J. theor. Biol. (2001) 208, 493}506doi:10.1006/jtbi.2000.2234, available online at http://www.idealibrary.com on

Complementarity of Ecological Goal Functions

BRIAN D. FATH*-, BERNARD C. PATTEN? AND JAE S. CHOIA

*;.S. Environmental Protection Agency, National Risk Management Research ¸aboratory, Sustainable¹echnology Division, Sustainable Environments Branch, 26=est Martin ¸uther King Drive, Cincinnati,

OH 45268, ;.S.A. ? Institute of Ecology, ;niversity of Georgia, Athens, GA 30602, ;.S.A. andADepartment of Oceanography, Dalhousie ;niversity, Halifax, Nova Scotia, Canada B3H 4J1

(Received on 17 July 2000, Accepted in revised form on 3 November 2000)

This paper summarizes, in the framework of network environ analysis, a set of analyses ofenergy}matter #ow and storage in steady-state systems. The network perspective is used tocodify and unify ten ecological orientors or extremal principles: maximum power (Lotka),maximum storage (J+rgensen}Mejer), maximum empower and emergy (Odum), maximumascendency (Ulanowicz), maximum dissipation (Schneider}Kay), maximum cycling(Morowitz), maximum residence time (Cheslak}Lamarra), minimum speci"c dissipation (On-sager, Prigogine), and minimum empower to exergy ratio (Bastianoni}Marchettini). We showthat, seen in this framework, these seemingly disparate extrema are all mutually consistent,suggesting a common pattern for ecosystem development. This pattern unfolds in the networkorganization of systems.

( 2001 Academic Press

1. Ecological Organizing Principles

From classical thermodynamics two principlesare "rmly established for systems near equilib-rium (Aoki, 1998). The "rst is the second lawwhich applies to isolated systems: entropy alwaysincreases with time and approaches a maximumat equilibrium. The second is for open systems(Nicolis & Prigogine, 1977): entropy productionalways decreases with time and approachesa minimum at steady state. Far from equilibrium,which is where many physical systems and allliving systems operate, these principles do notapply. The search for organizing principles thatdo apply has produced a variety of energy &&orien-tors'' (MuK ller & Leupelt, 1998).

The central idea of the orientor approach... refersto self-organizing processes, that are able to

-Author to whom correspondence should be addressed.E-mail: fath.brian@epamail.epa.gov

0022}5193/01/040493#14 $35.00/0

build up gradients and macroscopic structuresfrom the microscopic &&disorder'' of non-struc-tured, homogeneous element distributions inopen systems, without receiving directing regu-lations from the outside. In such dissipativestructures the self-organizing process sequencesin principle generate comparable series of con-stellations that can be observed by certainemergent or collective features. Thus, similarchanges of certain attributes can be observed indi!erent environments. Utilizing these at-tributes, the development of the systems seemsto be oriented toward speci"c points or areas inthe state space. The respective state variableswhich are used to elucidate these dynamics, aretermed orientors. Their technical counterparts inmodeling are called goal functions (MuK ller andFath, 1998, p. 15).

In a large part, these orientors follow from theseminal work of Odum (1969) in which hehypothesized on the trends to be expected in eco-system development. That paper formed thebasis for several of the long standing orientors

( 2001 Academic Press

494 B. D. FATH E¹ A¸.

investigated herein such as biomass, cycling, in-ternal organization, residence time, and informa-tion. More recently, Schneider & Kay (1994a),who take a thermodynamic approach, proposedseven ecosystem properties as basic orientors:exergy capture, energy #ow, cycling of energy andmaterials, respiration and transpiration, biomass,average trophic structure, and types of organ-isms. Except for the last two, each of these sevenproperties is addressed in this paper. Additionalthermodynamic goal functions have been pro-posed speci"cally in the context of ecologicalmodels. In particular, Bendoricchio & J+rgensen(1997) made the case that the primary ecosystemgoal function is exergy storage. Bastianoni (1998;Bastianoni & Marchettini, 1997) suggested min-imum empower to exergy ratio as the primaryecosystem goal function, and J+rgensen et al.(2000) suggested speci"c dissipation as the pri-mary pattern observed in growth phenomena.

A few authors have investigated the subject ofgoal function uni"cation. J+rgensen (1992, 1994;J+rgensen & Nielsen, 1998) found a strong cor-relation between several goal functions and sug-gested that perhaps their integration could leadto consideration of only one of them. Patten(1995), using an earlier development of networkanalysis, showed that many goal functions havea common basis in the path structure and asso-ciated microscopic dynamics of systems. Here, wetake this a step further by demonstrating consist-ency of ten goal functions through a singleexplicit notational scheme expressing networkorganization. We will make these ties initially byshort statements in italics at the end of eachnumbered section below, then amplify them later.Typically, a goal function refers to the &&max-imum'' or &&minimum'' value of a particularquantity. However, because open ecosystemsare self-organizing complex adaptive systems(Waldrop, 1992) responding to current environ-mental conditions, we view the organizingprinciples as &&orientors'' or &&attractors'' (MuK ller& Leupelt, 1998). The active descriptors &&maxi-mize'' and &&minimize'' refer then to the direc-tional nature of the network processes underlyingthe goal functions.

1. Maximize power. Lotka's (1922) maximumpower principle states that systems become organ-ized to maximize their energy throughput. Odum

has long championed this principle in ecology,beginning with Odum & Pinkerton (1955) whichargued that maximizing power produced themost energy to perform work and create order(&&pump out disorder''). In networks, power isre#ected in energy throughput or total systemthrough-ow (TST), the sum of #ows into, oralternatively out of, all compartments (Patten,1995). Thus, in the network context, maximizingpower is equivalent to maximizing total systemthrough#ow. This a!ects many of the other prin-ciples to follow. Maximizing power (through-ow)will be taken as reference condition 1.

