Systems Approach to the Concept of Environment · 2006-07-08 · SYSTEMS APPROACH TO THE CONCEPT OF...

Copyright © 1978 Ohio Acad. Sci. 0030-0950/78/0004-0206S2.00/0

SYSTEMS APPROACH TO THE CONCEPT OF ENVIRONMENT1' 2

BERNARD C. PATTEN, Department of Zoology and Institute of Ecology, University of Georgia,Athens, GA 30602

Abstract. A systems theory of environment formulates causal interactions betweenthings, including organisms, and their environments in terms of four system theoreticalabstract objects. Creaons receive stimuli and implicitly create input environments.Genons react to received causes and generate potential output environments as effects.A holon represents the combined input-output model of an entity consisting of a creaonand a genon. An environ is a creaon and its corresponding input environment, or agenon and its related output environment. The theory is presented in terms of threepropositions that: (1) recognize two distinct environments (input and output) asso-ciated with things, (2) establish things and their environments as units (environs) tobe taken together, and (3) partition systems into input and output environsassociated with intrasystem creaons and genons, respectively.

OHIO J. SCI. 78(4): 206, 1978

Ecology is the biological science of en-vironment. It considers environment asa derivative of physiology in the sensethat environment contains resources tobe mobilized by organisms, and condi-tions of life under which this mobilizationmust occur. The resource in least sup-ply at any given time is rate limiting (lawof the minimum), as is the factor, such astemperature, in greatest extreme (law oftolerance). Thus, the organism is seenby ecology to inhabit a physiological lifespace bounded by conservative and non-conservative elements of its environment—resources and factors, respectively.

The nature and composition of this lifespace varies according to the characterof the larger system of which the organ-ism is seen as a part. Population aspectsof environment encompass the intra-specific reproductive, genetic, demo-graphic and social worlds of the organ-ism. A community aspect refers to in-terspecific biotic associations. The eco-system aspect takes into account all fea-tures of the organism's biotic and abioticinteractions.

Although the strict ecological idea ofenvironment is based on the individual

1Manuscript received January 19, 1977 andin revised form June 15, 1978 (#77-6).

2University of Georgia Contributions in Sys-tems Ecology, No. 41.

organism, loose usage frequently extendsthe concept from individuals to groups{our environment), or suggests somethingabsolute (the environment). The dic-tionary defines environment variously as:"the surrounding conditions, influences orforces that influence or modify; the wholecomplex of climatic, edaphic and bioticfactors that act upon an organism or anecological community and ultimately de-termine its form and survival; the aggre-gate of social and cultural conditions thatinfluence the life of an individual or com-munity," (Merriam-Webster 1971). Thesignificant features of environment inordinary usage are that some defined sub-ject (individual or group) is immersed inor surrounded by it, and influenced byit through a causal relationship. Thiscausality, as developed below, is thebasis for the present attempt to expressenvironment in terms of system theory,which is the purpose of this paper.

SYSTEMS DEFINITIONSSystems ecology is a branch of ecology

that applies systems thinking and meth-ods to ecological problems. Several def-initions of basic system concepts are use-ful in prospect of a systems approach todefining environment. A system is apartially interconnected (interacting orcausally joined) set of components. In-teractions may be mediated by energy-

206

Ohio J. Sci. EIA-SYSTEMS APPROACH 207

matter through transactions, or by infor-mation through communications. Trans-actions and communications correspond,respectively, to transfers of conservativeresources and nonconservative factors inthe physiological account of environmentdescribed above.

In a hierarchical model of nature, anygiven system can usefully be abstractedas three discrete levels separated out of ahierarchical continuum: system, subsys-tem and supersystem. Subsystems arecomponents of the systems. Supersys-tems are composed of systems. Koestler's(1967) term "holon" for a hierarchicalsystem can be used to refer to any ofthese three levels of organization, accord-ing to the frame of reference.

A system is closed if it does not interactwith another system, and open if it re-ceives causes from or generates effects toanother system. A system boundaryprovides the interface with other systemsand is defined by specifying its componentset. Input is any movement of energy-matter or information from supersystemto system, and output is any similarmovement across the system boundary inthe opposite direction.

ECOLOGICAL CONCEPTS OFENVIRONMENT

Environment as a concept has not beentreated very seriously in ecological litera-ture and only a few explicit works areavailable. Mason and Langenheim (1957)defined environmental phenomena as thosethat have or may have an operational re-lation with any organism. The environ-mental relation of an organism is the sumof empirical relations between the en-vironmental phenomena and any indi-vidual organism. The operational en-vironment of an organism consists of thoseinstantaneous environmental phenomenathat actually enter a relation with an or-ganism; the concept applies to specific in-dividual organisms. Space and timeframes of the operational environmentare determined by the organism. Thelife span of the organism corresponds tothe existence time of its operational en-vironment. Potential environment con-sists of the set of environmental phenom-ena that may enter into an environmentalrelation at some point in the ontogeny of

an organism. Non-environment consistsof all phenomena (indirect, historical andorganism-caused) which never enter intoa direct environmental relation with theorganism.

Mason and Langenheim (1957) as-serted, "the environment of any organ-ism is the class . . . of those phenomenathat enter a reaction system of the organ-ism or otherwise directly impinge upon itto affect its mode of life at any timethroughout its life cycle as ordered by thedemands of the ontogeny of the organismor as ordered by any other condition . . .that alters its environmental demands."Only direct factors were considered partof environment. "[Indirect and histori-cal] factors both function to condition aphenomenon . . . to which an organismthen reacts. Important as this is to theecosystem the only [organism] reaction. . . is to an already conditioned phe-nomenon. The state of the phenomenonprior to its conditioning is outside thescope of operational . . . and . . . po-tential environment. . . . This may seemto rest upon trivial distinctions, but weare convinced that this is the preciseboundary between clarity and confusionin the problems of the environment."

