)/t i~=oXij/fl

(4)

(11) (15) (16)

The number in parentheses is the equation number in the text, where the symbols are explained. The expressions for exergy and ascendency are listed in Table 2, as is that for energy intensity. Both have been specified in optimizing principles: ecosystems behave in such a way as to optimize exergy or ascendency. This inclusion is important when one notes the degree of explicit dependence of the quantities on stock, flow, and time. For example, one might criticize exergy for its explicit non-dependence on intercompartmental flows: it can apparently be calculated for a system that is dead. 7 Similarly, ascendency 7 Jorgensen, Logofet and Svirezhev (1988) comment upon exergy's lack of explicit dependence on flows: "This [lack] contradicts one of the main paradigms of ecology, that it is the structure of matter and energy flows which determine the state of an ecosystem. Also, the basic concepts of ecology such as trophic levels, matter and energy cycling, all are primarily structural notions."

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND

ASCENDENCYIN DYNAMIC ECOSYSTEMS

35

depends explicitly on flows, with no apparent dependence on stocks. Neither exergy nor ascendency depends explicitly on time. Energy intensity, on the other hand, depends explicitly on stock, flow, and time. The lack of explicitly dependence is quickly seen not to be objectionable, however, because an optimization principle requires not only an optimand, but also an assumed (dynamic) model for the system. The model necessarily relates stock and flows over time (Appendix 2). Therefore exergy and ascendency depend implicitly on both stocks and flows. Exergy per liter (EX) is (Jorgensen and Mejer, 1983; Wall, 1986): n EX =

RT E Ci[ln(Ci/Ci,ref)-

(1 -

Ci,ref/Ci) ]

(14)

i=1 where R is the gas constant (units: kcal mo1-1 deg-1), T absolute temperature of ecosystem 8, Ci is concentration of compartment i's stock (units: mol/1), Ci,re e reference concentration of compartment i's stock, and n number of compartments. The term C~ ln(CJCi,ref) arises from the theoretical minimum work to be done on the ecosystem to change its concentration relative to the reference level, and the term C,(1 - C,,rcf/Ci) arises from the work done by the constant-concentrations surroundings. With this definition, exergy is always nonnegative, independent of whether the system is more or less concentrated than the reference state. 9 In practice the system is usually much more concentrated and the second term is negligible. Because of the logarithmic form of equation (14), exergy can be written as the sum of a size term representing the overall concentration of the system's stock relative to that of the reference level, and a structural term representing the distribution of stocks among the compartments relative to that of the reference level. This separation requires that summing the stocks is physically sensible:

Ex = RTC°[ ln(c°/c°'ref) - ( 1 - C°'ref/C°) + ~ xi ln(xi/xi'ref) where CO is summed concentrations of compartments, ence concentrations of compartments, and:

C0,re f

(15)

summed refer-

x, = CJCo Xi,re r = Ci,ref/fo,re f

The structural term R T C o Z,7=~ x~ ln(xi/Xi, ref) does actually depend on size, but now multiplicatively, such that a doubling of either size or the contris Assume that the temperature is uniform throughout. 9 In both cases the concentration difference can in principle be exploited to yield positive work.

36

R. H E R E N D E E N

bution from distribution gives a doubling of exergy. The choice of reference levels is important to the relative contributions of the size and structural components, a0 Exergy as defined by Jorgensen and coworkers is maximized by all stock being in one compartment. The structural term varies between R T C o ln(1/ Xi,ref,~n) and 0, corresponding to all stock in the originally least-occupied compartment, and to stock distributed in the same proportions as in the reference case, respectively. The principle drives towards one compartment, towards no system at all. As stated above, this surprising goal is not necessarily damning, because the maximization operates within the confines of the assumed system model. Ascendency (Ulanowicz, 1980, 1981) is defined exclusively in terms of flows. It uses direct flows only; there is no explicit accounting of indirect effects, such as the connection of autotrophs and carnivores via herbivores. Ulanowicz (1980) states that ascendency: ...perhaps...is best described as the coherence of the flow network, i.e., an indicator of the degree to which the flow system differs from either a homogeneous network or a collection of totally independent parts... [It] can also be interpreted as the average degree of unambiguity with which an arbitrary compartment communicates with any other compartment in the system.