2. Maximize storage. J+rgensen & Mejer (1979,1981) proposed a maximum storage principle inwhich energy systems maximize their distancefrom a thermodynamic reference point by storingusable energy (exergy). The associated accumula-tion of mass or energy is re#ected in structure,function, gradients, order, organization, and in-formation*all of which express in di!erent waysdeparture from the reference. For entire systems,this principle asserts that total system storage(TSS) is maximized. For biotic systems thismeans maximizing biomass. Maximizing storagewill be considered as reference condition 2.

3, 4. Maximize empower and emergy. Odum(1988, 1991) developed the concept of emergy(embodied energy) to describe energy quality asreferenced to solar radiation. As solar energypasses through a series of energy transforma-tions, its quality increases in proportion to theamounts of original solar energy required at eachstep. Emergy measures this stored energy quality,and empower the associated energy #ow. Theaccounting methods used to calculate emergyand empower re#ect the total direct and indirectenergy storage and through#ow in a system,respectively. Maximizing empower (EMP) is con-sistent with reference condition 1 and maximi-zing emergy (EMG) is consistent with referencecondition 2.

5. Maximize ascendency. Ulanowicz (1986,1997) proposed a maximum ascendency principle,where ascendency quanti"es network organiza-tion as the product of the total system through-#ow (TST) and average mutual information(AMI). AMI involves the individual #owand a complicated expression of the logarithmof various other #ow and organizational

i

COMPLEMENTARITY OF ECOLOGICAL GOAL FUNCTIONS 495

components (Ulanowicz & Norden, 1990). Aver-age mutual information is dimensionless and hasa restricted range of values (generally betweenabout 2.0 and 6.0). The total system through#ow,which scales this information quantity, can varywidely over the nonnegative real numbers. Asa result, through#ow dominates the ascendencymeasure such that power and ascendency givestrongly correlated results (J+rgensen, 1994).Maximizing ascendency, then, approximates refer-ence condition 1.

6. Maximize dissipation. Dissipative structuresfar from equilibrium have been suggested tomaximize entropy production (Prigogine &Stengers, 1984; Brooks & Wiley, 1986). Schneider& Kay (1990, 1994a,b, 1995, 1996) elaborated thisin exergy terms, stating that systems suppliedwith an external exergy source will respond by allmeans available to degrade the received exergy.This amounts to a maximize dissipation principle,and systems and processes satisfying it best gainfrom the implied work performed. Such gains inwork represent a source of selective advantage inphysical and biological evolutionary systems.For biological systems dissipation includes res-piration plus other usable or unusable exports.Total system export (TSE) is the sum of dissi-pative processes over all components. Maximiz-ing total system export (TSE) seems counter tothe maximize storage principle above. This in-congruence has produced a divergence in con-temporary discussion between proponents ofmaximizing storage (J+rgensen}Mejer) and maxi-mizing dissipation (Schneider}Kay). The net-work model developed below gives a commonbasis for both ideas because it allows, againstdissipation which must be bounded by prior en-ergy acquisition and utilization e$ciency, the in-de"nite development of total system storage(TSS) through increased organization. We showthat maximizing dissipation is consistent with bothreference conditions 1 and 2.

7. Maximize cycling. Morowitz (1968) con-sidered that energy #ow caused cycling and thisproduced organization:

The #ow of heat from sources to sinks can leadto an internal organization of the system... The#ow of heat can lead to the formation of cyclic#ows of material in the intermediate system.(p. 28).

The #ow of energy causes cyclic #ow of mat-ter. The cyclic #ow is part of the organizedbehavior of the system undergoing energy #ux.The converse is also true; the cyclic #ow ofmatter such as is encountered in biology requiresan energy #ow in order to take place. The exist-ence of cycles implies that feedback must beoperative in the system. Therefore, the generalnotions of control theory and the general prop-erties of servo networks must be characteristic ofbiological systems at the most fundamental levelof operation. (p. 120).

Control concepts involve negative (deviation-damping) feedback (Patten & Odum, 1981), butcycling also opens the possibility for positive (de-viation-amplifying) feedback. Ascendency theory(Ulanowicz, 1986) invokes the latter in &&auto-catalytic loops'' central to system development.Glansdor! & Prigogine (1971) and Nicolis &Prigogine (1971) hypothesized an order-through--uctuation principle. These authors noted thatsmall deviations in energy #ow exist statisticallyin any thermodynamic system. These are gener-ally damped out by dissipative processes, but asenergy gradients (including those re#ected instorage) increase, deviation ampli"cation be-comes more and more probable. Any damping#uctuations that can better dissipate the gradi-ents became selected for and ampli"ed over time.&&This'', Odum (1983, p. 574) writes, &&is probablyequivalent to the principle of selection throughmaximizing power with pulsing''. In far-from-equilibrium thermodynamics pulsing or-ganizations have been referred to as &&dissipativestructures'' (Glansdor! & Progogine, 1971;Wicken, 1980), and the signi"cance of their oscil-lations is the acceleration of energy #uxtoward the realization of maximizing power.Maximizing cycling contributes to both referenceconditions 1 and 2, and hence is consistent with allof the foregoing principles.

8. Maximize residence time. Cheslak & Lamarra(1981) proposed that ecological systems organizeto maximize the residence time of energy. Theydemonstrated this in an experimental investiga-tion of a simple aquatic ecosystem, showing alsothat the majority of the e!ect was due to system-level properties rather than molecular properties.The residence time of #ow in a particular com-ponent, q

i, is given by the reciprocal of the turn-

over rate, q~1. Total system residence time can be

496 B. D. FATH E¹ A¸.

found by summing the individual componentresidence times (q

i) and is the fraction of through-

#ow that remains as storage (TSS/TST). Maxi-mizing residence time is consistent with referencecondition 2.

