System-level indicators in dynamic ecosystems: Comparison based on energy and nutrient flows

Z theo~ BioL (1990) 143, 523-553

System-level Indicators in Dynamic Ecosystems: Comparison Based on Energy and Nutrient Flows ROHI:R r H E R E N I ) E L N

Illinois Natural History Surt'ev, Champaign, IL 61820, U~S.A. (Received on 13 Ju/y 1989, Accepted in revised lbrm on 7 September 1989) System-wide indicators of ecosyslem structure and function depend strongly on the choice of flow variable, or numeraire. I compare two such indicators, exergy and ascendency, using biomass energy or elemental nitrogen as flow variable. Comparison is for steady-state (data from ('one Spring and Tayozhny Log Bogt and for dynamic behavior (hypothesized. based on steady-state data from Tayozhny Log). In both systems, measured flows are available in terms of biomass energy; nitrogen flows are imputed by calculating dynamic nitrogen intensities ~analogous to energy intensities). I find that steady-state normalized ascendency differs as much for one ecosystem using these two flow variables it does for two different ecosystems using the same flow variable. This implies that the question of flow variable choice needs resolution before strong conclusions can be made about intersystem comparisons. For dynamic behavior, nitrogen-based exergy shows no more, and sometimes less, variation over time than energy-based exergy. Both nitrogen, and biomass-based ascendency follow variations in system stock and show little additional variation. Analyzing the size, concentration, and structural components of exergy and ascendency aids in understanding these behaviors. I also briefly compare trophic positions using the two flow variables.

1. Introduction Analysis of the distribution of energy and materials between ecosystem compartments, and of the distribution of energy and material stocks among compartments, is fundamental to the definition of "'ecosystem." Based upon these distributions, several system-wide summary indicators have been proposed. Such indicators represent a simplification, a loss of information, relative to the detailed flow and stock data, but they are more manageable and are purported to convey quantitatively the interdependence of system compartments. The indicators covered in this article are: (1) exergy (Jorgensen & Meier, 1983; Jorgensen, 1986), which is based on compartment stocks; (2) ascendency (Ulanowicz, 1980, 1986), which is based on intercompartment flows; and (3) (energy intensity). (net output), i.e. the scalar product of energy intensities and net output (Hannon, 1985). These indicators are defined explicitly in the text and Appendix 1. These particular indicators have been chosen because, in addition to their use for inter-system comparisons at steady-state (Wulff & Ulanowicz, 1989; Harmon & Joiris, 1989), they have been proposed as the optimands in optimizing principles for predicting dynamic response over time o f perturbed or evolving ecosystems (Jorgensen & Mejer 1983; Ulanowicz, 1980; Harmon, 1985). An important question in either application is what quantities to measure-biomass, energy, elemental 0022-5193/90/080523 + 31 $03.00/I)

~ 1990 Academic Press Limited

524

R. HERENDEEN

carbon, nutrients, trace elements, etc. For example, it has been shown that cycling index (Finn, 1976) and ascendency (Ulanowicz, 1986) differ markedly depending on whether biomass energy or nitrogen is considered (Finn, 1980; Wulff, 1986). While there are many flow variables that can be used, the extremes of the possibilities are represented by two classes of variable: ( 1) those that are dissipated, such as energy and (2) those that are not, such as elemental nitrogen. (Whether a quantity is dissipated or not will depend on, among other things, the choice of system boundary.) By comparing flow patterns for these two classes of variables, we cover the two extremes of describing the ecosystem, and lay the foundation for more complicated analysis involving simultaneous inclusion of several flow variables, as called for by Ulanowicz (1986: 145-146). In this article, I will compare the dynamic behavior of the two system-wide indicators-exergy and ascendency-based on both energy flows and nutrient flows. In so doing, I develop a method for imputing nutrient flows from energy flows (when the nutrient flows cannot be measured), first for steady-state and then for dynamic systems. I also compare the behavior of (energy intensity). (net output). In this paper, the dot will denote the scalar production of two vectors. I will ask not only how steady-state ascendency differs based on energy or nutrient, but also whether the dynamic behavior--the shape over time--differs. An underlying question will always be to what extent the dynamic behavior of any (calculated) system-wide indicator is different from the dynamic behavior of an observable quantity, such as stock. Application will be to two ecosystems at steady-state, Cone Spring (Ulanowicz, 1986: 32; original data from Tilly, 1968; expanded by Williams & Crouthamel, unpublished data) and Tayozhny Log Bog (Logofet & Alexandrov, 1983); and to one ecosystem undergoing hypothesized dynamic behavior, the same bog. Section 2 contains a derivation of imputing nutrient flows from energy flows at steady-state, and section 3 presents an application to steady-state results from Cone Spring and Tayozhny Log bog [Cone Spring has been analyzed--for energy only and only at steady-state--by Ulanowicz ( 1986)]. Section 4 contains a brief digression on the effect of choice of flow variable on trophic position. In section 5 the technique of imputing nutrient flows is extended to the dynamic case; in analogy to dynamic energy intensity (Herendeen, 1989), a dynamic nutrient intensity is developed. This then facilitates calculation and graphical presentation of exergy, ascendency, and (energy intensity). (net output) over time for a "real" system, i.e. a model dynamic system based on Logofet & Alexandrov's steady-state data for Tayozhny Log (section 6). Finally in section 7, I discuss what can be inferred about the choice between energy and nutrient-based analysis.

2. Imputing Nutrient Flows from Energy Flows at Steady-State Exergy (Jcrgensen & Mejer, 1983; Jcrgensen, 1986) is a system-wide indicator depending only on the distribution of stocks among compartments, while ascendency (Ulanowicz, 1980, 1986) depends only on flows between compartments, and energy intensity (Harmon, 1973; Bullard & Herendeen, 1975; Herendeen, 1989) depends

INDICATORS

IN DYNAMIC

ECOSYSTEMS

525

on both flows and stocks in the dynamic case. A detailed comparison and criticism is given by Herendeen (1989). To calculate all three indicators thus requires knowing flows and stocks vs. time. Table 1 presents steady-state flows in terms o f biomass energy for Cone Spring. The system is divided into five compartments: plants, detritus, bacteria, detritivores, and carnivores. The question I address here is: given that a nutrient is flowing in the system and that there must be an added c o m p a r t m e n t (the "nutrient pool"), what are the nutrient flows for the expanded n u m b e r of c o m p a r t m e n t s ? The obvious answer is to measure them, but if it is not possible or has not been done, we seek to impute a plausible, self-consistent flow pattern. To do this, we need to derive nutrient intensities, measured (say) in grams of nutrient per kilocalorie of energy for each o f the original compartments, and then to multiply the intensities times the flows in Table 1. In addition, we require the flows to and from the nutrient pool. Figure 1 summarizes these flows. Nutrient is assumed not to be dissipated, so that nutrient intensity gives the actual nutrient content, as would be measured by chemical analysis. Energy intensity, on the other hand, embodies some energy that has already been dissipated and therefore indicates more energy than one would recover in a calorimeter. The nutrient intensity of nutrient is unity. Energy intensity is notable for its indirect aspects: the intensity is higher for a top carnivore than for an autotroph because there has been much energy burned in the intervening food web. Nutrient intensity can also vary, not because of dissipation in our case, but because of accumulation. Accumulation can be accomplished mathematically by modifying a flow table such as Table 1 to account for the fact that some biomass flows carry proportionally more nutrient than others. For example, I will assume that basal metabolism by plants does not release nitrogen to the pool. Energy intensity is defined by, and calculated from, the embodied energy balance diagram o f Fig. 2: E = E(X

-

15st-X)

-~,

( 1)

where e = v e c t o r (1 . . . . . k) of energy intensities ( u n i t s = k c a l k c a l energy terms),

t if flows are in

E = vector (1 . . . . . k) of energy inputs to system (units = kcal time-t), X = matrix (1 . . . . . k, 1. . . . , k) of biomass flows (units = kcal time ~), .~ = diagonal matrix ( 1 , . . . , k, 1. . . . . k) containing total outputs,

Mst=diagonal matrix

( 1 , . . . , k, 1. . . . . k) o f "lost'" dissipation

Nutrient does not a p p e a r in eqn (1); its energy intensity is zero. Equation (1) differs slightly in form from previous expressions ( H a n n o n , 1973; Bullard & Herendeen, 1975; Herendeen, 1981, 1989) but not in consequences:

Plants Detritus Bacteria Detritivores Carnivores Gross imports Energy inputs

0 0 0 0 0 0 II 184

Plants

8881 0 1600 200 167 635 0

Detritus 0 5205 0 0 0 0 0

Bacteria 0 2309 75 0 0 0 0

Detritivores 0 0 0 370 0 0 0

Carnivores

1001-5 1554.5 1637.5 907 101.5 0 0

"'Lost" dissipation

1001.5 1554.5 1637.5 907 101.5 0 0

Basal

300 860 255 0 0 0 0

Gross exports

0 0 0 0 0 0 0

Stock change

I1 184 I I 483 5 205 2 384 370 635 II 184

Total output

Biomass energy flows in Cone Spring, at steady-state [units of flow = kcal (biomass) m +2 year-']. Energy enters via biomass production by plants and import of detritus. It is exported via exports of plants, detritus, and bacteria, and it is dissipated. On the assumption that dissipation is split 1 : 1 between "'lost" dissipation and basal metabolism, the energy intensities are [calculated using eqn (1)] 1.10, 1.56, 2-28, 2.55, and 3.52 kcalkcal I f or plants, detritus, bacteria, detritivores, and carnivores, respectively. Ulanowicz (1986) intends that gross exports, gross imports be used in calculating ascendency, but I use net exports only. See text

TABLE 1

m m Z

m Z

"r t~

Ix2 O~

IN

INDICATORS

DYNAMIC

527

ECOSYSTEMS

o.

,.a

TO "

-

From

1

Ehomoss energy kxk

From pool Gross Imports Nuhent :npuls Energy inputs

FIG. 1. Flow table incorporating energy and nutrient. S = stock and AS = change in stock. In a general accounting scheme, units for different rows need not be the same {but they must be across a row if total output, which is the sum of all row elements to the left of total output, is to have meaningt. In this article, we use either energy or nutrient for all flows, and the table differs d e p e n d i n g on w.hich is used. For example, for energy, dissipation = 0. but flows to and from pool = 0. For nutrient, the opposite is true. For energy, AS = S = 0 ; it is a flow resource. For nutrient, AS and S can be 20. "'Lost" dissipation is not included in net output, whereas basal is included. Reasons are given in Herendeen 11989}. Table I, the energy flow table for ( ' o n e Spring, thus contains no column "'to pool," or rows "'from pool" and "'nutrient imports." Table 1 incorporates the implicit assumpt i on that each c ompa rt me nt produces only one type of output. Multiple outputs may be dealt with by measures that reduce to this assumption.

,~ e,X,j._~l[ - - -

] E, (X,-"losr"

dissipationj)

t E,

j=l...k

FiG. 2. Balance diagram for calculating steady-state energy intensity.

energy intensities are unchanged. Previously, "lost" dissipation was added to both sides of the equation, being counted as an input. Here, to be consistent with Ulanowicz's definition of ascendency ( 1980, 1986), dissipation is not so considered (see Appendix 1). Nutrient intensity is calculated from the balance in Fig. 3, where the k + l t h compartment is the nutrient pool. Because I constrain the nutrient intensity of nutrient to equal unity, Fig. 3 shows a separation into a balance for the original k compartments and another for the nutrient pool. For the original k compartments, nutrient intensity is calculated analogously to energy intensity, with the inputs of nutrient from the pool taking the place of the energy inputs: "q = N ( X

- n~t - X + not)

',

(2)

528

R.

k i-!

r/,X~] --*

HERENDEEN

--* r/j (X~- "not carry'))

J l.X~+l.j

j = 1... k

and Yk r/,X,.k+ l--*

h

k+l

I'X,.I

1 • gross import of nutrient FIG. 3. Balance diagram for calculating steady-state nutrient intensity. "'Not carry" refers to biomass flows that do not carry along nutrient. For example, plant leaves eaten by herbivores carry nitrogen, but plant biomass c o n s u m e d for respiration does not release nitrogen, does not carry nitrogen. For compartments l . . . . . k, flows are measured in energy ~kcal t i m e - i l . For c o m p a r t m e n t k + 1, the nutrient pool, the flows are in nutrient (g t i m e ~1. X~ ~ ~., is thus flow from the nutrient pool to compartment j.

where, vl = vector ( 1. . . . .

k) o f nutrient intensities (units = g kcat-J),

N = vector ( 1 , . . . , k) of nutrient inputs to c o m p a r t m e n t s (from c o m p a r t m e n t k + 1, the nutrient pool) ( u n i t s = g t i m e ~), X, .,~ = energy flows, as above, not

= matrix ( 1 , . . . , k, I . . . . . k) o f those energy flows that do not carry nutrient and thus allow nutrient a c c u m u l a t i o n ( u n i t s = kcal time-~), and

nth = diagonal matrix (1 . . . . . k, 1. . . . . k) o f sums of all flows that do not carry nutrient. This is the row sum o f " n o t c a r r y " plus such flows in exports or stock change. If " n o t c a r r y " is equal to " l o s t " dissipation, and if N is p r o p o r t i o n a l to E, then is p r o p o r t i o n a l to e, i.e. nutrient and energy intensities are proportional. If, as in the e x a m p l e s used here, only one c o m p a r t m e n t , plants, takes in energy (light) and nutrient, then N is necessarily p r o p o r t i o n a l to E, and it is only b e c a u s e "'not c a r r y " is not equal to " l o s t " dissipation that Vl and r are not p r o p o r t i o n a l . Once eqn (2) has been used to obtain nutrient intensities, a nutrient flow table m a y be constructed in which ~1, ( X , , - " n o t carry", i) is the nutrient flowing from c o m p a r t m e n t i to c o m p a r t m e n t j. This also applies for i = k + 1, the nutrient pool, b e c a u s e r/k+~ = 1. E q u a t i o n ( l ) leads to total system energy b a l a n c e (Bullard & H e r e n d e e n , 1975): E=e

.Y,

(3)

where, k

E = s u m of all energy i n p u t s = ~ E,, t=l Y = vector

(1 . . . . .

k) o f net outputs,

Y, = basa b + gross exportj - gross import, + AS r

I N D I C A T O R S

IN

DYNAMI("

ECOSYSTEMS

529

For steady-state, AS, = 0. An analogous expression can be derived for nutrient balance: Rewrite eqn (2) using indices instead of vector notation. k

N,= V r / , ( . ~ ' - n 6 t - X + n o t ) , , i

1

N, the total flow of nutrient from the pool to the k original compartments is then: N = ~ N~= ~ 7.. r / , ( X - n 6 t - X + n o t ) , . I

I

I

I ~

I

Reverse the order o f s u m m a t i o n . Then; k

N = ~ 77, (lost, + basal, + gross export, - gross import, - ( n o t carry in net output),), /,

N = ~ rti[ Y , - ( n o t carry in net output),]. I

If we use primes to d e n o t e that portion o f the individual flows that comprise net output which carry nutrient, k

N=~ i

r/, ( l o s t ~ + b a s a t ~ + g r o s s export', - gross import',) I

or

N = - q . Y', where Y~ is defined can be extended as second, note that rl, the nutrient pool as

(4)

using lost~, basall, etc. E q u a t i o n (4) resembles eqn (2), but it follows. First, move gross imports to the left h a n d side, and (lost', + basall) is the nutrient delivered from c o m p a r t m e n t i to a result o f energy dissipation. Then eqn (4) becomes;

N + r l • g r o s s i m p o r t s ' = "q. g r o s s e x p o r t s ' + nutrient

into pool from c o m p a r t m e n t s i . . . . k.

(5)

Equation (5) applies to c o m p a r t m e n t s i . . . . . k. Recalling that rtt.j = 1, it can be written as: ~1 • g r o s s imports' = 11. g r o s s e x p o r t s ' ,

(6)

now a p p l y i n g to c o m p a r t m e n t s 1 , . . . , k + 1. E q u a t i o n (6) expresses nutrient balance in the most general form, but it c a n n o t be used to derive the nutrient intensities rl. That must be d o n e using eqn (2). Equation (6) is a direct c o n s e q u e n c e o f eqn (2). A c o m p a r a b l e expression can be derived for energy: . gross imports = E. (basal + gross exports).

(7)

This applies to 1 , . . . , k + 1 c o m p a r t m e n t s a n d assumes that the energy intensity o f energy is unity.

530

R. H E R E N D E E N

Both eqns (6) and (7) apply to steady-state. Equation (6) says that no nutrient can be imported if none is being exported, as expected for a non-dissipatable flow variable. On the other hand, eqn (7) shows that basal metabolic dissipation allows a steady-state system to have energy inputs, as it must.