Thus, chains and networks of historicalcausation, which condition direct factors,are excluded from Mason and Langen-heim's (1957) concept of environment:" . . . we must reject the implication that. . . [causal] chains constitute a unitaryevent playing a significant role in the en-vironmental relation even though thesteps are very important to the ecosys-tem. . . . There is also a philosophicalreason for removing indirect factors fromthe concept of environment. To intro-duce indirect factors into causal relationswithin the environment is to introduce aninfinite regress into the system of expla-nation. Every cause has in turn itselfa cause which becomes an indirect causeof the most recent effect. The regress istoward the limbo of ultimate cause alongan infinitely reticulating path; for thiswe have neither finite description norfinite explanation. . . . To include suchrelations in environment is to confuse en-vironment with its history."

A systems ecology concept of environ-ment must take issue with the Mason and

208 BERNARD C. PATTEN Vol. 78

Langenehim theory. The whole thrustof a systems understanding of nature isto reconstruct the main patterns of causa-tion in models. Within the confines of afinite model forming a whole from inter-connected parts, an expanded concept ofenvironment of the parts is possible,which includes both direct and indirectfactors. The intrasystem causal net-work is never an unknown infinite regress,but is explicit to the model boundarywhich constitutes the limit of finite de-scription and explanation which werelacking in Mason and Langenheim's time.While the conditioning of direct causes byindirect effects may be temporally ante-cedent, ecosystems and their models arepersistent or recurrent organizations sothat historical patterns of causation arerelevant, with perhaps small correctionsfor evolution, to present and future pat-terns as well.

Such a systems view of environmenthas precedent in an ecological work byHaskell (1940), who focused on events inthe universe that may eventually in-fluence an organism during its lifetime.Their influence is limited by how fastcausality can be propagated, no fasterultimately than the speed of light. Thus,corresponding to each instant in the lifeof an organism is a light cone. (Haskell1940, fig. 1) bounding the spatiotemporalextent of possible causes. The cones di-minish in time as the universe that canpossibly affect the ageing organismcontracts: "The cones prepresent . . .a steadily shrinking region . . . withinwhich the fastest moving process—light,traveling at about 300,000 km a second—can start, at any point-instant . . . dur-ing the organism's existence, and effect(sic) it before its end. . . . This region isequal to a geometric hyperbody, dennedbelow as 'habitat', and, constitutes partof 'environment.' Habitat is the "im-mediate environment" (Haskell 1940, p.7), taken as Weaver and Clements (1929)denned it: "Every part of the environ-ment that exerts directly or otherwise[i.e., indirectly] a specific influence uponthe life of the plant is a factor of thehabitat."

Thus, Haskell's concept of environmentincludes not only the direct causes ofMason and Langenheim, but indirect

causes as well, so long as their eventualinfluences can be propagated to a sub-ject, such as an organism, during itsexistence interval. Systems ecology mod-els that represent complex intrasystemwebs of direct and indirect causationmake it possible to implement such anexpanded concept of environment. Aformal approach to such implementationis described below.

HOLONSGeneral systems theory defines a sys-

tem to be a partially interconnected set ofobjects, then proceeds to describe the ob-jects and various aspects of their inter-active coupling. Formal details differwith the specific theory, but most generalsystems objects have in common that insome sense they perform a double map-ping of time into state, then state intooutput. Examples are "finite state ma-chines" of Gill (1962), "abstract objects"of Zadeh and Desoer (1963), Wymore's(1967) "formal systems," the "generalsystems" of Klir (1969) given accordingto five definitions, "T-processors" ofWindeknecht (1971), and "general timesystems" of Mesarovic and Takahara(1975). All such units may be madecausal, and can be generalized under thenonspecific hierarchical object, holon(Koestler 1967). An extensive theory ofthe causal holon as the basis for a systemsconcept of environment has been pre-sented elsewhere (Patten et al 1976),based on Zadeh's model (Zadeh andDesoer 1963, Zadeh 1969). This theoryis outlined below, with notation modifiedaccording to Mesarovic and Takahara(1975).

To model a causal link between twoentities requires some kind of process orobject whose action converts cause toeffect. Such an object, H, is a relationon attributes, VeA, that are time func-tions in a time domain, T. For each aeAa is a behavior V eT, a(t) is the value of aat time teT, a,* is the segment of a priorto /, and at is the behavior segment of abeginning at and following /. This ob-ject definition provides latitude in select-ing the set A of behavioral attributes.

The holon becomes oriented when itsset of attributes is partitioned into inputs,Z, and outputs, Y. The relation H on


A is then expressed as a set of input-output time segments: (z,y)eH, zeZ andyeY. The oriented holon associates re-sponse (output) time sequences withstimulus (input) histories. In develop-ing the holon as a causal object, givenoutput sequences must be uniquely as-sociated with given input segments. Thisproperty is incorporated in the notion ofa functional holon, where H is construedas a map (function) of inputs, z, into out-puts, y. Such an object is said to bedeterminate, i.e., a time series of inputsfrom its environment uniquely deter-mines a corresponding time series ofoutputs.