9. Minimize speci,c dissipation. Internal con-straints, such as caused in living systems by inef-"cient energy transfer or limited availability ofmetabolites, modulate through#ow maximiza-tion and divert free energy (exergy) to storage aschemical potential. This tends to minimize dissi-pation per unit mass or volume, which expressesthe least speci,c dissipation principle (Onsager,1931; Prigogine, 1947, 1955). Although this least-speci"c dissipation principle was developed forsystems near thermodynamic equilibria, wecontend that even if the global system is &&far fromequilibrium'', sub-systems at "ner spatio-tem-poral-organizational scales may be considered tobe in some proximity to a quasi-local steadystate*close enough at least for the principle toprovide an understanding of &&how a systemshould change.'' A question for future research isto determine whether a system is too far or nearenough for this to be valid. Choi et al. (1999) usedthe respiration/biomass ratio (TSE/TSS in ournotation) of lacustrine communities as an empiri-cal measure of least-speci"c dissipation at theecosystem level. With the dimensions of recipro-cal time, TSE/TSS (to be minimized) approxim-ates how e$ciently structure (TSS) can be createdfor a given amount of work performed (re#ectedin the unusable released heat portion of TSE).Minimize speci"c dissipation complements maxi-mize residence time (or equivalently, minimizeturnover rate) because TSE/TSS has units ofreciprocal time. The two measures di!er only inthe proportions of through#ow (power) which isdissipated. Minimizing speci,c dissipation isconsistent with reference condition 2.

10. Minimize empower to exergy ratio.Bastianoni & Marchettini (1997) proposed thatsystem organization can be measured by a ratioof empower to exergy (in their paper they inad-vertently label empower, a through#ow metric, asemergy, a storage metric). The empower toexergy ratio measures the total environmentalcost (through#ow) required to produce a unit oforganization (structure). This di!ers from mini-mizing speci"c dissipation in that speci"c

dissipation is only a fraction of TST. The metricwas tested on three lagoon systems and showedthat the &&natural'' system had the lowest em-power/exergy value. Bastianoni & Marchettini(1997) concluded that it was the most e$cient ofthe three at processing through#ow to maintainstructure. In network notation this goal func-tion is expressed as TST/TSS and has units ofreciprocal time. Minimizing empower to exergyratio supports reference condition 2, and is con-sistent with maximizing residence time. Also, notethat it is not necessarily inconsistent with refer-ence condition 1 so long as total system storageincreases more rapidly than total systemthrough#ow.

Employing ecologically oriented variables,these ten extremal principles can be estimated bya set of metrics to be optimized: power and em-power by total system through#ow, max(TST);storage and emergy by total system storage,max(TSS); ascendency by the product of totalsystem through#ow an average mutual informa-tion, max(ASC); dissipation by total systemexport, max(TSE); cycling by total system cycl-ing, max(TSC); residence time by the ratio of totalstorage to through#ow, max(TSS/TST); speci"cdissipation by the ratio of total system exportand total system storage, min(TSE/TSS); andempower to exergy ratio by the ratio of totalsystem through#ow to total system storage,min(TST/TSS). We employ established notationfor total system through#ow (TST) and introduceTSS, TSC, and TSE as total system storage,cycling, and export, respectively. Note, at steadystate the complement of total system export, totalsystem import (TSI), also follows from thisderivation. The extremal principles apply to sys-tem-wide properties, whereas most basic networkmetrics address pair-wise interactions betweensystem compartments. We describe below howpair-wise interactions are summed to determinewhole system contributions that are compar-able to the ecological interpretations. Basicnetwork fundamentals are sketched below sothat the ten extremal principles can be des-cribed in network notation. For a morecomplete treatment of network environ analysissee Patten (1978, 1981, 1982, 1985), Higashi &Patten (1989), Higashi et al. (1993), and Fath& Patten (1999).

COMPLEMENTARITY OF ECOLOGICAL GOAL FUNCTIONS 497

2. The Environ Model: Setup

Environs (Patten, 1978, 1982; see Fig. 1) area!erent and e!erent networks leading to and awayfrom open systems that are components of systemsat higher scales. Both the systems and their com-ponents are &&holons'' (Koestler, 1967). Givena mathematical description of the system in termsof its components, the latter's input and outputenvirons bounded within the system can also bedescribed. With this, it is possible to partition theinterior conservative (energy or matter) #ows andstorages of an n-th-order dynamical system witha di!erential or di!erence equation descriptioninto n input environs or n output environs, wheren is the number of components whose storages x

i,

i"1,2, n, serve as state variables. A system withn such storage components, or compartments, willhave 2n environs running within it, half of theminput environs traceable backward in time toboundary inputs, z

j, from boundary outputs, y

i,

and the other half output environs driving forwardin time from boundary inputs, z

j, to boundary

outputs, yi. The set of input environs forms one

partition of the storages and #ows, and the set ofoutput environs another (Patten, 1978).

3. Functional Analyses:For Through6ows and Storages

To identify the direct and indirect contribu-tions of #ow and storage, taking account of all

modes are described in more detail below. Thisexplicit network accounting is used to showconsistency and complementarity of the goalfunctions described in Section 1.

3.1. FLOW COMPONENTS OF

THROUGHFLOW*LEONTIEF MODEL

Network #ow analysis is predicated upon di-mensional #ow information for the system underinvestigation. Here, f

ijare elements of a square

matrix F denoting #ows from column elements j torow elements i and ¹

j"+n

i/0fij

where fij

repres-ents boundary out#ow. The motivation for #owpartitioning begins with nondimensional #ow in-tensities (that is, through#ow-speci"c #ows)which result when #ows are divided by through-#ows of originating compartments: g

ij"f

ij/¹

j.