3. Examples for Energy and Nutrient Flows at Steady-State: Ascendency for Cone Spring and Tayozhny Log Bog The available data on flows in the Cone Spring system (Table 1) were measured in biomass energy. Five biotic compartments are used; plants, detritus, bacteria, detritivores, and carnivores. Ulanowicz (1986) has calculated ascendency for these flows. Let us now impute plausible nutrient flows for Cone Spring. I assume that nutrient from the pool is taken up only by plants. Each of the five original compartments is a potential contributor of nutrient to the pool, as, for example, a by-product of metabolism. If each compartment's dissipation releases to the pool as much nutrient per kilocalorie as is contained in that compartment's stock, then all nutrient intensities are equal, indicating no nutrient accumulation. To allow for accumulation, we can explicitly vary the amount of nutrient released during dissipation of biomass. I therefore assume that dissipation of biomass by plants, detritus, and carnivores releases no nutrient, but that dissipation by bacteria releases all nutrient incorporated in the biomass, and that dissipation by detritivores releases half of it. (Any fraction between 0 and l is possible.) This manipulation results in carnivores having a nutrient intensity approximately six times that of plants, which is in the observed range (Larcher, 1975; table 15). Having no data on nutrient inputs, I normalize by assuming that nutrient/carbon in aquatic plants is 1:20 (Wetzel, 1975: tables 17-19; dry weight basis for nitrogen/carbon). This leads to; r/pj.m~=4"O0.10 ~gkcal tt Applying eqn (2) yields: r / = 4 - 0 0 . 1 0 ~gkcal+',

plants

6-81 . 10 --~g kcal-',

detritus

6.81.10 ~ g k c a l - ' ,

bacteria

11.0.10 ~g kcal ',

detritivores

24.3.10 3gkcal ',

carnivores

1.0gg-~,

nutrient.

11 shows an increase with trophic level, reflecting the assumptions about accumulation. These intensities are then multiplied by the energy flows, modified for "not carry", to produce the nutrient flows in Table 2. t One gram C is equivalent to 30/12 g of C~H,,O~, or approximately 12.5 kcal. Then nitrogen (g)/carbon (kcal) = ( 1/20)(I/12.5) = 0.004.

2

Plants Detritus Bacteria Detritivores Carnivores Nutrient pool Gross imports Nutrient inputs

0 0 0 0 0 3.672E + I 0 0

Plants

3.552E 0 1.089E 2. 1 9 3 E 4.065E 0 4.322E 0

+0

+ I +0 +0

+ 1

Detritus

0 3-543E + I 0 0 0 0 0 0

Bacteria 0 1.572E+ 1 5-105E - I 0 0 0 0 0

Detritivores 0 0 0 4.065 E +0 0 0 0 0

Carnivores 0 0 2.229E + I 9-965E +0 0 0 0 4-467E +0

Nutrient pool 0 0 0 0 0 0 0 0

"Lost" dissipation

0

0

0 t)

0 0 0

0 ()

0 0 0 0 0 0 0 0

1 - 2 0 0 E +0 5-854E +0 1.736E + 0 0 0 l) t)

Stock change

Gross exports Basal.

3.672E 5.70OE 3.543E 1.623E 4.065E 3.672E 4-322E 4.467E

+ I + I + I + I +0 + I + I +0

Total output

Imputed nutrient flows.[or Cone Spring, at steady-state. Units of flow = g (nitrogen) m z year t. Table 2 entries are obtained from energy flows in Table 1, as modified for flows that do not carry nutrient, using the nutrient intensities calculated from eqn ( 2 ) . Table 2 has one more compartment than Table 1, the nutrient pool Dissipation o f nutrient is zero; "'lost'" dissipation and basal columns are zero. As a result, nutrient input + gross imports = gross exports, as discussed in text. Ulanowicz (1986) intends that gross exports and gross imports be used in calculating ascendency, but I use only net exports, See text

TABLE

532

r . HERENDEEN

There is one more compartment (the pool) than for energy flow, and the two

dissipation columns are zero. By assumption, only bacteria and detritivores contribute nutrient to the pool, and the pool contributes only to plants. One can verify explicitly that the system is in nutrient balance, satisfying eqn (6). Tables 1 and 2 are in the form required for calculating ascendency (except for the minor conversion to net exports as discussed in Appendix 1 ). Table 3 lists results of the calculation. For Cone Spring, ascendency based on energy flows is -'~ -I 5-86. l04kcal-bit m - y e a r , or 1.422 bit when normalized for (i.e. divided by) system throughflow. Based on nutrient flows (with an extra compartment for the nutrient pool), ascendency is 328 g-bit m 2year-~, or 1.760 bit when normalized. Using of nutrient instead of energy as flow variable thus increases normalized ascendency by 24%. | apply the same approach to the four-compartment Tayozhny Log bog data, assuming that only plants take nutrient from the pool and that only detritivores release nutrient during energy dissipation (Tables 5 and 6). I assure that plants have nutrient:carbon of l :30 (Larcher, 1975: table 15). Equation (2) yields: r t = 2 " 6 7 . 1 0 ~gkcal -~,

plants

3 . 5 2 . 1 0 -~ g kcal-t,

animals

2-79.10 -3 g kcal-~,

decomposers

2 . 7 9 . 1 0 - 3 g kcal-~,

detritus

1.0 g g-~,

nutrient,

indicating a small degree of accumulation. Further manipulation of the nutrient-tocarbon ratio in the intercompartment flows (not just dissipation) would likely result in the desired accumulation, but this would be arbitrary, and I do not do it. Ascendency based on energy flows for Tayozhny Log is 4 . 9 8 . 1 0 ~ kcal-bit m -"year (normalized, 1.492 bit), and based on nutrient flows, 9 . 8 4 . 1 0 -~ g-bit m -'year -~ (normalized, 1.625 bit). For Tayozhny Log, use of nutrient flows increases normalized ascendency by 8-9%. The increase in normalized ascendency is attributable to both increasing the number of compartments and to changes in the relative magnitude of the flows (relative to the biomass energy flow pattern) arising from nutrient accumulation. We can remove the second effect by letting "not carry" equal zero, so that there is no accumulation and nutrient intensities for compartments 1 , . . . , k are equal. Results of this exercise are also given in Table 3. There is little change in normalized ascendency compared with that based on biomass energy flows. Compared with the 24% and 8.9% increases above, the increases in this case are 0.7% and 0.5%, respectively. On the other hand, we can remove the first effect, that of adding the nutrient compartment, by using the new, unequal nutrient intensities and accounting for "'not carry", but limiting the calculation to k compartments. This yields much larger changes, 23% and 6%, respectively.

Biomass energy, nutrient pool not included (k compartments) Nutrient. with accumulation (k + I compartments) Nutrient, with no accumulation (k + I compartmentsJ Nutrient. with accumulation but nutrient pool excluded k compartments)

Flow variable 1-422

1,760 (+24%J 1.432 (+0.7%1 1'743 (+23%1

3.28E + 2 g - b i t m "year ~

Normalized (bit)

5.86E +4 kcal-bit m " year ~

Ascendency

( ' o n e Spring (5 biotic compartments)

9,84 g-bitm : y e a r '

4,98E + 3 g (biomassl-bit m : year ~

Ascendency

1.625 (+8.9°/oJ 1,500 (+0.5%} 1.582 (+60%)

1.492

Normalized (bitl

Tayozhy Log Bog (4 biotic compartments)

Ascendency based on energy and nutrient flow for Cone Spring and Tayozhny Log bog. Normalized ascendency = ascendency/throughflow. Because base 2 logarithms are used, normalized ascendency is measured in bits. Ascendency is calculated here on the basis of net exports (see text); values would therefore disagree slightly using Ulanowicz's method (1986). For example, Ulanowicz obtains normalized ascendency for biomass flows for Cone Spring of l " 3 3 6 bit; using his approach, I confirm this. My approach yields 1 . 4 2 2 bit, a 6 . 4 % difference. Numbers in parentheses indicate changes relative to normalized ascendency for energy, nutrient pool not included