The expression for ascendency (A) is (Ulanowicz, 1986):

A=BE

E

In

Xi/B

(16)

k=O j = 0

where/3 is total system output (throughflow): n+2

/3= Ex, i=0

Again, it is assumed that the sum makes sense physically. The ratio X k j / X k is the probability that output (throughflow) from compartment k will go to compartment j. In contrast to exergy, the size and structural terms of ascendency are not added, but multiplied. Ascendency is also defined at a single instant of time. Ascendency is maximized when each compartment sends all inputs, undiminished, to only one other compartment (Ulanowicz, 1986, p. 101), in a lossless, unbranching circulation that is highly unrealistic, a0 Exergy also includes a term for the chemical potential of compounds which do not occur in the reference state. For example, the exergy of COH1206 formed from CO 2 and H2 O is typically dominated by this binding energy. If the reference level is defined to contain already-formed C6Ha206, there is zero contribution to exergy. J~rgensen (1986) makes this latter assumption.

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND ASCENDENCY IN DYNAMIC ECOSYSTEMS

37

and physically impossible if the flow variable is energy. This is as unattractive as the all-eggs-in-one-'basket of exergy maximization, but ascendency maximization is also subject to " . . . hierarchical, thermodynamic, and environmental constraints." (Ulanowicz, 1980) The range of subscripts indicates that the boundary for calculating ascendency extends beyond the n compartments used for energy intensity and exergy: i = 0 corresponds to system inputs, such as energy, n + 1 refers to dissipation, and n + 2 to exports (Ulanowicz, 1986). About this extension: The use of dissipation strongly suggests that energy is meant to be the flow variable. This bias towards energy was mentioned above with respect to materially closed ecosystems. - It assigns a kind of donor-control determinism to exports by assuming that the distribution of exports among compartments is fixed. It does the same thing for ir~puts. 11 It 'triple counts', adding the summed outputs of the n compartments (which already represents double counting in the same sense that in economics gross output greatly exceeds gross national product) to the inputs that allowed those outputs to occur. -

-

5. D Y N A M I C BEHAVIOR O F E N E R G Y I N T E N S I T Y * N E T O U T P U T , EXERGY, A N D ASCENDENCY

I have suggested (Herendeen, forthcoming) that carefully planned simulation 'experiments' are a proper next step in testing the reasonableness of the optimizing principles. While somewhat expensive, they are much less so than actual data collection. The only attempt until now has tested the first optimand, ~, Y, and has been inconclusive (Herendeen, forthcoming). A more modest step is to investigate how the quantities vary in a dynamic ecosystem. The implicit hypothesis is that the quantities vary significantly during transitions between steady states, and that differences in their transient behavior (for example, the degree to which they track the perturba-

11 A compartment's excess, i.e., output - (that consumed by other compartments) dissipation, can be removed (cropped) a n d / o r contribute to growth of stock. If cropping is determined exogenously, then what a compartment perceives as (potential) export must be this excess, i.e., the sum of cropping and stock change. There is no justification for separating the two portions, given the assumed homogeneity of the compartment. Export so defined should be included in equation (16), unless it < = 0, in which case it is left out. The relative shakiness of this argument emphasizes how difficulties arise in a dynamic situation as compared with steady state: at steady state stock change is zero, and the excess is all cropping.