[Note that the original Leontief (1966) approachnormalized the #ows by the through#ows of thereceiving compartments. The #ow-forward ori-entation used here was independently introducedby Augustinovics (1970) and Finn (1976)]. Theelements of matrix G"(g

ij) give the transfer ef-

"ciencies corresponding to each direct #ow, fij.

Powers Gm of this matrix give the indirect #owintensities associated with paths of lengthsm"2, 3,2. Due to dissipation these #ows tendto zero as mPR so that the power series+=

m/0Gm representing the sum of the initial, di-

rect, and indirect #ows converges to an integral#ow intensity matrix, N:

N

integral

"

"

Ihij

initial

#

#

Ghij

direct

#

#

G2#G3#2#Gm#2

hggggggiggggggj

indirect

"(I!G)~1. (1)

possible pathways, several variants of input}output analysis as originated by Leontief (1936,1966) and introduced into ecology by Hannon(1973) are employed. These methodologicalextensions were developed to implement theenviron concept as a quantitative system theoryof the environment. The objective here is to dem-onstrate how #ow components of storage andthrough#ows can be partitioned into "ve distinctstages or modes: boundary input (mode 0),"rst-passage (mode 1), cycled (mode 2), compon-entwise dissipative (mode 3), and systemwisedissipative boundary output (mode 4). These

N maps the steady-state input vector z into thesteady-state system through#ow vector (Pattenet al., 1990):

¹"Nz"(I#G#G2#G3#2#Gm#2)z.

(2)

Term by term, #ow intensities Gm of di!erentorders m are propagated over paths of di!erentlengths m. These paths can be enumerated bypowers of the corresponding adjacency matrix(Patten, 1985). The "rst term G0"I brings theinput vector z across the system boundary as

FIG. 1. Depiction of the environment of any focal entity at any level of organization, including (left to right):(a) a!erent input environment from an ultimate source, partitioned successively into (b) input environs de"ned within k-th,k#1-th, etc. level systems in which the focal entity is a compartment, (c) internal state-de"ning milieu (not shown) of the focalentity, (d) e!erent output environs de"ned within k-th, k#1-th, etc. level systems of focal-entity membership, and (e) e!erentoutput environment extending to an ultimate sink. The input and output environments are shown as light cones, which bound(because nothing moves faster than light) possible cause and e!ects, respectively, that can in#uence and be in#uenced by thefocal entity at any given moment. The environs are restrictions of the light cones to within scaled systems (levels k, k#1,2)of de"nition. Environs, as partition elements of described systems, have quantitative descriptions available; environmentsexternal to described systems have no such descriptions, and cannot be speci"ed. ( ) k-th level environs; ( ) k#1-th levelenvirons; ( ) k#2-th level environs.

498 B. D. FATH E¹ A¸.

input zj

to each initiating compartment, j. Thesecond term, G, produces the "rst-order (m"1)direct transfers from each j to each i in the system.The remaining terms where m'1 de"ne m-thorder indirect #ows associated with lengthm paths. As stated before, these go to zero in thelimit as mPR, which is necessary for series con-vergence. In the above developments F, ¹, andz represent matter or energy #uxes, and G andN are dimensionless intensive #ows.

A heuristic point to be made from eqn (2)is that the steady-state (far-from-equilibrium)through#ow consists of #ow contributionsarriving at each terminal compartment i afteroriginating at various source compartments j andbeing transferred over all paths of all kinds (acyc-lic or cyclic in di!erent permutations) and lengths(m). In other words, the steady-state through-#ows are distributed quantities, not only withrespect to the #ows that add directly to them, butalso in relation to the shorter or longer, directand indirect, histories of these #ows back to theirpoints of introduction into the system. What arecalled and appear in digraphs as &&direct #ows''fij

in environ analysis, are really not direct at all.They are actually fractions of antecedentthrough-#ows ¹

j, f

ij"g

ij¹j, distributed to

di!erent destinations:

¹j,¹ (065)

j"

n+

i(Oj)/0

¹ij, (3)

where each distribution element, ¹ij, is derived

historically [eqn (4)] from boundary inputs zj:

¹ij"n

ijzj"

=+

m/0

g(m)ij

zj. (4)

¹ij, as the i-th component of the j-th element of ¹,

shares the same direct and indirect decomposi-tion elements as given in eqn (2) for ¹:

¹ij" (g(0)

ijhij

initial

# g(1)ijhij

direct

#g(2)ij#g(3)

ij#2#g(m)

ij#2)

hgggggggigggggggj

indirect

zj. (5)

From eqns (4) and (5) it follows that

fij"g

ijCn+

i(Oj)/0

=+

m/0

g(m)ij

zjD . (6)

COMPLEMENTARITY OF ECOLOGICAL GOAL FUNCTIONS 499

This demonstrates that each &&direct'' #ow fij

atsteady state is actually composed of #ow ele-ments of all orders, m"1, 2,2, and is, therefore,a doubly distributed quantity (re#ected in thedouble sum) derived from a large number set ofdirect and indirect paths leading from the origin-ating inputs, z

j. This is no real surprise if one

thinks about it. When a herbivore i in an ecosys-tem eats a primary producer j to generate a directfood #ow f

ij, clearly the energy and matter em-

bodied in this #ow have had di!erent historieswithin the encompassing ecosystem since the

the indirectness involved, C must "rst be non-dimensionalized. This is accomplished in discretetime: P"I#CDt, where Dt is selected so that0)p

ij(1, ∀i, j. Speci"cally, the diagonal ele-

ments become pii"1#c

iiDt"1!q~1

iDt, thus

making P a one-step Markovian transitionmatrix (Barber, 1978). P"(p

ij) de"nes dimen-

sionless storage-speci"c #ow intensities repres-enting the probability that substance in j at timet will be in i at time t#Dt. The di!erent ordersm of #ow contributions to storage can then beexpressed, analogously to eqn (1), as

Q

integral

"

"

Ihij

initial

#

#

Phij

direct

#

#

P2#P3#2#Pm#2

hggggggiggggggj

indirect

"(I!P)~1. (8)

energy was originally photosynthetically "xed atdi!erent times at the boundary as z

j. These di!er-

ent histories imply di!erent pathways, and thusdi!erent degrees of indirectness, even though thebulk food #ow is &&direct''. The formulationpoints, in fact, to little real &&directness'' at all inthe #ow phenomenology of steady-state connec-ted systems, and leads to the conclusion thatnature is organized more around dominantindirect e!ects (Higashi & Patten, 1989) andholistic determination (Patten, 2001) thanaround direct causes and their immediate e!ects.