TABLE 3

U'l

-I m

O ..<

z 3,

z

O

r~

7_

Energy

1.00 1.00 2-00 2.03 3.03

Fiow variable

Plants Detritus Bacteria Detritivores Carnivores

1.00 1.00 2.00 2-03 3.03

Nutrient

Plants and detritus

Cone Spring

1.00 2.46 3-46 3.50 4.50

Energy 1.00 3.09 4.09 4. t 2 5.12

Nutrient

Plants only

Plants Animals Detritivores Detritus

Flow variable

1.00 2.00 2.00 1-00

Energy

1-00 2.00 2.00 1.00

Nutrient

Plants and detritus

C o m p a r t m e n t assigned trophic position = I

Tayozhny Log Bog

1-00 3.43 4.90 3.90

Energy

1-00 3.52 5.00 4-O1

Nutrient

Plants only

Trophic positions for energy and nutrient flow for Cone Spring and Tayozhny Log Bog. Calculations are based on net rather than gross exports. See Appendix 1 and comments in section 4. Trophic position exceeding the number of compartments is possible

TABLE 4

Z

FI3

Z

;¢

"r

4~

Plant.,, Animals Decomposers Detritus Gross imports Energy inputs

(1 0 0 0 0 987.55

Plant.,,

38.05 0 0 36-9 0 O

Animab, 0 (1 (1 5~4.9 0 0

Decompo~,ers 337.4 58,21 305 0 35.93 0

l)etritu,, 304 g-37 139.95 0 0 0

"Lost'" dissipation 304 g.37 139.95 0 t) 0

Basal

4. I tl 0 114.735 0 0

Gross export.,,

0 0 o 0 0 o

Stock change

9X7.55 74,9'; 584.9 736535 35.93 9~7.55

Total output

~49(1 1.246 35.0 8836 ---

Stock

Biomass energy flows f o r Tavozhn), . Log . bog. Units . o["flow = g . ( biomass ) m z ),ear ~ (dry weight). Units o f stock = g m " Small adjustments have been m a d e to Logofet & A l e x a n d r o v ' , (1983) Fig 1 to produce a steady-state. Respiration has been arbitrarilv split 1 " 1 between "'lost'" dissipation and basal metabolism. The energv intensities are ! . 4 5 , ! . 9 4 , 2 . 6 3 , and 2 . 0 0 g g t .for plants, animals, decomposers, and detritus, respectively. Ulanowicz (1986) intends that gross exports, groxs imports be used in calculating ascendeno, , but 1 use net exports only. See text

TABLE 5

~. .< ~. -4

'~

r~

Z >

z

>~" "~

z

TABLE

6

Plants Animals Decomposers Detritus Nutrient pool Gross imports Nutrient inputs

0 0 0 0 1.013E+0 0 0

Plants

1-016E - I 0 0 1.031E-I 0 0 0

Animals

0 0 0 1.635E+0 0 0 0

Decomposers 9.009E - I 2.047E-I 8-524E - I 0 0 1.004E - 1 0

Detritus 0 0 7.822E - 1 0 0 0 2.312E - 1

Nutrient pool 0 0 0 0 0 0 0

"Lost" dissipation 0 0 0 0 0 0 0

Basal

1.095E - 2 0 0 2.202E-1 0 0 0

Gross exports

0 0 0 0 0 0 0

Stock change

1.013E +0 2.047E-I 1.635 E + 0 1.958E+0 1.013E+0 1.004E - I 2.312E - I

Total output

2-267E + l 4.382E-3 9.781E - 2 2.469E-~ 1 1.000E+2 ---

Stock

Imputed nutrient flows in Tayozhny Log Bog. Units o f flow = g (nitrogen) m -~" year ~. Entries are obtained from energy flows in Table 3, as modified for flows that do not car D, nutrient, using the nutrient intensities calculated from eqn ( 2 ) . Dissipation of nutrient is zero. As a result, nutrient input + gross imports = gross exports, as discussed in the text. Ulanowicz (1986) intends that gross exports, gross imports be used in calculating ascendency, but I use net exports only. See text

INDICATORS

IN I ) Y N A M I (

F('OSYNTI:MS

537

In these two examples, therefore, the major portion of the difference between normalized ascendency for energy flows with k compartments, and to nutrient flows with k + 1 compartments, is the changed relative flows and not the addition of the nutrient c o m p a r t m e n t per se. These changes in normalized ascendency resulting from changing flow variable for one system can be c o m p a r e d with Wutff & Ulanowicz's (1989) result for different ecosystems (Chesapeake Bay and the Baltic Sea) using the same flow variable (biomass). Their table 6 allows one to calculate that the Baltic's normalized ascendency is 20% higher than Chesapeake's, which is bracketed by our results of 24% and 9%. Thus the choice of flow variable can produce differences as large as the choice of ecosystem. 4. Trophic Position for Energy and Nutrient Flows at Steady-State: Cone Spring and Tayozhny Log Bog While this article is primarily concerned with system-wide indicators-exergy, ascendency, and energy intensity-let us also check if trophic positions shift with choice of flow variable. Trophic position for c o m p a r t m e n t j is defined as the average number of compartments the (input-weighted) inputs t o j have already been through plus 1. It is usual to define one or more compartments to have a trophic position of unity. This is done by zeroing all inputs to those compartments, in effect treating the inputs as having a t r o p h i c position of zero. For example, light and inorganic nutrients usually are assigned trophic position =0, so that producers with no other inputs than those have trophic position = 1. Wulff & Ulanowicz (1989) also assign trophic position = 1 to dead organic compartments, in effect assigning trophic position =0 to feces and dead organisms. The expression for trophic position is then: T=u[l - (X.lf*-')']-',

(8)

where, T = vector ( ! , . . . , k) of trophic positions, u=unit vector(1,...,k), X = matrix of flows ( 1. . . . .

k, 1. . . . . k ),

,~* = diagonal matrix ( 1 , . . . , k, 1. . . . , k) of total inputs: that is, column sums of X, and the prime denotes that certain columns of X.~'* a have been zeroed as explained a b o v e . . ~ * depends on c o l u m n sums of X, while Y~, which is used to calculate energy and nutrient intensities, depends on r o w sums of X. Finn (1976) established the concepts that lead to eqn (8). Table 4 contains results of applying eqn (8) to Cone Spring and to Tayozhny Log for two cases regarding detritus: (1) assigning it trophic position = 1, and (2) not assigning it an a priori trophic position+. + Trophic positions calculated on the basis of net exports are equal to or greater than those based on gross exports. By the argument of Appendix I, I use net exports. This practice is equivalent to assuming

that the trophic position of imports is the same as that of the corresponding compartment.

538

R. HERENDEEN

Because o f the relative simplicity o f these two flow tables, calculated trophic positions are the same for either biomass or nutrient if both plants and detritus are assigned trophic position = 1. If, however, detritus is not assigned a position, then results for biomass and nutrient differ for both Cone Spring and Tayozhny Log. First, trophic positions are higher (as much as three trophic levels) when the detritus compartment "'floats," indicating that in both systems there is considerable cycling through detritus. Second, trophic positions are higher for nutrient than for biomass as flow variable; nutrient does indeed cycle more. However the differences are relatively small, never exceeding 0.6 trophic level. There are no reversals of trophic position, which is possible in principle.

5. Imputing Nutrient Flows from Energy Flows in a Dynamic System To impute dynamic nutrient flows requires obtaining a dynamic nutrient intensity, which can be derived analogously to dynamic energy intensity (Herendeen, 1989). Figure 4 shows the nutrient balance, which now depends on stocks as well as flows, and on the time step At. The resulting expression for nutrient intensity is: l t *.~l

11

A

^

= ( N ' A t + I I ' S ' ) ( X ' A t - n ~ t ' A t - X ' At+not' A t + S ' )

',

(9)

where in addition to terms already defined, t = time =diagonal matrix ( ! , . . . , k, 1. . . . . k) of stocks. As with dynamic energy intensity, future dynamic nutrient intensity depends explicitly on its current value. Equation (9) can also be manipulated to yield a balance equation, l do not give details here, except to note that S' +AS' = S '*'~', which expresses accumulation of stock. The result is: .q, ~a, gross imports"At =.q, ~a, gross exports' ' At +'q'+a'S '~a' - ' q ' S ' ,

(10)

for compartments 1, . . . , k + 1. rt,"a"X'-{ ,jar - not~jAt).

~s{X,at'

~,'s;:[ ~ X~.,.sat

-not;At)

k.Th,.~,,~'s

j=l...k

and

t

I "gross import of nutrient+At FIG. 4. Balance diagram for calculating dynamic nutrient intensity. Xk+~.~ is the flow of nutrient from the pool to the original compartments; it is also called N i.

INDICATORS

IN

DYNAMI("

ECOSYSTEMS

539

Equation (10) is the same as eqn (6) except for the added terms " q ' * a ' S " a ' - - ~I'S' i which account for the changing nutrient stored in the stocks of the k compartments, and in the nutrient pool itself. At steady-state, the added terms sum to zerot. The analogous expression for energy intensity is e '÷a'

gross imports' At = e ''a' gross exports' At + e ' * " basal' At +e'"X'S"a'-e'S

'.