Dynamic behavior of a determinate ob-ject occurs in response to the object's en-vironment's behavior, which is receivedas input. This is modeled by introduc-ing a third set, X, of object variables,states. Heuristically, inputs zeZ serve tomap time teT into states xeX, and thestates take inputs zeZ into outputs yeY.States are generated by a state transitionfunction:

0:ZXX->X,and outputs are generated by a responsefunction:

The only other requirement for a de-terminate holon to be causal is that it notrespond at time / to inputs received aftert. That is, the object cannot anticipateits future environment; it is nonanticipa-tory. If a determinate object were togenerate more than one output sequencecorresponding to a given input sequence,the only way it could do this (since it isdeterminate) would be based on informa-tion about the future. This possibilityis precluded for the causal object.

The full theory (Patten et at 1976)should be consulted for details. Thecausal holon may serve at either thesystem or subsystem level. The focusof the original work was on intrasystempropagation of causes between subsystemlevel holons. As a result, consequencesof the theory for a system concept ofenvironment were not as clearly per-ceived as they are now. Environment isnormally a supersystem level concept.Causation was considered to be intro-duced as inputs from an environment at

the supersystem/system interface, thenpropagated through the interactive net-work connecting subsystem holons, andfinally dissipated as output effects gene-rated to the environment across the sys-tem boundary. The key to recognizingthe main features of the theory and itsimplications for an improved concept ofenvironment lay in focusing on intra-system environments associated with sub-system level holons. These environ-ments may be explicitly identified andmeasured as a causal reticulum within asystem model, with consequences thatemerge as three main points of the theory.These points are developed as specificpropositions in the next three sections.

FIRST PROPOSITIONProposition 1: Every object H defines

two environments: an input environmentH', and an output enviroment H". Theprerogative of environment definition isthat of the object.

The causal model of subject/environ-ment interaction leads to not one, buttwo equally plausible and useful conceptsof environment. The first is input en-vironment H\ defined by holon H in theact of receiving energy-matter or perceiv-ing information. Behavioral attributesof the real world that do not impact H asinput during its existence interval cannotinfluence the state of the object. Theygo unrecorded by H and consequentlyare not part of its environment. Sobasic is this environment defining func-tion that this aspect of the holon is given(Patten et al 1976) a special name, creaon,to signify an implicit act of environmentcreation. Mason and Langenheim (1957)restrict the concept of (input) environ-ment to phenomena that "directly im-pinge" upon the organism, whereas Has-kell (1940) includes, in addition, the indi-rect causes from which direct ones are gen-erated. The latter, and the present ap-proach, are more consistent with a sys-tems view, and in the context of finiteecosystem models do not produce the in-finite causal regress to which Mason andLangenheim objected. That is, when His a subsystem level component, W istraceable only to the model boundary, be-coming beyond this merely undiffer-entiated input to the system level. The


within system portion of H} is thus ex-plicit in the concept of input environment.

Reciprocally, the second concept of en-vironment is that of an output environ-ment Hn. This begins as a set of potentialenvironments embodied in the states ofH. These states are converted to out-puts through interaction of H with otherobjects (creaons). This is, to produce anactual output environment from potentialenvironment implicit in the state struc-ture of H requires holon production ofpotential attributes, then sequential cre-aon selections to achieve realization ofthese potentials. Output environmentHn is the resultant causality propagatedfrom H as a network of direct and indirecteffects. This environment generatingproperty of holons is equally basic to thecreaon function, and to distinguish it thename genon is given (Patten et al 1976).As in the creaon case, an infinite progres-sion of effects from // is implied, but atthe component level in the context offinite models, the progression terminatesat the system level boundary beyondwhich only undifferentiated output isrecognized. The within system portionof Hn is thus explicit in the concept ofoutput environment.

Neither Haskell (1940) nor Mason andLangenheim (1957) considered outputenvironment as a proper component ofthe general concept of environment.However, an older physiological theoryprovides explicit justification for the out-put environment, von Eexkiill (1926)presented a picture of environment as anorganism surrounder in terms of the fol-lowing set of concepts:

World-as-sensed: "Every animal is asubject, which, in virtue of the structurepeculiar to it, selects stimuli from thegeneral influences of the outer world, andto these it responds in a certain way."

World-of-action: "These responses, intheir turn, consist of certain effects onthe outer world, and these again influencethe stimuli."

Function-circle: "In this way therearises a self-contained periodic cycle,which we may call the function-circle ofthe animal. The function circles . . .connect up . . . in the most variousways, and together form the function-world of living organisms, within which

plants are included. For each individualanimal, however, its function-circles con-stitute a world by themselves, withinwhich it leads its existence in completeisolation."

Inner world: "The sum of the stimuliaffecting an animal forms a world in itself.The stimuli, considered in connectionwith the function circle as a whole, formcertain indications which enable the ani-mal to guide its movements. . . . Theanimal itself, by the very fact of exercis-ing such direction, creates a world for it-self, which I shall call the inner world."

Surrounding world: "World-of-actionand world-as-sensed together make acomprehensive whole, which I call thesurrounding world."

World-as-sensed and world-of-actioncorrespond to input and output environ-ments, respectively, and the latter is thusclearly distinguished. Moreover, vonUexkull's view of the organism/environ-ment relation is unitary: "The entirefunction circle formed from inner worldand surrounding world . . . constitutes awhole which is built in conformity withplan, for each part belongs to the others,and nothing is left over to chance . . .where there is a foot, there is also a path;where there is a mouth, there is also food;where there is a weapon, there is also anenemy. . . . If this circle is interruptedat any point whatsoever, the existence ofthe animal is imperilled. . . . continuityof the complete whole must never be lostsight of." Output and input environ-ments are continuous through the func-tion circles of the organism, and that con-tinuity erases, in theory, any distinctionbetween them. However, there is thematter of practicality to be considered:"All the [function] circles, however farthey lie separated from one another in theworld-as-sensed, intersect in the steeringapparatus of the inner world, and thenseparate from one another again in theworld-of-action." World-as-sensed (in-put environment) and world-of-action(output environment) are, for all practicalpurposes, separate by virtue of the enor-mity of reality compared to the identifi-able sphere of influence of any singleorganism (holon).