3.2. FLOW COMPONENTS OF

STORAGE*MARKOV MODEL

Identifying direct and indirect contributionsto storage follows the same basic logic as forthrough#ows. In storage analysis, #ows arenormalized by steady-state storage values of thedonating compartments, x

j, giving c

ij"f

ij/x

j,

with

cii"!

n+

k(Oi)/0

cki"!q~1

i. (7)

Here, q~1i

is the turnover rate of storage at i andqiis the turnover time. C"(c

ij) is the Jacobian

matrix in the standard linear system formulationof the input-driven form. The state vector x"(x

j)

is a storage, and C contains #ow rates. To obtainan input}output power series formulation com-parable to eqns (1) and (2), which explicitly shows

The series in eqn (8) converges so long as columnsums of P are less than one (Matis & Patten,1981). Since the systems in question are energeti-cally and materially open, the convergence condi-tion can be realized by making the time step, Dt,su$ciently small. And, corresponding to eqn (2),this series maps steady-state boundary inputs, z,in discrete time Dt, into a steady-state internalstorage vector:

x"Q(zDt)

"(I#P#P2#P3#2#Pm#2)zDt . (9)

This series shows explicitly the direct and indirect#ow contributions to storage, and makes appar-ent the basis for dominant indirectness in the#ow}storage phenomenology. Just as through-#ow and #ows are distributed quantities,[eqns (3)}(6)], so is storage:

xj"

n+

i(Oj)/0

xij, (10)

where

xij"q

ij(z

jDt)"

=+

m/0

p(m)ij

zjDt (11)

so that

xij"p

ijCn+

i(Oj)/0

=+

m/0

p(m)ij

zjDtD . (12)

As eqn (6) does for #ows, this shows steady-statestorages are doubly distributed quantities also,

500 B. D. FATH E¹ A¸.

the resultant of inputs from all sources ("rst sum-mation) being subsequently distributed to com-partments over all paths of all lengths m (secondsummation).

3.3. FLOW AND STORAGE PARTITIONING INTO MODES

As stated earlier, #ow and storage contribu-tions can be partitioned into "ve modes (0, 1, 2, 3,and 4) using network analysis (based on an ear-lier two- and three-mode partitioning presented,respectively, by Higashi et al., 1993; Patten& Fath, 1998). This partitioning is key to demon-strating consistency of the orientors in Section 1.Mode 0 is the boundary input into the system.Mode 1 accounts for all #ow in which substancemoves from node j to a terminal node i for the"rst time only. Mode 2 is #ow cycled at terminalnodes i of each (i, j ) pair. Mode 3 is compon-entwise dissipative #ow in the sense that it exitsfrom node i never to return again. Mode 4 is theboundary output from i constituting systemicallydisspative #ows exiting the system. Gallopin(1981) independently proposed a similar, butnon-mathematical, classi"cation in whichwithin-system #ow is partitioned into three cat-egories: strictly in#uenced (1), both in#uen-cing and in#uenced (2), and strictly in#uencing(3). These categories encompass and are concep-tually equivalent to modes 1}3 as indicated

TABL

Network representation of -ow and storage partitio

Flow

Equation (pair-wiseinteractions)

Notatiowide co

Mode 0 (boundary input) f (0)j0

"zj

f (0)"

Mode 1 ("rst-passage)f (1)ij

"Anij

nii

!d*ijB z

jf (1)"

Mode 2 (cyclic)f (2)ij

"

nij

nii

(nii!1)z

jf (2)"

Mode 3 (compartment-wise dissipative) f (3)

ij"A

nij

nii

!dijB z

j

f (3)"

Mode 4 (boundaryoutput)

f (4)0i

"yi

f (4)"

*dij

is the Kronecker delta de"ned by dij"1 for i"j and d

parenthetically. The "ve modes can be quanti"edand notated for both #ow and storage contribu-tions for each (i, j ) pair using the equations inTable 1, where superscripts refer to the modes.System-wide mode contributions are obtained bysumming all pair-wise combinations. Notationswithout subscripts ( f (k), x(k), for k"0,2, 4)represent single- (for boundary #ows) or double-summed (for internal #ows and storages) system-wide quantities.

Note from Table 1 that just as boundary in-puts, f (0), and outputs, f (4), are equal at steadystate, mode 3 is numerically equal to mode 1 forboth #ow and storage: f (1)"f (3) and x(1)"x(3).These equivalences are implicit in the "rst law ofthermodynamics and mass conservation sinceany matter or energy which crosses a system orcompartment boundary for the "rst time mustalso be dissipated from that system or compart-ment regardless, in the compartment case, of howmany nodes it passes through en route to its "naldestined exit.