(11)

An instantaneous nutrient flow table is produced by multiplying ~q" a, by the biomass flows at time t, as modified for "'not carry". Instantaneous nutrient stocks are obtained by multiplying 11' by biomass stocks S'.

6. Exergy, Ascendency, and (Energy Intensity). (Net Output) for an Example Dynamic System These three quantities have been proposed as potential maximands in ecological optimizing principles (J~rgensen & Meier, 1983; Ulanowicz, 1981; Harmon, 1979, 1985). Exergy and (energy intensity). (net output) (called e . Y) are much more useful in the dynamic case than at steady-state, because: (l) reference levels for calculating exergy are controversial, so that any steadystate value will be questioned. In a dynamic system, exergy is changing, and it is possible to investigate only the change of exergy, in effect treating the beginning situation as the reference state. This is not without problems, but more tractable than dealing with "'absolute" levels at steady-state. (2) E. Y at steady-state is identically the total energy input to the ecosystem [by eqn (3)]. Maximizing e . Y means maximizing energy inputs E, so that at steady-state, the principle does not explicitly mention energy intensities, with their indirect effects, at all. e . Y is more complicated in the dynamic case. Rewriting eqn (10), recalling that Y is in terms of net exports and includes AS: E ' At = e"-~'Y ' A t + ( e ' + ' ~ ' - e ' ) S ' . Now maximizing e . Y means maximizing energy inputs minus a term proportional to the time rate of change of energy intensities. Exergy and ascendency can be calculated using either energy or nutrient, and the statement of the optimization principles is sufficiently general that either is admissible. The maximization of e . Y is, however, explicit; no corresponding principle for 11 has been offered. Thus, we can compare the dynamic behavior of five whole-system indicators, exergy and ascendency for energy and nutrient, and e . Y (for energy). This is done for a dynamic model based on the steady-state data for Tayozhny Log Bog (Logofet & Alexandrov, 1983). The interactions of the four c o m p a r t m e n t s and the nutrient pool with each other and with light and cropping levels are postulated to have nonlinearities of two types: ( 1 ) a c o m p a r t m e n t ' s total output per unit of stock is a decreasing function o f relative scarcity of inputs, and (2) a c o m p a r t m e n t ' s "lost" dissipation, expressed as a fraction of total output, is an t Notation: -q' the nutrient intensity at the beginning of the time step. After flows have occurred, a new intensity is calculated at the end of the time step At; this will also be the intensity at the beginning of the next step. Therefore it is called .q,,a,. Thus "q'÷~' is calculated on the basis of the flows in the time step between t and t+At.

540

R. H E R E N D E E N

increasing function of relative scarcity of inputs. Input scarcity can refer to stocks of other compartments as well as light level. Details of the nonlinear functions are given in Appendix 2. Two perturbation regimes are used: Regime 1: starting at steady-state, light intensity is doubled, then returned to original level. During the increased light period, nutrient is removed from the pool. After the light level is restored to its original value, nutrient is added to the pool. Regime 2: same as regime 1 except that instead of nutrient, plant biomass is removed and then added. Detailed specifications of the regimes are given in Table 7. The two regimes are similar in that the assumed changes in light and cropping result in changes in system biomass on the order of ±50%. Because there are comparable amounts of nutrient in the pool and incorporated in biomasst, the two regimes exhibit comparable changes in pool concentration. In regime 1, approximately one-half of the nutrient is removed and then returned directly. In regime 2, the removal and return of plants produces the same effect on nutrient. Figures 5-8 show the response of the model system to the perturbations thus: (5) regime 1 (nutrient inputs varied), flows of biomass, k compartments used; (6) regime 1 (nutrient inputs varied), flows of nutrient, k + 1 compartments used; (7) regime 2 (plant inputs varied), flows of biomass, k compartments used; and (8) regime 2 (plant inputs varied), flows of nutrient, k + 1 compartments used. (a) Compartment stocks over time (b) Exergy, ascendency, e . Y over time (c) Size, concentration, and structural terms of exergy over time. (d) Size and structural terms of ascendency over time. The basic question is "what are the differences between biomass-based, fourcompartment analysis, and the nutrient-based, five-compartment analysis?" "'Difference" will refer more to the shape of the graph of the quantity over time and less to absolute values. 6.1, S T O C K S

Stocks vs. time [Figs 5(a), 6(a), 7(a), and 8(a)] roughly track the perturbations. In response to increased light or increased nutrient concentration, plants increase, and the other compartments follow suit, with all approaching a new steady-state. When light level is increased, plants overshoot slightly. If light or nutrient is decreased, stocks are reduced. However, a comparison of Figs 7(a) and 8(a) shows that stocks expressed in nutrient are not proportional to those expressed in biomass. When plants are stocked (in regime 2), their biomass increases, but the nutrient in ,~ These numbers are chosen for illustrative purposes, but they have some justification. For example, for an average deciduous forest ( Larcher, 1980 ), there is 156 ton ha- I of plant biomass above and below ground (p.142), which is 15.6.103gm -:. Soil contains approximately l gkg -~ of nitrogen (p. 173). Assuming a soil density (dry) of 0.9 g ml ~ and an effective depth of 20cm, we obtain 180 g m -2 or 1.8 tons N ha -~. Thus this deciduous forest has 15.6. t0~gm .2 of biomass in plants and approximately I 8 0 g m "~ of soil nitrogen. Tayozhny Log has 8 4 9 0 g m : of plant biomass, or roughly half that of Larcher's example. If we scale down the soil nitrogen proportionally, we obtain a nutrient pool concentration of approximately 100 g m - : . The N / C ratio for biumass has already been discussed.

TABLE 7

0 100 1000 5000 6000 10000 II000 1500O 16000 2000O 21000 25000 26 000 31001

Regime 2

0 3 33 167 200 333 367 500 533 667 700 833 867 1000

Regime I

Year

I '0 I '0

I '0

I'0 1'0 2"0 2'0 2'0 2'0 2"0 2'0 1-0 I'0 I '0

I'0 1'0 2'0 2"0 2'0 2'0 2'0 2'0 1"0 1"0 1'0 1"0 1.0 1-0

Light

4.1 4.1 4" I 4- I 204' I 204" I 4"1 4" I 4-1 4. I - 195-9 - 195.9 4-1 4. I

4"1 4"1 4. I 4- I 4.1 4"1 4. I 4-1 4. I 4.1 4.1 4-I 4.1 4-1

Plants (kcalm -'year ~D

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

Animals (kcalm 2year t)

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

Decomposers ( k c a l m - 2 y e a r ')

Cropping

78-81 78.81 78,81 78'81 78'81 78.81 78'81 78.81 78,81 78.81 78'81 78.81 78.81 78'81

78'81 78"81 78"81 78"81 78'81 78'81 78'81 78'81 78'81 78'81 78.81 78.81 78.81 78-81

Detritus (kcalm "year ')

-0"2311785 -0-2311785 -0-2311785 -0.2311785 -0.2311785 -0.2311785 -0.2311785 -0-2311785 -0" 2311785 -0"2311785 -0'2311785 -0"2311785 -0-2311785 -0.2311785

-0.2311785 -0-2311785 -0"2311785 -0-2311785 0.0 0'0 -0.2311785 -0"2311785 -0-2311785 -0'2311785 -0.462357 -0-462357 -0.2311785 -0"2311785

Nitrogen ( g m - : y e a r ')

Perturbation regimes. Values between the indicated years are obtained by linear interpolation. Negatioe cropping is stocking, i.e. positioe net import

H m

0

m

r~

.< z

0

G

z

542

R. H E R E N D E E N Nutrient

"°k

(.Y

Ascendenc~

Plants Antmots

Decomposers DGtritls

Exergy

2-4

0-8

.........