Thus, the first proposition. Every in-teracting thing in nature defines two


separate and distinct environments, bothtaken to include the network of causesand effects as far as these are traceable inany particular model in which the defin-ing object serves as a component.

SECOND PROPOSITIONProposition 2: The internal cause prop-

agating structure of systems cannot be com-pletely determined, i.e., all causal paths inthe interactive network accounted for, with-out input or output reference to an externalenvironment. The prerogative of realiza-tion of internal system structure is that ofenvironment.

This proposition, developed in detail inPatten et al (1976), can best be presentedhere in terms of an example. Figure 1illustrates a simple steady state model ofmarine coprophagy (Cale and Ramsay1970, description in Patten et al 1976,Appendix). The model consists of fourholons in series, with a feedback loopconnecting H3 and H4. Hi is a mud crab,Callianassa major; H2 is the feces of thisanimal; H3 includes all other benthic in-vertebrates of the marine community un-der consideration; and H^ is defined asthe feces of this latter group of animals.The model is simple in its interactivestructure, and for that reason, quite in-

structive. Causality is expressed as car-bon flow (gC m~2 y^1) and system state isrepresented by carbon storages (gC m~2).

FIGURE 1. Steady state marine coprographymodel (Cale and Ramsay 1970). Holon inputsand outputs represent carbon flows in gCmT2 y~', and states represent carbon storagesin gC m~2.

Hi Callianassa majorHiC. major feces#3Benthic invertebratesiJ^Benthic invertebrate fecesCarbon flow: x's and y's in gC m~2 y"1

z's in gC m~2

Environmental inputs are received at Hiand H3, and outputs from the system aregenerated (respiration) by all four holons.Table 1 presents the model in tabular(matrix) form.

To account for all possible holon inter-actions within such a model, a property

TABLE 1

Steady state marine coprography model H, as shown in figure 1.

''Entries denote carbon flows in gC m 2 y 1. The state variables for Hi, . . . , H\ are xi, . . . , x4,respectively; zi0 is input from the system's input environmentH] to holons H\ in rows i (i = l, . . . , 4);yoj is output to the output environment Hn from holons Hs in columns j (j = l, . . . , 4). z and y are

input and output vectors, and T is the throughput vector. Correspondences with figure 1 are obvious.

from


of mathematical graphs, transitive closure(Ore 1962), is required. This property isillustrated for the marine coprophagymodel by the set of matrices shown intable 2. Let B = (bij) be a binary

TABLE 2

Boolean matrices for the marinecoprophagy model.

Referring to figure 2, let: (1) Hi andrepresent subsystem level components

Row and column headings are state vari-ables xi, . . . , X4 for holons Hi, . . . , HA. Orien-tation is such that column elements propagatecausality to row elements.

Boolean adjacency matrix denoting di-rect causal coupling (paths of length one)from Hj to Hiy i, j = l, . . . , 4. Per-forming matrix multiplication, B2 entriesidentify indirect couplings via paths oflength two, B3 via paths of length three,and in general Bk via paths of length k.The table 2 matrices B, B2 and B3 maybe readily verified by reference to figure 1.

oo

The matrix 2 Bk denotes all causalk = l

paths of all lengths in the system, includ-ing diverging, converging and feedbackpaths. This is the transitive closureproperty, meaning that all causality prop-agated within the system network is ac-counted for. B* is a transitive closurematrix. This matrix for the marinecoprophagy model is the last of the setthat appears in table 2.

Leontief (1936) developed a methodfor steady state analysis of economicsystems that requires the transitive clos-ure property. The procedure, as modi-fied and extended by Finn (1976), in ef-fect defines within system input and out-put environments of each componentlevel holon. The more complicated non-steady state case is discussed in Pattenet al (1976).

(a)

(b)

FIGURE 2. Derivation of transitive closureinput and output matrices, (I-Q1)"1 and(I-Q11)"1, respectively, (a) Creaon case; (b)genon case.

of an n component system H when i,j = 1, . . . , n; (2) Hj denote system inputenvironment H' when j = 0; and (3) Hibe system output environment H" wheni = 0. Input from W to Hj (j = 1, . . . , n)is ZJO, and output to H]] from Hi is yOi.In figure 2a, if output yoi is received orperceived from Hi by some observer, thenthe input environment Hi] required toproduce yOi is of interest. Reciprocally,in figure 2b the environment Hj" of in-fluence generated in response to Zj0 is theconcern.

In deriving these environments it isconvenient to introduce two sets of iden-tity constraints.

(1) Interaction constraints: Zij = yij= F i i , i, j = 0, . . . , n .

(2) Steady state constraints: Zi = yi= Ti, i = l , . . . ,n

The first identities allow a direct causalflux Fij from Hj to Hi to be recognizedwithout distinguishing whether it is aninput Zjj to Hi from Hj (fig. 2a) or anoutput yij from Hj to Hi (fig. 2b). Thesecond constraints make it possible torecognize the total throughput Ti of Hiwithout considering whether it corre-sponds to total input (fig. 2a) or total out-put (fig. 2b) from the holon in question.Intrasystem environments of componentlevel holons may now be derived.