38

R. H E R E N D E E N

tions over time) will suggest that one is more sensitive at transitions which are known to be destabilizing, such as abrupt decreases in availability of light. Specifically, I define sensitivity of an indicator as a combination of two attributes: - exhibiting a large change over time under DRP, with that change having a time pattern different from that of the DRP, i.e., one that does not track the DRP. Figure 6 shows behavior of the three quantities - cY, exergy, ascendency for the four-compartment Russian bog. 12 Ascendency seems to track the perturbations the most closely, with no contrary transients. Energy intensity* net output shows an undershoot when light is reduced, a consequence of the fact that net output can actually be negative for organisms which suddenly have reduced availability of inputs. For example, plants' output is reduced but the demands of grazers are not reduced until after a lag, and overgrazing can result. Exergy's sensitivity depends strongly on the assumed reference level. Two possible choices are presented. If only the structural term in equation (15) is used, representing the deviation of the relative stocks of the four compartments from their original proportions, exergy is extremely sensitive to DRP, being large only at transitions between steady states. If the full equation (15) is used, exergy is dominated by the size term, and it shows the same pattern as stocks, with no remarkable behavior at transitions. In both figures it is assumed that the reference level for calculating exergy is the original steady state. If, as argued by Jorgensen (1986), the reference level is some very low concentration corresponding to the non-living aquatic background, there would be an additional term, easily on the order of one million times as large as the variations in Fig. 5, in which case the effects in Fig. 5 would be undetectable on the scale fo the graphs. The question of -

12 Robert Ulanowicz, author of the maximum ascendency principle, has (personal communication, 1988) pointed out the limits of this demonstration. It is performed for a fixed-form model in which inter- and intracompartment interactions are specified beforehand. This implicitly requires a short time scale compared with evolution, and yet ascendency maximization is more suited for the longer time scale, in which functional relationships between compartments can change, even to the point of disappearing completely or appearing where none existed before. Cheung (1985) has performed such maximization; he does not investigate dynamic behavior but rather compares possible steady states when links are allowed to be added to, or removed from, known systems. Ulanowicz has also said (personal communication, 1987) that knowing ascendency at any given time is "...useful .... even if the conditions for the hypothesis of increasing ascendency do not prevail." On the other hand, Jorgensen and Mejer (1981) have stated that exergy maximization applies at all time scales, and Hannon (1985b) has not specified a time scale for maximizing (energy intensity)* (net output).

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND ASCENDENCY IN DYNAMIC ECOSYSTEMS

39

1.0 0.9

~"~1!

~- ascendency

0.8 >. t~ 0.7 o 0.6 ~

0.5-

-

0.4-

~0) x 0.30.20.1

.

0.0 light

1.

detritus cropping $

plants cropping

o

0 plants stocking --> i

-2 1(~0 2()~ 3(30 4(30 5()0 6(}0 7(30 8()0 900 1000 year

Fig. 6. Energy intensity* net output, exergy, and ascendency in dynamic model ecosystem. Regime and normalization as in Fig. 4. Exergy (structural term only) uses only the structural term in equation (15). Exergy (size and structural term) uses the full equation (15). In both cases, the reference level is assumed to be the initial steady state. reference level thus remains important and vexing, and has led at least one researcher to abandon using exergy (Ulanowicz, 1986). The conclusions above also apply to two similar exercises which are not presented in this article, which I have performed on the five-compartment Silver Springs ecosystem (Odum, 1957) and the six-compartment oyster reef system of D a m e and Patten (1981). For both, the original data (biomass energy) for stock and flows are for steady state and are treated dynamically using the nonlinearities assumed here, and are subjected to similar D R P . From these three examples, in terms of decreasing sensitivity to D R P , the quantities are ranked thus: - exergy (structural term only) - energy intensity* net output - exergy (size + structural term) - ascendency.