0'0

~

Nutrient

~

~

Nutrient

--0-8 --0-6 -2.4

I, 4 O0

I I I .... I i I 600 800 1000 0 200 400 600 800 I000 Year Fie;. 5. D y n a m i c behavior of Tayozhny Log. All plots are normalized arbitrarily for appearance. Regime 1, biomass flows, k c o m p a r t m e n t s used for calculating ascendency and exergy. Light is doubled and then reduced to ils initial value. During the period o f increased light, addition o f nutrient is decreased to zero; during period of original light intensity, nutrient addition rate is doubled. Actual values are given in Table 7. (a) Stocks vs, time. Plants, animals, decomposers, and detritus are measured in g (biomass) m ' y e a r '; nutrient, in g (nitrogen) m : y e a r -j. Actual values can be inferred from initial data given in Tables 5 and 6. (b) E. ¥, ascendency, and exergy. Initial values for e . Y, 987.55 g (biomass) m--" y e a r ~ ; ascendency, 4.980 g (biomasslbit m " year-~ (c) Size, concentration, and structural components of exergy. For comparison, the size term was divided by 80 000 RT; the concentration, by 0.07 RT; and the structural, by 0-007 RT. (d) Size and structural terms of ascendency. For comparison, the size term was divided by 7000 and the structural term by 1.7. Time step = 0.033 year.

0

200

_

plant biomass increases much more. This occurs because nutrient intensity (which I do not show graphically) increases significantly at that point. The increase in nutrient intensity is justified thus: (1) stocking of plants increases their numbers relative to light and nutrient levels, so that input scarcity increases [by eqn (A.2.1)]; (2) this increases "lost" dissipation [by eqn (A.2.2)]; (3) because I assume that dissipation in plants does not release nutrient, the increased dissipation results in nutrient accumulation, and a higher nutrient intensity. 6.2. E X E R G Y (TOTAL, A N D SIZE VS. C O N C E N T R A T I O N VS. S T R U C T U R A L C O M P O N E N T S )

The reference level for exergy is chosen as the original steady-state. Exergy there is equal to zero.

INDICATORS

IN I)YNAMI(."

(c')l

I'0 E

0.8 0-6

~

~Size

543

ECOSYSTEMS

L

(d) --Structural

S~ze

0-4 O2

~

~

Concen trohon

I

0"0

I .............

I

I

1.6 0"8

I

F

2"4

"•_•"

Nutrient

-0-8 --I.6 -2.4

I

0

200

400

600

800

O

I000

200

I

I

I

400

600

800

IOOO

Yeor FIG, 5--continued

For regime 1, exergy vs. time looks dramatically different for nutrient- and biomass-based analysis. For biomass [Fig. 5(b)], exergy increases when light is doubled, then returns almost to zero when nutrient is removed. The effects of increased light and decreased nutrient concentration work oppositely, so that the resulting steady-state has approximately the same biomass stock levels as the initial state [see Fig. 5(a)]. Nutrient level varies, but nutrient stock is not included in a biomass-only analysis. The same pattern appears when the light is reduced to its original level; exergy increases until nutrient is returned. Then biomass stocks return roughly to their original levels and exergy again drops to zero. Thus, regime 1 yields a double-peaked exergy, using biomass as flow variable. Using nutrient as flow variable [Fig. 6(b)], exergy vs. time has only a single peak (though a bumpy one), being non-zero only when nutrient stock is reduced. The reason for the difference is evident if we break exergy into size, concentration, and structural components (see Appendix 1). The size term by definition tracks the stock and is relatively uninteresting. For biomass, the concentration term is at least ten times larger than the structural term [Fig. 5(c)]. The concentration term depends on total biomass, and this differs from the steady-state value only when light is

544

R. H E R E N D E E N

I-0 Nutr ~ent

(b

)

0.8

Plants

0.4

Ammols Decomposers Detrd,s

O.Z 0.0 2.4

Ascendency

--Exergy I

I

I

l

1.6

---•.

0.8 Nutr,ent

0.0

"

Nutr,ent

-0.8

7

0

200

I 400

t 600

I 800

I000

0

,l 200

~ 400

I 600

l 800

I000

~or

FIG. 6. Dynamic behavior of Tayozhny Log. All plots are normalized arbitrarily for appearance. Regime 1, nutrient flows, k + I c o m p a r t m e n t s used for calculating ascendency and exergy. Light is doubled and then reduced to its initial value. During the period of increased light, addition of nutrient is decreased to zero; during period of original light intensity, nutrient addition rate is doubled. Actual values are given in Table 7. {a) Stocks vs. time. All are measured in g (nitrogen) m -'~ year t. Actual values can be inferred from initial data given in Tables 5 and 6. Ibl ~ . Y, ascendency, and exergy. Initial values for . Y, 987-55 g (biomass) m - " year '; ascendency, 9.84 g (nitrogen)-bit m- " year L (ct Size, concentration, net export, and structural c o m p o n e n t s of exergy. For comparison, the size term was divided by 6000 RT; the concentration, by 0.13 RT; the structural, by 0.13 RT. (d) Size and structural terms of ascendency. For comparison, the size term was divided by 12 and the structural term by 2. Time s t e p = 0 . 0 3 3 year.

increased with cropping unchanged, or when light is decreased with cropping unchanged. Each situation occurs once; hence the double peak. For nutrient, concentration and structural terms are of like magnitude [Fig. 6(c)]. We expect the structural term to be relatively more important for nutrient than for biomass because the inclusion of the nutrient pool in the expression for exergy is much more likely to produce changes in the fractional concentrations [the z, in eqn (A.1.3)]. The concentration term is expected to be non-zero when nutrient has been removed (which would produce a single peak)l. The structural term, being based on the imbalance between compartments, is expected to be non-zero when transitions occur, and could be multiply peaked. In Fig. 6(c) the structural term is t Another consequence of nutrient-based analysis is that for a materially closed system, the size term is constant, the concentration t e r m is zero, and the variation in exergy is all structural,

INDICATORS I-0_/

IN DYNAMIC

(c

ECOSYSTEMS

545

I

(d)

1

Siructurol S~ze

0-4 lilt /

Concen trotlon

/.

~,~. "1"~-..~k

]St"ruciuro,

I

I

I

l

1

2.4

I-6

/~L~ght~

l~LIgh~

0'8

0"0

~

Nutrient

---~~

'

Nufrlent

-0-8

-I,6 -2,4

0

200

I 400

I 60(~

~ 800

. iooo

0

Yeor FI(;.6--cominued

t I 200 400

1 600

I 800

Iooo

single peaked and in Fig. 8(c) it is triple peaked. In Fig. 6(b) the concentration and structural terms add to produce a single-peaked exergy. For regime 2 (in which light level and cropping of plants are varied), exergy shows a behavior similar to that under regime 1 [Figs 7(b), 8(b)]. With biomass as flow variable, the familiar double-peaked structure emerges (as before the concentration term is about ten times the structural term), With nutrient as flow variable, the triple-peaked structural term [Fig. 8(c)] adds small peaks to the single-peaked concentration term. In general, the structural term shows more variation under perturbations than the concentration term, as has been discussed by Herendeen (1989)t. We can now comment on a recent claim by Jorgensen (1988:117) that " . . . changes in species composition and structure as a consequence of changes of forcing functions (i.e. perturbations) always are accompanied by an increase in exergy. The changes in the forcing functions (themselves) may cause an increase or d e c r e a s e in exergy, but the changes in species composition always give an increase in the exergy." "~ In Herendeen (1989) a two.term breakdown was used; in the language o f this paper, the terms were (size. concentration) and (size. structural).

546

R. H E R E N I ) E E N Plonts E,Y

0-8 Nutrient 0-6

Ascendency

0.4 Ammols Decomposers Oetrd~s

0'2

F xercJy

O'0 2.4 I-6 0'8

/

l

0-0 --

Lighf

Light

Plonts

Plonts

0'8 --t.6

L

-2.4 0

200

400

600 Yeor

8OO

I000

0

l 2O0

l 400

I

I

600

800

~6oo

FIG. 7. Dynamic behavior o f Tayozhny Log. All plots are normalized arbitrarily for appearance. Regime 2, biomass flows, k compartments used for calculating ascendency and exergy. Light is doubled and then reduced to its initial value. During the period of increased light, removal of plants is increased; during period of original light intensity, plants are added. Actual values are given in Table 7. (a) Stocks vs. time. Plants, animals, decomposers, and detritus are measured in g (biomass) m -'~ year-~; nutrient, in g (nitrogen) m-" year -~. Actual values can be inferred from initial data given in Tables 5 and 6. (b) • . Y, ascendency, and exergy. Initial values for e . Y, 987.55 g (biomass) m -2 year-~; ascendency, 498 g (biomass)-bit m -2 year-t. (c) Size, concentration, and structural components of exergy. For comparison, the size term was divided by 80000 RT; the concentration, by 0-1 RT; the structural, by 0,01 RT. (d) Size and structural terms of ascendency, For comparison, the size term was divided by 7000 and the structural term by I-7. Time step = 0-033 year.