CREAON CASE

In figure 2a, let total output yj fromHi be

n(3) yj= 2 yij, j = l, . . . , n,

i = 0where i = 0 denotes output to H". Thislatter output yoj to the sysem's environ-ment can be isolated:

n(4) y j = 2 yii+yoj, j = l, . . . , n ;

i = lit is illustrated as yOi for Hi in figure 2a.Applying constraints (1) and (2), thelast expression (4) can be rewritten

(5)

The direct cause Fij from Hj to Hi can beexpressed as a fraction of the throughput.Ti of Hi-.

(6) F i j = q' i jT i , i, j = l, . . . ,n ,which, substituted into (5), gives

(7) Tj= 2 q'uTi+yoi, j = l, . . . .n.

In matrix notation this becomes

(8) T = TQ' + ,y

where T is a 2n-dimensional vector of the

n holon throughputs Tj, y is a 2n-dimen-sional vector of holon outputs to W\ andQ' is a 2n x 2n matrix of fractional directcauses q'ij from H} to Hi per unit ofthroughput Ti [eq. (4)]. The outputvector y and throughput vector T are in-

cated in table 1 for the marine coprophagymodel. The Q1 matrix for this model isshown in table 3a. Correspondence ofthe intrasystem submatrix with theBoolean matrix B in table 2 should benoted. Equation (6) can be solved for T:

(9) ( ' )

where(10) (I-Q I)ir1 = 0ii/yoi,iJ = l , - - - , n .

Here, </>ij represents the total causal flux(direct, Fij, plus indirect) from Hj to Hioverall possible pathways of propagation

TABLE 3

(-4) Fractional input -matrix Q] and (B) fractional output matrix Qv for the marine coprophagy model.

(A)

\Hi \

X i

x2x3x4

Z i o

Z20

Z30

Z40

(B)\ Hi

\Hi \

Xi

x2X3

X4in

y o iyo2yo3yo4

X i

01.000

0000

X l

00.176

00

0.824000

x2

00

0.0470

0000

x2

00

0.7000

00.300

00

x3

000

1.0

0000

X3

000

0.246

00

0.7540

from

x4

00

0.1630

0000

from

x4

00

0.6640

000

0.336

Zio

1.0000

0000

y o i

0000

0000

Z 2 0

0000

0000

yo2

0000

0000

Z30

00

0.7890

0000

yo3

0000

0000

Z40

0000

0000

yo4

0000

0000


through the interconnection network ofH, and (I-QOij"1 represents the amount ofthis flux normalized to one unit of out-put yOi observed from Hi (fig. 2a). Thus,the matrix (I-Q')~x must be a transitiveclosure matrix, and conditions to guaran-tee this are to be established. The inputenvironment defining (I-Q1)""1 matrix forthe marine coprophagy model is depictedin table 4a.

Just as entries in Q1 represent directcausal links of length 1, (Q1)2 denotescausality propagated indirectly overpaths of length 2, (Q1)3 over length 3paths, and in general (Q')k over paths oflength k. From the identity

(11) ( I+Q+Q 2 + . . . ) ( I -Q) = I,it follows that

£(12) lim S (Q'^HI-Q1)-1

- ^ c o k = 0if the limit exists. For the series to con-verge, (Q')k—>0 as k—»oo; that is, allcausal paths of all lengths must be ac-counted for. If this (transitive closure)

occurs, the convergence is to an inversematrix of the form (I-Q1)""1- Suchmatrices are therefore transitive closurematrices, provided the limit exists.

Existence conditions are well known inlinear algebra (e.g., Faddeev and Fad-deeva 1963). Ortega (1972), cited byHannon (1973), gives the following con-vergence theorem. Block diagonalize Q',

[x 1 0 . . . 0

(13) Q' =0 Q-. 0

.(0 0 . . . Q, ,forming m irreducible block diagonalsubmatrices such that det Qi' • det Q21 •. . . • det Qm' = det Q1. In each blocksubmatrix sum the state variable entriesin each state variable row. (I-Q')-1

exists if and only if for each block sub-matrix the sum of state variables in eachrow is strictly <1 for at least one statevariable row. The significance is thatat least one component level holon in

TABLE 4

(A) Transitive closure input environment matrix (I-Q])~1 and (B) output environment matrix (/-()")"for the marine coprophagy model.

(A)

Hi \

X i

x2x3x4

Z i o

Z20

Z30

Z40

(B)\ Hi

\Hi \

Xi

x2x3x4

yoiV02

yo3yO4

Xi

1.01.00.0570.057

0000

Xi

1.00.1770.1480.036

0.8240.0530.1110.012

x2

01.00.0570.057

0000

x2

01.00.8370.206

00.3000.6310.069

x3

00

1.1951.195

0000

x3

00

1.1950.294

00

0.9010.099

from

x4

00

0.1951.195

0000

from

X4

00

0.7941.195

00

0.5980.402

zio

1.01.00.0570.057

1.0000

yo i

0000

1.0000

z20

0000

0000

yo2

0000

01.000

Z30

00

0.9430.943

00

1.00

y03

0000

00

1.00

Z40

0000

0000

yo4

0000

000

1.0


TABLE 5Block diagonal forms of (A) creaon matrix Q' {table 3a) and (B) genon matrix Qv (table 3b)

for the marine coprophagy model.

(A)\ Hi

\

Hi \

x,x2

Z i o

toX; j

x4Z2oZ30

Z40

(B)

\Hi \

yo ix2

y 0 2

to x3

y 4

X4

yosXl

statevariablescolumn 2

zio

1.0*00

00000

X l

0.824*0.176

000

000

0.176

z2o

000

00000

y o i

00

000

000

—

X l

01.00

00000

x2

00

0.300*0.700

0

000

0.700

from

Z30

000

0.789*0000

from.

x4

00

00.6640.336*

000

0.664

Z40

000

00000

yo2

00

000

000

—

x2

000

0.0470000

X3

00

000

0.2460.754*

0

0.246

x4

000

0.1630000

yos

00

000

000

—

X3

000

01.0000

yo4

00

000

000

—

statevariables

row 2

01.0

0.21.0———

*See text below.

each submatrix must have input contactwith the system's input environment H,and that this must be true for all of them subsystems formed by the matrix di-agonalization procedure. Thus, to ac-count for all causal propagation within asystem H, it is necessary to refer to anenvironment H outside of H. This isProposition 2, for the creaon case.