Total #ow into i derived from j, ¹ij

forj"0,2, n, is the sum of modes 0, 1, and 2,¹ij"f (0)

ij#f (1)

ij#f (2)

ij, as is node storage, x

ij"

x(0)ij#x(1)

ij#x(2)

ij. Because of mode 0}mode 4 and

mode 1}mode 3, equivalences, these relationscan also be written as, ¹

ij"f (2)

ij#f (3)

ij#f (4)

ijand x

ij"x(2)

ij#x(3)

ij#x(4)

ij. These relations can

E 1ning into ,ve modes for any (i, j) pair in a system

Storage

n (system-ntribution)

Equation (pair-wiseinteractions)

Notation (system-wide contribution)

+ f (0)j0

x(0)j0"z

jDt x(0)"+ x (0)

j0

+ + f (1)ij

x (1)ij

"Aqij

qii

!dijB z

jDt x(1)"+ + x (1)

ij

+ + f (2)ij

x (2)ij

"

qij

qii

(qii!1)z

jDt x(2)"++x (2)

ij

+ + f (3)ij x (3)

ij"A

qij

qii

!dijB z

jDt x(3)"+ + x (3)

ij

+ f (4)0i

x(4)0i"y

iDt x(4)"+ x (4)

0i

ij"0 for iOj.

COMPLEMENTARITY OF ECOLOGICAL GOAL FUNCTIONS 501

further be written using the system-level notationas TST"f (0)#f (1)#f (2)"f (2)#f (3)#f (4) andTSS"x (0)#x (1)#x (2)"x (2)#x (3)#x (4),where the left-hand sums in each case representa!erent relations and the right-hand sums e!er-ent relations. The network mapping ¹

ij"n

ijzj

[eqn (2)] and TST"++ nijzj

verify the modepartitioning for through#ows as follows:

¹ij"f (0)

ij#f (1)

ij#f (2)

ij

"zjdij#A

nij

nii

!dijBz

j#

nij

nii

(nii!1)z

j

"Adij#

nij

nii

!dij#n

ij!

nij

niiBz

j"n

ijzj

(13)

and similarly for storages:

xij"x(0)

ij#x(1)

ij#x(2)

ij

"zjDtd

ij#A

qij

qii

!dijBz

jDt#

qij

qii

(qii!1)z

jDt

"Adij#qij

qii

!dij#q

ij!

qij

qiiBz

jDt

"qijzjDt. (14)

It is the recognition that internal #ows andstorages are comprised of input, "rst-passage,cyclic, locally dissipative, output portions,that makes possible the demonstration ofgoal function complementarity. Notationalconventions across scales are summarized inBox 1.

4. Goal Function Uni5cation

At the end of Section 1 the ten extremal prin-ciples power, storage, empower, emergy, ascend-ency, dissipation, cycling, residence time, speci"cdissipation, and empower/exergy ratio werelisted using simple ecological notation. Networkparameter equivalents of these principles aregiven in Table 2, including the formulation usedto generate the parameters. The mode partition-ing in Table 1 is for a speci"c (i, j) pair. Therefore,a double sum over all (i, j) pairs is needed toconvert Table 1 quantities to the total system-wide properties of Section 1. In Table 2 the de"ni-tion of turnover time as storage divided bythrough#ow (Higashi et al., 1993) is used to mapthrough#ows and inputs into storages: x

ij"q

i¹i

"qinijzj. The objective is to use the network

understanding and explicit notation to show that

Box 1. Clari"cation of notation for the three levels for each #ow and storage:

(1) Pair-wise interactions*the contribution to any i from any j (i, j"0, 1,2, n);(2) Compartmental level*the total contribution to i from all j (i, j"0, 1,2, n); and(3) Total system level contribution to all i from all j (i, j"0, 1,2, n)

We assume that these levels are additive, ¹i"+¹

ijand TST"+¹

i, and x

i"+ x

ijand TSS"+ x

i, leading to doubly

distributed through#ows and storages: TST"+¹i"++¹

ijand TSS"+ x

i"++x

ij. The mode distinction indicates

that total #ow and storage are partitioned into input (vector), "rst-passage (matrix) and cycled (matrix) portions.Table B1 presents the notation incorporated herein.

Table B1. Notation representing various hierarchical levels of #ow and storage in the environs of (i, j) pairs usingnetwork analysis

Level Flow Storage

(1) Contribution to any i from any j plusboundary input (pair-wise)

¹ij"f (0)

i0#f (1)

ij#f (2)

ijxij"x (0)

i0#x (1)

ij#x (2)

ij

(2) Contribution to i from all j plus boundaryinput (compartmental) ¹

i"f (0)

i0#

n+j/1

( f (1)ij

#f (2)ij

) xi"x (0)

i0#

n+j/1

(x (1)ij

#x (2)ij

)

(3) Total system level*the contributionto all i from all j plus boundary input(system-wide)

TST"

n+i/1

f (0)i0

#

n+i/1

n+j/1

( f (1)ij

#f (2)ij

) TSS"n+i/1

x(0)i0

#

n+i/1

n+j/1

(x (1)ij

#x (2)ij

)

TABLE 2Energy organizing extremal principles with associated network formulations. Superscripts denote modes.¹hese are system-wide properties so the appropriate notation is used (Box 1): ¹S¹"total systemthrough-ow, ¹SS"total system storage, ¹SE"total system export, ¹SC"total system cycling,EMP"empower, EMG"emergy, ASC"ascendency, AMI"average mutual information. ¹he termn*ij

used in the emergy and empower formulations denotes network transformities which convert energy -owand storage to empower and emergy and di+ers from the n*

ijof eqn (1) is the extent that transformities do

not account for cyclic feedback.