This claim would seem immediately to have exceptions. For example, assume that, by changes in light level, a system is perturbed away from an initial steady-state to a new steady-state with a changed exergy, and then back to the starting point with the original exergy. Because light intensity is not included in calculating exergy, the perturbation itself does not cause an immediate change in exergy. All the changes are due to "changes in species composition", which are claimed to be positive. The claim then requires that exergy undergo a net zero change as a sum of two positive changes, an impossibility. In addition to this counter-argument, Figs 7(b) and 8(b) serve as empirical counterexamples for biomass energy and nutrient, respectively. Both positive and negative changes in exergy are seen. Three general trends that emerge for exergy are:

IN

INDICATORS

547

E('OSYSTEMS

DYNAMIC

,

(d)

.......~ - - - - ~

0.8

.......

"-../- . . . . . .

I -~ StruCturoI

I

Size

i

p 0.6

0-4-

0.2

Concentration Structuroi

0-0 2-4

I

l

1.6 0"8

Light

0-0

- -

-

----

L~ght

4plants

Plonts

-0"8 -t.6

-2"4 0

I

l

200

400

t 600

l 800

I

I000

0

2O0

J 400

t 600

1 800

iO00

Yeor F t ( ; . 7--continued

(1) Nutrient-based exergy (for five compartments) shows the same or less variation over time in a perturbed ecosystem than energy-based exergy (for four compartments). (2) The concentration term tends to be larger than the structural term for biomass, but not for nutrient, especially when the chosen variable's stock remains roughly constant, as with nutrient in a materially closed system. (3) For any regime, for either energy- or nutrient-based analysis, the structural term varies as much as or more than the concentration term in response to perturbations.

6.3. A S C E N D E N C Y

For both regimes and flow variables, ascendency roughly tracks total stock [Figs 5(b), 6(b), 7(b) and 8(b)]. Ascendency vs. time shows no features more sensitive to transitions in response to perturbation. Inspection o f the size and structural terms in ascendency explains the lack o f sensitivity [Figs 5(d), 6(d), 7(d) and 8(d)]. The size term is just system throughflow, which is roughly proportional to system stock. The structural term shows no more

548

R. H E R E N D E E N

I-0 Nutrlent

~.Y

0'8 0.6

Ptonts

0-4

Ammal Decomposers

Ascendenti

Oetr~hs 0'2 E~ergy 0-0 2.4

i

I

|

I

t

.6 I

Light

0-8

0-0

-0"8 I -0-6 -2.4

200

bght

Plants

Plonts

ll/ I

0

i

1

400

...............

I

I.

600 Year

1300 I000

0

2O0

400

I

I

600

800

I000

FIG. 8. Dynamic behavior of Tayozhny Log. All plots are normalized arbitrarily for appearance. Regime 2, nutrient flows, k + 1 compartments used for calculating ascendency and exergy. Light is doubled and then reduced to its initial value. During the period of increased light, removal of plants is increased; during period of original light intensity, plants are added. Actual values are given in Table 7. (a) Stocks vs. time. All are measured in g (nitrogen) m -2 year -=. Actual values can be inferred from initial data given in Tables 5 and 6. (b) e . Y, ascendency, and exergy. Initial values for e . Y, 987-55 g (biomass) m -2 year -a ascendency, 9.84g (nitrogen)-bit m--" year -s. (c) Size, concentration, net export, and structural components of exergy. For comparison, the size term was divided by 7000 RT; the concentration, by 0.25 RT; the structural, by 0-15 RT. (d) Size and structural terms of ascendency. For comparison, the size term was divided by 12 and the structural term by 2. Time step = 0.033 year.

than 10% variation for both perturbation regimes, and usually less. The maximum variation occurs for regime 1, for nutrient as flow variable, and it appears that variation for nutrient is greater than that for biomass. Thus structural exergy seems much more sensitive to perturbations than structural ascendency. It could be argued that this is an unfair comparison because exergy is normalized to the initial steady-state while ascendency is not. However, it seems likely that at this time the intended reference level for ascendency is that for which structural ascendency =0 (examples are given in Ulanowicz, 1986). Should there emerge an argument for changing the reference level, the conclusion could change. 6.4, (ENERGY INTENSITY).(NETOUTPUT)

The quantity (nutrient intensity). (net output) does not play a role in any proposed optimizing principle; therefore, it is inappropriate to compare it with (energy

INDICATORS

IN

DYNAMIC

1.0

549

ECOSYSTEMS

: i C '{

!

_

'0

i

S~ze

0-8

0.6

0.4

0.2 C o n c e n t r o t ion L

Stuciurol

0.0 2.4

I

I

l

.....

1

1.6 Light

0.8

/

0.0 -0.8 -I,6

P}onts

l

L

-2.4 0

200

400

600

800

--

0

IO00

~

.......

2oo

t___

40o

I

600

! .....

800

IOOO

Yeo r FIG. 8--continued

intensity). (net output). We can, however, comment on the dynamic behavior of the latter, e . Y in Figs 5(b), 6(b), 7(b), and 8(b) tends, like ascendency, to track total biomass stock, but with an exception noted previously (Herendeen, 1989). It tends to undershoot at instances o f abrupt light reduction, even though this does not occur to stock. The undershoot is expected; a rapid decrease in light reduces production of plant biomass, but basal energy dissipation continues, and Y can easily have negative entries, tending to reduce e . Y rapidly. 7. What Does the Comparison of Biomass- and Nutrient-based Analysis Tell Us? The results obtained here do not, unfortunately, indicate whether biomass or nutrient-based analysis is "'better". The original question was to what extent biomass and nutrient-based analysis provide different results. At steady-state, we have seen that for ascendency, the choice of flow variable is as important quantitatively as the choice of ecosystem, which confounds inter-system comparisons. For the dynamic case, ascendency tracks total stock quite faithfully, due to the relative insensitivity of the structural term. The difference between biomass and nutrient reflects only the difference in stocks already mentioned.

550

R. H E R E N D E E N

Dynamic exergy does not track stocks, and is different for the two flow variables. The difference is better illustrated by breaking exergy into size, concentration, and structural components. For biomass, the concentration term is larger than the structural term, the latter varying rapidly as the system is perturbed. For nutrient, concentration and structural terms are comparable. (In a materially closed system the concentration term is zero, with the initial steady-state chosen as reference). The structural term again shows more sensitivity to perturbation than the concentration term. There still is a rather large multiplicity of ( 1 ) choice of flow variable, and (2) choice of reference level (for exergy and, possibly, for ascendency). This, coupled with other long-standing questions of level of disaggregation and of choice of nonlinear terms in the model, continues to make the choice and application of system-wide indicators vexing. An often-quoted response to this type of question-the search for "'the true, fundamental numeraire"--is that both (or all) approaches are useful and complementary. A set of rules or guidelines on how to combine the different approaches is, however, still lacking.

I a m g r a t e f u l for h o s p i t a l i t y , d a t a , a n d d i s c u s s i o n s to S v e n - E r i k J ~ r g e n s e n ( C o p e n h a g e n ) , A n n - M a r t J a n s s o n a n d F r e d r i k Wult~ ( S t o c k h o l m ) , a n d Yuri S v i r e z h e v a n d Dmitrii L o g o f e t . This w o r k was partially s u p p o r t e d by t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r g r a n t I NT-8713261.