Block diagonalization of Q1 for themarine coprophagy model is illustratedin table 5a. Row sums appear in theright hand column. For both Qi' andQ2' the sum of state variable rows is < 1for at least one such row, namely the rowfor xi in Qi' due to input zw to Hi (in-dicated by an asterisk), and the row forX3 in Q2' due to input z30 to Hz (asterisk).Existence of the transitive closure matrix(I-Q1)"1 for this model is thus established,and the matrix in fact is illustrated intable 4a. The input environments Hi},

. . . , Hi] that it defines will be clarifiedlater.

GENON CASE

A parallel development is required toestablish Proposition 2 with respect tooutput environment. In figure 2b, letthe total input z\ to H.x be

n(14) zi= 2 zij+Zio, i= l , . . . , n,

i = 1

where z i0 (fig. 2b) is input from the sys-tem level environment W. Applyingequations (1) and (2) as before gives

n(15) T i= 2 Fij+Zio, i = l, . . . ,n .

3 = 1Fij can be expressed as a fraction of thethroughput Tj of H}:

(16) Fij^q^ijTj, i, j = l, . . . , n ,which, substituted into (15), results in


(17)

In matrix notation this becomes(18) T = Q"T+z,

where T is a 2n-dimensional throughputvector, z is a 2n-dimensional vector rep-resenting inputs from H\ and Q" is a2n x 2n matrix of fractional direct effects)qij" from Hj to Hi per unit of Tj [eq.116c].Input z and throughput T vectors for the

marine coprophagy model are indicatedin table 1. Table 3b shows the Q" ma-trix. Solving eq. (18) for T:

where(20) (I-Q")ir^ = 0ii/Zio,i,j = l , . : . , n .

0ij is the total (direct, Fij, plus indirect)effect of H} on Hi transmitted over allpossible paths interconnecting the com-ponents of H. (I-Q") ij""1 is the same totaleffect normalized to a unit of input zj0 toHj (fig. 2b). Therefore, (I-Q")-1 re-quires the transitive closure property,for which conditions must be established.This output environment denning ma-trix for the marine coprophagy modelappears in table 4b.

As before, Q" denotes direct effectsand (Q")k indirect effects over paths oflength k. From identity (11), series con-vergence is to an inverse matrix,

£(21) lim 2 (Q")k=(I-Q")r1

^ c o k = 0if the limit exists. Again, diagonalizeQ" into m irreducible block sub matricessatisfying det Qi" • det Q2" . . . • det Qm".In each block submatrix sum the statevariable entries in each state variablecolumn. (I-Q")-1 exists if and only iffor each submatrix the sum of state vari-ables in each column is strictly <1 forat least one state variable column. Thatis, at least one holon in each subsystemrepresented by a block diagonal matrixmust have output contact with the out-put environment Hn of H; no subsystemso defined may lack such contact. Hence,to account for all propagation of ef-

fects within a system H it is necessary toreference, as output, an environmentalsystem Hn external to H. This is Prop-osition 2 expressed for the genon case.

Block diagonalization of Q" for themarine coprophagy model is shown intable 5b. Column sums appear in thebottom row. For submatrix Qx" the sumof the only state variable column, xi, is< 1 due to out put yOi form Hi (shown by*). In Q2" both state variable columnsums are < 1 because H2 and H4 both gen-erate output to Hn (asterisks). And inQ3", column x3 sums to < 1 because of out-put y03 from H3 (asterisk). The existencecondition for (I-Q")-1 is met for thismodel, and the matrix is shown in table4b. The output environments H\\ . . . ,Hi] defined by this matrix will be demon-strated in the next section.

The second proposition has been estab-lished. The internal interactive struc-ture of systems cannot be fully specified,with all causal pathways of all lengthsaccounted for, without reference to anexogenous input or output environment,or both. The systems must be opensystems. The causal pattern withinclosed systems cannot be completelyspecified, from which it may be con-cluded that it is a function of environ-ments to validate the internal nature oftheir defining systems.

As Patten et al (1976) indicate, Propo-sition 2 can also be realized from Markovchain theory. Its ultimate generality,however, is probably conferred by thefact that it may be a manifestation ofGodel's famous theorem (e.g., Nagel andNewman 1956) on incompleteness oflogical systems. Godel in what is con-sidered one of the mathematical land-marks of this century, showed that theconsistency of any deductive system can-not be established without reference tosome external system of logic whose ownconsistency is in question without refer-ence to a further external system, etc.If logical systems have logical "environ-ments" which must be consulted todemonstrate internal consistency of theformer, then it should be no surprise thatnature as comprehended by the samemind that created logic should possessthe same characteristic inherent in theobject/environment relationship.


Propositions 1 and 2 together signifythat the object (organism)/environmentpair is an inseparable, mutually definingunit. In the next section, a system isformulated as a composition of such sub-system level units.

THIRD PROPOSITION

Proposition 3: A system can be con-structed as a set union of mutually dis-joint and exhaustive object/ environmentelements {environs). The within systemobject/environment units of Propositions 1and 2 form a partition at the system level oforganization.