Principle Extremal principleNetwork parameter

(system level) Network analysis formulation

Maximize power max (TST) TST"f (0)#f (1)#f (2) TST"++ (nij)z

jMaximize storage max (TSS) TSS"x(0)#x(1)#x(2) TSS"++ q

i(n

ij)z

jMaximize empower max (EMP) EMP"f (0)#f (1)#f (2) EMP"+ +(n*

ij)z

jMaximize emergy max (EMG) EMG"x(0)#x(1)#x(2) EMG"+ +q

i(n*

ij)z

jMaximize ascendency max (ASC) ASC"AMI*[ f (0)#f (1)#f (2)] ASC"AMI*++ (n

ij)z

jMaximize dissipation max (TSE) TSE"f (4) TSE"++ e

i(n

ij)z

jMaximize cycling max (TSC) TSC"f (2) TSC"++ (n

ij/n

ii)(n

ii!1)z

jMaximize residence time max (TSS/TST) TSS/TST"q TSS/TST"++ q

i(n

ij)z

j/(n

ij)z

j"+q

iMinimize speci"c dissipation min (TSE/TSS) TSE/TSS"f (4)/(x(0)#x(1)#x(2)) TSE/TSS"++ e

i(n

ij)z

j/q

i(n

ij)z

j"++ e

i/q

iMinimize empower to exergyratio

min (TST/TSS) TST/TSS"1/q TSS/TST"++ (n*ij)z

j/q

i(n

ij)z

j"++ (n*

ij)/q

i(n

ij)

502 B. D. FATH E¹ A¸.

these extremal principles are internally consistentand complementary.

Much debate and confusion have centered onthe appropriateness of these various goal func-tions because, at "rst glance, the simultaneousrealization of max(TST), max(TSS), max(EMP),max(EMR), max(ASC), max(TSE), max(TSC),max(TSS/TST), min(TSE/TSS), and min(TST/TSS) seems contradictory. Further inspection,however, shows that all these goal functions arein fact mutually consistent. They are all gener-ated by network processes and they givecomplementary perspectives on the spon-taneous directions of ecological growth anddevelopment.

Maximize power has been taken as the "rstreference condition against which to evaluate theothers. Maximize power, as represented by totalsystem through#ow using network parameters(TST"f (0)#f (1)#f (2)), is a combination ofinput, f (0), "rst-passage #ow, f (1), and cycling, f (2).Each of these three basic building blocks contrib-ute to overall through#ow. Using the networkderivation as described in eqn (2) and Table 2,TST is composed of the products of integral #owintensities, n

ij, and inputs, z

j. Maximize power

appears to be the primary orientor and founda-tion for the complementarity of ecological goalfunctions because the combination n

ijzj"gures in

the network formulation of all the others (as seenin the last column of Table 2). However, a closerlook reveals a more subtle situation in whichthere are tradeo!s between the goal functionsparticularly regarding the rate at which theyoccur.

Maximize storage, TSS"x(0)#x(1)#x(2), isthe second reference condition and it also com-bines input, "rst-passage, and cyclic contribu-tions. In the network formulation, x

ij"q

inijzj,

storage is directly proportional to power by thefactor of the turnover time, q

i, at each compart-

ment. Storage is a measure of #ow impedance,therefore, greater #ow and higher capacitance (inan analogous sense to electrical networks) resultin greater storage. Maximizing power supportsthe maximize storage principle and maximizingstorage reinforces the maximize residence timeprinciple.

Maximize empower (EMP) and emergy(EMG), consider total system through#ow andstorage as expressed in terms of the sourceenergy, EMP"f (0)#f (1)#f (2)"n*

ijzj

and

COMPLEMENTARITY OF ECOLOGICAL GOAL FUNCTIONS 503

EMG"x(0)#x(1)#x(2)"qi(n*

ijo

)zj, where n*

ijis

a network-based transformity which converts en-ergy (in joules) to emergy (in emjoules). [Fora thorough treatment comparing network andemergy analyses see Brown & Herendeen, 1996.]These principles are conceptually equivalent tomaximizing total system through#ow and totalsystem storage, but within the context of theprocess transformations from the initial source.These goal functions are consistent with referenceconditions 1 and 2.

Maximize ascendency is the product of totalsystem through#ow (TST) and average mutualinformation (AMI), ASC"AMI*[ f (0)#f (1)#f (2)]"AMI*n

ijzj. The AMI scales TST according to

the system organization, but the measure is prim-arily dominated by the through#ow because thecontribution of average mutual information isusually small in relation to that of through#ow.Thus, this goal function is consistent and highlycorrelated with the "rst reference condition(J+rgensen, 1994).

Maximize dissipation, TSE"f (4), is to increasethe total boundary #ow exiting the system. Dissi-pation is expressed here as compartment-speci"cfractions, e

i, of through#ows at i derived from

each source input zj: TSE"+ e

i¹i"++ e

inijzj;

since at steady state, total system export equalstotal system import ( f (4)"f (0)). Therefore, maxi-mizing dissipation is equivalent to maximizinginput which is one component of total systemthrough#ow. Maximizing input contributes tomaximizing power.

Maximize cycling, f (2), occurs when the mode 2portion of total system through#ow increases.Cycled #ow contributes to TST separately frominput or "rst-passage #ow. The diagonal ele-ments n

iiof N give the total number of times

a resource will exit a particular compartment.When the cycled portion, n

ii!1, is weighted by

the "rst passage #ow, (nij/n

ii)z

j, this gives the

total system cycling. (Note, total system cycling(TSC) is an absolute measure of the amount ofcycled #ow in the system. Finn (1976) developeda cycling index which calculates the portion ofsystem #ow that is cycled). There is a seemingdiscrepancy between maximizing cycling andmaximizing dissipation because it appears dissi-pation is limited by cycling. If TST were "xed, i.e.zero-sum, then there would be a tradeo! among

mode 4 (boundary dissipation), mode 2 (cycling),and mode 1 ("rst-passage #ow). However, TST isnot "xed but is itself being maximized. Maximiz-ing cycling is, therefore, consistent with the "rstreference condition since TST is comprised ofboth dissipation and cycling.