40

R. HERENDEEN

This is comforting from an experimental standpoint, as it is so m u c h easier to measure stocks than flows, but the demonstration here is unfortunately not compelling enough to cause one to embrace exergy to the exclusion of the other quantities. It is not impossible that the observed ordering is model-dependent. 13 The applicability of the various principles to the time scale used here is, as mentioned, still contested. Realizing this, I forward these preliminary conclusions, based on the strong belief that studying dynamic systems is the best way to pursue the question of optimizing principles in ecology. APPENDIX 1

Time step in dynamic simulation This appendix covers several constraints on the time step At used in dynamic simulations and in calculating the various system-wide indicators. These are: (1) Ensure that At exceeds transit times of material and organisms, and relaxation times for temperature inhomogeneities, so that system-wide quantities can be defined. The economic analog of this is the "clearing of the market" (Amir, 1987, and personal communication, 1987). This is relatively easy to satisfy. (2) In order to justify the assumption of homogeneity within a compartment, ensure that At exceeds the lifetime of organisms in the slowest compartment. (3) A large At tends, however, to drive fast compartments extinct when energy and other inputs are changed. I f we desire to avoid this, ensure that perturbations are gradual enough. (This last requirement is somewhat arbitrary; perhaps we wish to simulate such an extinction, in which case the constraint should not be applied.) A semi-quantitative definition of 'gradual' is sketched following. In response to a change in availability of light or other inputs in a time At, a compartment's output X and its flows to other compartments are changed. For this not to result in extinction, the absolute value of the product of the flow imbalance (AB) and the time step between production 13 For one example, the model used here does not contain a nutrient loop (or submodel). Sven-Erik Jorgensen (personal communication, 1988) argues that nutrient should be included in the calculation of exergy. Work on a model containing a nutrient loop is under way.

ENERGY INTENSITY, RESIDENCE TIME, EXERGY, AND ASCENDENCY IN DYNAMIC ECOSYSTEMS

41

and outputs should be small compared with the stock, i.e.

i(dB/dt)l(At)2
or

L stands for whatever is being perturbed, such as light. By dividing dB by X we suggest that there is some proportionality between balance and output, which is not always true. Let 1/[dL/L)/dt] be called P, the characteristic time of the perturbation. Then equation (AI-1) becomes:

(dB/X) (At) 2 <1 (dL/L) rsse

(A1-2)

Equation (A1-2) implies a reciprocity between the compartment's residence time, and the characteristic time of the perturbation: any combination yielding the same product would be equally likely to lead to extinction of the compartment. Usually a compartment's balance B depends in a complicated way on stocks and flows of all compartments which have inputs into or outputs from it, so that this reciprocity does not hold exactly; (dB/X)(dL/ L) for a compartment depends on the entire system. Equation (A1-2) also shows a quadratic dependence on the time step At. Reducing the time step is thus more effective than reducing the abruptness of the perturbation by a proportional amount. As an illustration of equation (A1-2), consider a human; At is roughly the generation time, 30 years; S is roughly 20 kg (dry carbohydrate), X is roughly 300 kg/year (food intake), and % - - S / X = 1/15 year. Of the annual ingestion, 2/3 is for basal metabolism and 1/3 for searching, etc. Now assume that the food supply is halved: basal metabolism continues unchanged and search activities are also unchanged (the latter would likely increase in the face of scarcity). Then dB is 150-200-100 = - 150 kg/year, and (dB/X)/(dL/L) = 1. Equation (A1-2) gives that the characteristic time of perturbation, P, must be on the order of at least (30)(30)(15)= 13 500 years! Because of the quadratic dependence on At, reducing At has a large impact on the required P. If At were reduced to 1 year, P would decreased by a factor of (30.30) to 15 years. In the dynamic simulation of the Russian bog shown in Figs. 4, 5 and 6, the time step is 0.033 year, well below the lifetimes of many plants and animals, and violating constraint (2) above. I found that the reciprocity between At and P did break down in this case, and that even extremely gradual perturbations produced extinction with At = 1 year. To obtain stability, I arbitrarily reduced At: the simulations thus have an additional 14 A t s t e a d y state, B = 0.