This final proposition can be illustratedadvantageously with the marine co—prophagy model. First, the formal state-ment. Let H^ i = l , . . . , n, be a sub-system level component of an n-com-ponent system H, with input environ-ment W and output environment Hn atthe supersystem level. The within sys-tem input environment of Hi is Hi\ andthe corresponding output environment isHin. The creaon/input environment andgenon/output environment units havebeen well enough established by Propo-sitions 1 and 2 that they can be regardedas entities in their own right. They willbe termed input and output environs(within system object/environmentunits), Ei] and £;" respectively, i = l ,. . . , n. This is consistent with normalusage in which the word environ refersot nearby surroundings. Here, nearbymeans within the boundary of the de-fined system. Proposition 3 can be for-mulated in terms of these units: inputenvirons do not overlap,

(22) £ iV\Ej ' = <k i, j = l, • • • , n,4> the empty set; output environs also arenonintersecting,

(23) £ i " n £ j " = <£,_i, j = l, . . . , n ;and system H is a union of input or out-put environs,

n n(24) H= VJ Ei}= \J Ei".

i = l i = lThe sense of these statements will now beclarified.

Table 4a shows the (I-Q1)"1 transitiveclosure matrix for the marine coprophagymodel. This matrix defines the inputenvirons Ei of this model normalized to

one unit of output yoi from each com-ponent holon Hi ( i = l , . . . , 4). Thesenormalized input environs are depictedin figure 3. Each environ is relative to aunit output (heavy arrows) from thecomponent holons. Numbers within theholon symbols denote throughputs re-quired to generate the unit outputs;numbers associated with arrows representpropagated causes that sum to thethroughputs. Correspondences betweenfigure 3 and table 4a are obvious. Toexpress the normalized environs as car-bon flows (gC m~2 y"1) numbers in thefigure and table must be multiplied bythe corresponding output flux as given infigure 1. The normalized versions (fig.3) will be used for interpretation.

Consider £4 ' in figure 3. Observation(measurement) of one unit of carbon out-put from H^ specifies the indicated causalnetwork as input environment H±. Cau-sation is traced back through the networkto its origins at the system boundary.Most of the output from Hi derives frominput to Hs (94.3%), and only a smallamount (5.7%) originates with Hi input.The relations shown for the remainingthree input environs are self evident. Ifthese four normalized environs E\\ . . . ,£4

? are scaled to actual carbon flows andsummed, the original figure 1 system isreconstructed. That is,

4(25) H = S E{\

i = lThus, the input environs of figure 3 arenonintersecting [eq. (22)] and also exh-haustive [eq. (24)], establishing Proposi-tion 3 for the creaon case.

Table 4b presents the (I-Q")"1 matrixfor the marine coprophagy model. Thismatrix defines output environs .Ej" nor-malized to one unit of input zj0 to eachcomponent holon Hj (j = 1, . . . , 4). Thesenormalized environs are depicted in figure4, each in relation to a unit input (heavyarrows) to the member holons. Numberswithin holon symbols denote through-puts generated by the unit inputs, andnumbers associated with arrows indicatepropagated effects which sum to thethroughputs. To express the environsin terms of absolute carbon flows, figure4 values should be multiplied by the as-sociated inputs in gC nr~2 y~l as given in


1.0 '

.057

• 057

• 195

• 9^3FIGURE 3. Normalized input environs £ / , . . . , £4' which partition the steady state marinecoprophagy model.

figure 1. The normalized environs (fig.4) will again be interpreted.

In the upper diagram of figure 4 de-picting £1", 82.4% of Hi input exits thesystem at Hh 5.3% at H2, 11.1% at H3and 1.2% at H4. The within systempropagated effects leading to these out-puts are shown. The other environs pro-vide similar information about the fateof other inputs. If these environs aredimensionalized to actual carbon flows(gC m~2 y"1) and the results summed, theoriginal figure 1 system is again recom-posed. That is,

4(26) # = 2 £ / ,

indicating that the output environs E^,. . . , £4" are mutually exclusive [eq. (23)]and exhaustive [eq. (24)]. Proposition 3is therefore established for the genon case.

Thus, for general systems, but espe-cially for ecosystems which motivate thistheory, within system object (organism)/environment units (environs) form setpartitions at the system hierarchicallevel. Two such partitions are possible,one by input environs and the other byoutput environs. Both are distinct anddifferent as the input and output en-virons defined by a given holon are dis-tinct and different (EiM-Ei", i= l , . . . ,n). von Uexkull (1926) apparently ap-preciated the disjoint property of such


partitions when he wrote, "For each in-dividual animal, . . . its function-circlesconstitute a world by themselves, withinwhich it leads its existence in completeisolation."

DISCUSSIONEcology was stated previously in this

paper to take a fundamentally physio-logical view of environment. This isconsistent with ordinary usage in whichliving or nonliving systems are influencedby external surroundings. The physio-logical concept is manifested in Masonand Langenheim's (1957) theory, whichlimits environment to direct causes only.This is the normal ecological view ofenvironment, although other viewpoints

(e.g., Haskell 1940) have been offered.The systems concept outlined abovediffers from the normal one in fourparticular ways: two environments arerecognized instead of one; indirect cau-sality is included; the object (organism)/environment complex is unitary; andthe units (environs) partition reality.TWO ENVIRONMENTS

The causal holon H is a general systemsobject that originates not one, but two,environments, H (input) and H" (out-put). If H is a system level object, Hand Hn are supersystem concepts andcannot be further described. If H is asubsystem, then its within systemenvironments can be specified to the

1.0

.069

.012

.099

.402

FIGURE 4. Normalized output environs Ei", . . . , £4" which partition the steady state marinecoprophagy model.


system boundary as input and outputenvirons, £ ' and E", respectively. Theenviron is a new class of object in systemtheory. What it may contribute to theunderstanding of ecological or generalsystems remains to be seen. For example,where a holon is a biological object,inheritance and evolution of its environsmay be reasonable to consider as outwardprojections of known genetic mechanisms.The necessity for objects to interactconsistently within environs providesconstraints that almost certainly guar-antee coevolution to be an ecosystemlevel phenomenon (Patten et at 1976;Patten 1977). Prospects for an organ-ismic representation of environment arequite real in this theory.