REFERENCES Bur LLARD, C. & H t. rE N I>EE N, R. ( 1975 ). The energy costs of goods and services. Energy Policy 3, 268-278. EHrLI( H, P. R., HOLt;,REN, J. & COM~,~ONf:.r, B. ( 1972,1. Review of Commoner's book, The Closing Circ'&, and response by Commoner. Ent, ironrnent 14, 23-52. FIN N, J. T. (1976). Measures of ecosystem structure and function derived from analysis of flows. J. theor. BioL 56, I 15-124. F l i ' ~ , 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. H,~,NNON, B. (19791. Total energy cost in ecosystems..1, theor. BioL 80, 27t-293. HANNON, B. (19851. Ecosystem flow analysis. Can. Bull. Fish. Aquat. Sci. 213, 97-118. HANNON, B, & Jolt. IS, C. ( 1989,1. A seasonal analysis of the southern North Sea ecosystem. Ecology 70, 1916-1934. HEr~-Nt~EEN, R. ( 1981 ), Energy intensity in ecological and economic systems. J. theor. Biol. 91,607-620. H e r l : N t>EE N, R. ( 1989,1. Energy intensity, residence time, exergy, and ascendency in dynamic ecosystems. Ecol. Modelling 48, 19-4.4. JORGENSLI',4, S, E. (1986). Structural dynamic model, Ecol, Modelling 31, I-9. JORGENSEN, S. E. X, MEJER, H. (1983,1. Trends in ecological modelling. In: Analysis o1 Ecological Systems: State of the Art in EcohJgical Modelling (Lauenroth, W., Skogerboe, G. & Flug, M., eds,1 pp. 21-26. New York: Elsevier. J~RGENSEN, S. E. (1988,1. Use o f models as experimental tool to show that structural changes are accompanied by increased exergy. EcoL Modelling 41, 117-128. LARt. HER, W. (1975). Physiological Plant Ecology. New York: Springer. LARC'HER, W. (1980). Physiological Plant Ecology 2nd ed. New York: Springer. LOGOFET, D. O. & ALEXANDrOV, G. A. (1983). Modelling of matter cycle in a mesotrophic bog 1. Linear analysis of carbon environs. EcoL Modelling 21,247-258. TILt.Y, L. J. (1968). The structure and dynamics of Cone Spring. EcoL Monogr. 3g, 169-197. ULANOWlCZ, R. E. (1980). An hypothesis on the development of natural communities. J. theor. Biol. 85, 223-245. Ut..a.nrowt('z, R, ( 1986,1. Growth and Development: Eco.wstems Phenomenology. New York: Springer.

INDICATORS

IN D Y N A M I ( "

551

ECOSYSTEMS

WALL, G. (1986L ExerKv--A useful concept. Thesis. Physical Resources Theory Group. Chalmers University of Technolog)', G6teborg, Sv~eden. W~ TZl~t, R. (19751. Limnology. New York: V,'. B. Saunders. WULFF, F. 119861. A flow analysis of nitrogen and carbon budgets of the pelagic c o m m u n i t y during a nodularia bloom. Manuscripl, Ask6 Laboratory, Institute of Marine Ecotog), University of Stockholm, Sweden. Wt,t_FF, F. & ULANO~A'ICZ, R. (1989t. A comparative a n a t o m y of the Baltic Sea and Chesapeake Bay ecosystems. In: Flow Anal)'xis in Marine Eco~v~tem~ IWulff, F., Field, J. & Mann, K.. edsl Lecture Notes in Coastal and Estuarine Studies. New York: Springer. APPENDIX

I

Calculating and Analyzing Exergy, Ascendency, and Energy Intensity. Net Output Exergy per liter

(EX) is (J0rgensen & Mejer, 1983; Wall, 1986): k

EX=RT~

C,(InIC,/C,.r~r)-(I-C,.r,~,/C,I],

(A.I.I)

I

where, R= T= C, = C,.r~r = k=

the gas constant ( u n i t s = k c a l m o l d e g ~), absolute temperature of ecosystem, concentration of c o m p a r t m e n t i's stock (units = mol I *J, reference concentration of c o m p a r t m e n t i's stock, and n u m b e r of compartments.

The term C, In (C,/C,.r¢,.) arises from the theoretical minimum work to be done on the ecosystem to change its concentration relative to the reference level, and the te, m [C,(1-C,.r~r/C,] arises from the work done by the constant-concentrations surroundings. With this definition, exergy is always non-negative, independent of whether the system is more or less concentrated than the reference state. The expression for ascendency (AI is (Ulanowicz, 1986):

A=t~ v I • (~ I

_v IXl, f ~ l l o g (I

cx,,IX,

l

X,,lf3 t

,

(A.1.2)

I

where, k,2

/3 = total system output (throughftow) = V X,. It is assumed that the sum makes sense physically. The ratio X~,/XI is the probability that output (throughflow) from c o m p a r t m e n t I will go to compartment). Ascendency is also defined at a single instant of time. The range of subscripts indicates that the boundary for calculating ascendency extends beyond the c o m p a r t m e n t s used for energy intensity and exergy. The i = 0 corresponds to system inputs, such as energy, k +1 refers to dissipation, and k + 2 to exports (Ulanowicz, 1986). Using both input (import) and export compartments avoids the use of negative flows in calculating ascendency. For example, a negative net export is the difference between a (positive) gross import and a smaller (positive)

552

R. H E R E N D E E N

gross export. The value o f ascendency depends directly on the gross import and export flows, while the energy and nutrient intensities do not. In general, 1 believe it is the net export that is measurable, and significant. These imports and exports are by definition the same "'item" that is produced within the system, so that a large import and export of, say, diatoms could well correspond to a purely physical transport process unconnected to the biological activity. My approach is therefore to consider only the net import in calculating ascendency. I still use the gross import/gross export format to avoid negative flows: a negative net export appears as a positive gross import with zero gross export, a positive net export appears as a positive gross export with a zero gross import. See further notes in Tables 1-5. I also use this adjustment for the dynamic case. Exergy, ascendency, and e . Y combine the effects o f (1) overall system size, and (2) the distribution of stocks or flows; it is of interest to investigate these terms independently, to the extent that this is possible. The commonly used "sensitivity analysis" does not serve well here, because it applies to infinitesimal changes, and we are interested in large changes. The idea has been applied to a number of issues, such as the debate over the contributions of population growth, per capita consumption, and pollution intensity of industry to overall pollution releases (Ehrlich et al., 1972). Answering the question is complicated by the fundamental form of the indicator. (1) e . Y (energy intensities times net output). This can be written as e . Y . Y, where Y = Y~=~ Y~ and y, = Y~/Y. This is a product of energy intensities times the fractional distribution of net outputs times the total net output Y. (2) Ascendency. From eqn (A.1.2), we see that ascendency is a product of total system throughput times a dimensionless information theory-based term expressing an aspect o f the "'structure" o f the system. This can be expressed as size. structure. If the log is to the base 2, the structure term is given in bits. (3) Exergy. Exergy [eqn (A.I.1)] can be rewritten:

E X = RTCo{[ln(Co/Co.r~,)-(1-Co.r~r/C,,)]+,=,~

z, ln(zJz,.r~r)},

where,

C,, = summed concentrations of compartments, Co.~er= summed reference concentrations of compartments,

z~ = CJCo = fractional concentration of compartment i, z;.r~r= k

C~.rcr/Co.r~r, and k

z, = E z,.r,f= 1. i=1

i=l

Equation (A.I.3) can be considered a function of three variables, size. {[concentration] + structure}, where the terms correspond to those in eqn (A.1.3).

(A.1.3)

INDICATORS

IN

DYNAMIC

ECOSYSTEMS

553

The dimensionless concentration term [In (C,/C,.,,f) - ( 1 - C,.,~d C,,)] can automatically be dominatingly large if, for example, the reference level is that of the primordial soup (Herendeen, 1989). The dimensionless structure term ~=1 zi In (z,/z,.,cf) could in principle be as large, but in practice this is unlikely because the z's add to unity. In this paper, the magnitude of the concentration term is deliberately minimized by defining the reference state to be the initial steady-state. APPENDIX 2

Nonlinear expressions used in the dynamic model Let input scarcity be defined relative to that at an initial steady-state:

("fi' scarcity~ =

-,~,~r~,~,,o,

.~;,i.~,,,/nw, -J" o - !o/

S,/S,]

,

(A.2.1)

where, S~ = stock of compartment j at time t, S ° = stock of compartment j at start, and

n(j)

= number of compartments that provide inputs to compartment j.

The product is over the n(j) compartments that do have inputs to compartment j, including the nutrient pool. (The underlying dynamic model includes the nutrient pool.) If a compartment utilizes light, the product will also include the term L,/L,, the ratio o f the light level at time t to that at the start. The model used here began as a linear, recipient controlled one, which is unstable. Stability is added through use of two types o f nonlinear interaction: (I) The ability o f a given stock to support output (throughflow) diminishes as input scarcity increases:

X'IS' :,' ~,= Xi/S i 211 - ( 1 / 2 ) '~..... ',',] .

(A.2.2)

This fraction thus varies from zero (for scarcity-* infinity) to twice the original steady state value (for scarcity~ 0). (2) The fraction o f energy output (throughflow) that goes to "'lost" dissipation increases as inputs become scarcer:

lost j / X ; lost','/X(,' = scarcity;.

(A.2.3)

This fraction varies from zero (for scarcity -~ 0) to infinity (for scarcity--, infinity). It is possible for lost dissipation to exceed total output, in which case the compartment must draw down its stock to survive. These nonlinearities apply to all living compartments. They do not apply to detritus and the nutrient pool, both of which do not produce anything, and which in principle have no dissipation.