42

R. H E R E N D E E N

aspect of unreality. A more proper resolution would employ modifications of the nonlinear relations in Fig. 3. APPENDIX 2

Dynamic model For the purposes of this article the details of the model are relatively unimportant compared with overall structure. As shown in Fig. 7, stated broadly, for one time step the model: - begins with specified stock in each compartment; determines what maximum output (throughflow) each stock can sustain (which depends nonlinearly on abundance of inputs); determines, via optimization of some optimand, actual outputs; optimization is not used in this article, and the actual output is assumed to be the maximum; allocates inputs to other compartments, according to their outputs, and to self-use. Self-use has two parts: basal metabolism (maintenance), which depends on stock; and a 'search' term, which depends nonlinearly on -

-

-

Stock(St)

I stook-~,ow I I M~e, I

/ / ~

/ rePleat

\

NewStock 1 St* At__st+Ast

I IOpUmizel r A ? t

t ,

L otuaou°u, j / k =, /

[~...-I~ C. . . . . . d

whichj~eaves .c,u.,

(.2

)

Fig. 7. Schematicdiagram of dynamic model.

E N E R G Y INTENSITY, R E S IDENCE TIME, EXERGY, A N D A S C E N D E N C Y IN D Y N A M I C ECOSYSTEMS

-

-

-

43

abundance of inputs. This is recipient control except for the detritus compartment, whose inputs are determined by mortality in the other compartments; removes cropping or adds stocking; determines the stocks at the end at the beginning of the next time step as a residual = (stock at beginning of this time s t e p ) + output (inputs to other compartments and to self) cropping; and then repeats the sequence.

ACKNOWLEDGEMENTS

I am grateful for discussions and correspondence with Robert Ulanowicz and Dmitri Logofet and for the stimulating hospitality of Sven-Erik Jorgensen during a visit supported by the National Science Foundation under Grant NSF-8713261. REFERENCES Amir, S., 1987. Energy pricing, biomass accumulation, and project appraisal: a thermodynamic approach to the economics of ecosystem management. In: G. Pillet and T. Murota (Editors), Environmental Economics - The Analysis of a Major Interface. R. Leimgruber, Geneva, pp. 53-108. Barber, M.C., Patten, B.C. and Finn, J.T., 1979. Review and evaluation of input-output flow analysis for ecological applications. In: J.H. Matis, B.C. Patten and G.C. White (Editors), Compartmental Analysis of Ecosystem Models. Statistical Ecology Series, 10. International Co-operative Publishing House, Fairland, MD, pp. 43-72. Bullard, C. and Herendeen, R., 1975. The energy costs of goods and services. Energy Policy, 3: 268-278. Cheung, A., 1985. Network optimization in ecosystem development. Dissertation, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, MD (forthcoming). Costanza, R. and Harmon, B., forthcoming. Multicommodity ecosystem analysis: dealing with apples and oranges in flow and compartmental analysis. In: B. Patten and S. Jorgensen (Editors), Progress in Systems Ecology: Mid-1980's Issues and Perspectives. Dame, R.F. and Patten, B.C., 1981. Analysis of energy flows in an intertidal oyster reef. Mar. Ecol. Progr. Ser., 5: 363-380. Denton, R.V., 1975. The energy costs of goods and services in the Federal Republic of Germany. Energy Poficy, 3: 279-284. Dorfman, R., Samuelson, P. and Solow, R., 1958. Linear Programming and Economic Analysis. McGraw-Hill, New York. Finn, J.T., 1976. Measures of ecosystem structure and function derived from analysis of flows. J. Theor. Biol., 56: 115-124. Finn, J.T., 1980. Flow analysis of models of the Hubbard Brook ecosystem. Ecology, 6: 562-571. Hannon, B., 1973. The structure of ecosystems. J. Theor. Biol., 41: 535-546. Hannon, B., 1979. Total energy cost in ecosystems. J. Theor. Biol., 80: 271-293. Hannon, B., 1985a. Linear dynamic ecosystems. J. Theor. Biol., 116: 89-110.