Maximize residence time, q"TSS/TST, is asingle parameter goal function that is clearly con-sistent with the maximize storage goal functionas previously discussed. Unlike the previous en-ergy organizing principles, this one is indepen-dent of the input z

j. Maximizing residence time is

also foundational to the two following goal func-tions minimizing speci"c dissipation and minim-izing empower to exergy ratio as discussed above.

Minimize speci,c dissipation, f (4)/(x(0)#x(1)#x(2)), states that the ratio of total system export tototal system storage decreases. In network nota-tion, this simpli"es to e

i/q

i, which has units of

reciprocal time. There is an important distinctionhere because the through#ow term ¹

i"n

ijzj

isnot present. Minimizing speci"c dissipation isdependent on turnover time q

iand the fraction of

through#ow that is exported, ei. Turnover time

is generated from the internal storage transfere$ciencies [eqn (7)] and fractional dissipationfrom boundary input. Therefore, this principlecaptures two basic systems properties and tooptimize it residence time should increase.

Minimize empower to exergy ratio, notated asTST/TSS, simpli"es to minimizing the turnoverrate which is equivalent to maximizing residencetime. This goal function combines both referenceconditions and measures the system e$ciency atmaintaining structure. To the extent that thetransformity, N*, approaches the integral #owmatrix, N, this ratio approximates the maximizeresidence time principle. The implicative loop ap-pears to be close around three fundamental prop-erties*through#ow, storage, and residence time.

The two principles that seem most contradic-tory are maximize dissipation (max( f (4))) and min-imize speci,c dissipation (min( f (4)/(x(0)#x(1)#x(2)))). However, both can co-occur if total systemstorage increases faster than total system export.That is, if f (4) is maximized, then the ratio,f (4)/(x(0)#x(1)#x(2)), can still be minimized iftotal system storage, (x(0)#x(1)#x(2)), maxi-mizes more rapidly. Minimizing speci"c dissipa-tion combines output (and by equivalence, input)

504 B. D. FATH E¹ A¸.

and storage into one organizing principlesuch that both dissipation and structure aremaximizing while at the same time their ratio isminimizing. J+rgensen et al. (2000) have arguedthat this is in fact the pattern observed in allgrowth phenomena. System dissipation rises rap-idly to near-theoretical maxima in early growth,but storage continues to increase inde"nitelythoughout middle and late developmental stages.In biological systems, the tendency of storage toincrease faster than dissipation is well known inthe power scaling of respiration rates to organismsize, R+B3@4 (von Bertalan!y, 1957). It has beenspeculated by various authors that this may besimply due to dimensional constraints (e.g. Stahl,1962; Economos, 1979; Platt & Silvert, 1981;Barenblatt & Monin, 1983; Patterson, 1992). Thepresent analysis adds thermodynamic and organ-izational constraints as well.

5. Conclusion

The consistency of the goal functions investi-gated here by network methods is more than justthe sharing of several notated variables*z

j, n

ij, and

qi. Only those pertaining to empower, emergy,

and ascendency were originally conceived in anexplicit network context, yet it is global systemicorganization that is behind the similaritiesinherent in all the studied goal functions. Theimplication is that the network perspective isfundamental, and somehow the originators ofdi!erent orientors managed to capture this intu-itively in their concepts.

At steady state all ten energy organizing prin-ciples are founded on increasing boundary #ow(import or export since f (0)"f (4)), and three pri-mary internal properties: "rst-passage #ow ( f (1)),cycling ( f (2)), and residence time (q

i) Boundary

#ow along with "rst-passage and cycling #ows allcontribute to increasing total system through-#ow and give a complete picture of #ow par-titioning. Boundary #ow follows from exogenousinputs (z

jand z

jDt) and "rst-passage #ow from

endogenous transfer e$ciencies (nij/n

ii!d

ijand

qij/q

ii!d

ij). In addition to the inputs and trans-

fer e$ciencies, cycling is also a function of systemconnectivity and organization. Retention timedepends on cycling and system structure becausecycling retains and stores #ow, thus increasing

the turnover time. Cycling at one scale is struc-tural storage at another. The primary boundaryand internal properties that are common to theseorganizing principles can be summed up in thefollowing maxim: Get as much as you can (maxi-mize input and "rst-passage #ow), hold on to itfor as long as you can (maximize retention time),and if you must let it go, then try to get it back(maximize cycling).

The ten energy organizing extremal principlesare all consistent with these properties. Not onlyare all these orientors mutually consistent, butthey are interdependent for ful"llment. Maximiz-ing boundary dissipation and maximizing cyclingboth contribute to maximizing through#ow.Maximizing through#ow contributes to maxi-mizing storage, subject to turnover consider-ations. The intuition of J+rgensen, Bendoricchio,and Patten to show consistency of the goalfunctions was correct. However, contrary toBendoricchio & J+rgensen (1997), but in agree-ment with J+rgensen et al. (2000), we "nd thatspeci"c dissipation rather than storage per se isthe primary goal function. Minimizing speci"cdissipation is most encompassing because itcaptures all three properties above and is depen-dent on maximizing storage faster than maximiz-ing dissipation, which is empirically observed. Inconclusion, we support the use of a plurality ofgoal functions because each organizing principlere#ects a slightly di!erent aspect of overallsystem function. In fact, it is probably theircomplementarity and interdependency that hasmade the identi"cation of a single universal ex-tremal principle di$cult.

Dedication

The prior work of Masahiko Higashi, who perishedwith four other ecologists in the "eld during lateMarch 2000, runs all through this paper and much ofour previous investigations. With sadness and fondremembrance we dedicate this small increment of newknowledge in the conviction that his &&indirect e!ects''will continue to propagate into the long future of eco-logy, wherever networks become its objects of study.

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