44

R. HERENDEEN

Hannon, B., 1985b. Ecosystem flow analysis. Can. Bull. Fish. Aquat. Sci., 213: 97-118. Herendeen, R., 1978. Total energy cost of household consumption in Norway, 1973. Energy, 4: 615-630. Herendeen, R., 1981. Energy intensity in ecological and economic systems. J. Theor. Biol., 91: 607-620. Herendeen, R., forthcoming. Do economics-hke principles predict ecosystem behavior under changing resource constraints? Presented at 1986 ISEM Annual Meeting, August 1986, Syracuse, NY. In: T. Burns and M. Higashi (Editors), Network Perspective in Ecology. Cambridge University Press, London. Herendeen, R., 1988. Network trophic dynamics. Letter in response to B.C. Patten, Energy cycling in the ecosystem. Ecol. Modelling, 42: 75-78. Higashi, M., 1986. Residence times in constant compartmental ecosystems. Ecol. Modelhng, 32: 243-250. Jorgensen, S.E., 1982. Exergy and buffering capacity in ecological systems. In: W. Mitsch, R. Ragade, R. Bosserman and J. Dillon (Editors), Energetics and Systems. Ann Arbor Science Publishers, Ann Arbor, MI, pp. 61-72. Jorgensen, S.E., 1986. Structural dynamic model. Ecol. Modelling, 31: 1-9. Jorgensen, S.E. and Mejer, H., 1981. Exergy as key function in ecological models. In: W.J. Mitsch, R.W. Bosserman and J.M. Klopatek (Editors), Energy and Ecological Modelling. Developments in Environmental Modelling, 5. Elsevier, Amsterdam, pp. 587-590. Jorgensen, S.E. and Mejer, H.F., 1983. Trends in ecological modelling. In W.K. Lauenroth, G.V. Skogerboe and M. Flug (Editors), Analysis of Ecological Systems: State-of-the-Art in Ecological Modelling. Developments in Environmental Modelling, 1. Elsevier, Amsterdam, pp. 21-26. Jorgensen, S.E., Logofet, D.O. and Svirezhev, Y.M., 1988. Exergy principles and exergical systems in ecological modelling (manuscript). Leontief, W., 1970. The dynamic inverse. In: A. Carter and A. Brody (Editors), Contributions to Input-Output Analysis. American Elsevier, New York, pp. 17-47. Logofet, D.O. and Alexandrov, G.A., 1984. Modelling of matter cycle in a mesotrophic bog. 1. Linear analysis of carbon environs. Ecol. Modelhng, 21: 247-258. Odum, H.T., 1957. Trophic structure and productivity of Silver Springs, Florida. Ecol. Monogr., 27: 55-112. Patten, B.C., 1985. Energy cycling in the ecosystem. Ecol. Modelling, 28: 1-71. Patten, B.C. and Finn, J.T., 1979. Systems approach to continental shelf ecosystems. In E. Halfon (Editor), Theoretical Systems Ecology. Academic Press, New York, pp. 183-212. Pielou, E.C., 1977. Mathematical Ecology. Wiley-Interscience, New York, 385 pp. Ulanowicz, R.E., 1980. An hypothesis on the development of natural communities. Ecol. Modelling, 85: 223-245. Ulanowicz, R.E., 1981. A unified theory of self-organization. In W.J. Mitsch, R.W. Bosserman and J.M. Klopatek (Editors), Energy and Ecological Modelling. Developments in Environmental Modelling, 1. Elsevier, Amsterdam, pp. 649-652. Ulanowicz, R., 1986. Growth and Development: Ecosystems Phenomenology. Springer, New York. Wall, G., 1986. Exergy - a useful concept. Thesis, Physical Resources Theory Group, Chalmers University of Technology, Goteborg, Sweden.