The normal one-environment conceptincludes only input environment, vonUexkull (1926) provided a precedent foroutput environment in the notion offunction circles that fail of closure (out-put affecting input) due to complexityof the external world. Propagated effectsbecome lost in the general flux of causa-tion before they can return as identifiableinputs to the original generating organ-ism. By explicitly recognizing two en-vironments, an analytical potential isgained that is absent in a one-sided the-ory. Creaon and genon partitions (eg.,figs. 3 and 4) are never the same, and pat-terns of how they differ are foreseeablesystem properties of interest. For ex-ample, Patten (1978) has analyzed con-trol relationships in ecosystem models bycomparing input and output environs ofcomponent holons.

INDIRECT CAUSALITY

Mason and Langenheim (1957) wrotethat to include indirect factors in environ-ment is to confuse environment withhistory. In the two-environment ap-proach the future enters as a similarobjection. How should time be regardedin a concept of environment? Twoaspects of the question are dynamic andstatic.

Let H be a component of a systemthat exists with respect to a cause during[t\ t\ t}<t<t", teT. (Symbols [, ],( and ) mean >, <, > and <, respec-tively, in denoting time intervals.) t1 isthe time the cause initially enters the

system as input, / is present time, and tn

is the time at which a correspondingeffect is generated as system output.Dynamically, H defines its input andoutput environments Hl and H" instan-taneously at time t through direct inter-active coupling to other holons of thesystem. In input environs E\ indirectcausality, which conditions the directcoupling events at /, has already occurredduring the past [t\ i\. Thus, an instan-taneous input environ defined at /encompasses a historical network ofcausation extending backward to thesystem boundary at /'. Similarly, in-direct effects in an instantaneous outputenviron E" are propagated from thedirect coupling events at time t duringthe future, [t, tn]. The instantaneousoutput environ contains the succession ofindirect causes and effects extendingforward to the system boundary at t}\Note that the system exists with respectto a cause introduced at t] only duringthe interval [/', t]l] required for it togenerate a corresponding effect at tn, andthis is true "vA tn e T. The role of holonH in the system relative to the samecause is similarly restricted to the sameinterval. Without a temporally finitemodel Mason and Langenheim's (1957)objection of infinite regress, and acounterpart infinite future progression ofthe two-environment theory, would bevalid. So long as a holon's memory ofthe past and horizon to the future arerelatively small, so that its systemappears relatively permanent comparedto itself, this permanent organizationshould be represented in its environs.Environment as a concept is not instan-taneous. It is natural history, a windowon the relatively near past and future,and to make it so, indirect causality mustbe included. Therefore, instantaneousinput environs E] defined at time tproperly span intervals [t\ t], and cor-responding output environs En spanintervals [t, tn].

The static case reflects this. Staticmodels depict, usually, steady statecharacteristics of systems over timespans that are long compared to the timescales of dynamic properties. For exam-ple, the marine coprophagy model offigure 1 represents a persistent steady


state organization expressed as meanannual carbon storages and flows. Finertime resolution is not desired, and notime difference is implied between inputsand outputs. Each static environ (figs.3 and 4) represents average relationshipsinherent in the system organization yearafter year. Historical aspects are sup-pressed in such static abstractions. Thisobscures the fact that the commonsenseconcept of environment is actually asystems concept. It includes indirectcausation implicitly, because in its stasisit presumes relatively constant ecologicalorganization over relatively long timescales.

To illustrate, the immediate physicaland informational environment of myoffice here as I write is not the environ-ment of concern when I consider environ-mental management or protection. Thislocal direct environment is well managedby lights, windows and thermostats. Tocontinue to guarantee these devices andmy personal well-being, without whichthey would be meaningless, I must anddo consider phenomena at the far reachesof my environs that never will touch thisoffice directly. DDT, mercury, radio-activity or a thousand other hazards andother aspects may or may not everdirectly impinge on me, but they alreadyaffect me and my management of thisplace. This knowledge is implicit inmy working approach to environment,based on a static model in my mind ofboth direct and indirect factors. Man asa species (i.e., as an aggregate holondenning aggregate input and outputenvirons) takes account of indirect factorshabitually. Only recently, with the ad-vent of computers, has this systemsreflex begun to be implemented in non-static models. Indirect causality is anintegral part of environment, and in bothdynamic and static cases is correctlyincluded in the systems approach to theconcept.

HOLON/ENVIRONMENT UNITY

In Proposition 1 a holon defines apair of environments, and in Proposition2 these environments confer completenessupon the holon's internal organization.Input and output environments may beconsidered outward extensions of physi-

cal, chemical or biological characteristicsof the holon's inner organization, mechan-ism and law. The holon similarly maybe regarded as an inward projection ofthe properties of its environments, thecreaon a reflection of input environmentand the genon a reflection of outputenvironment. An unbroken continuumof causes and effects streams across theholon/environment boundary. Proposi-tions 1 and 2, with probable support fromGodel's theorem (Nagel and Newman1956), strongly portray the holon/envi-ronment complex as a unit.

The nature of the relationship betweena defining holon and other holons withwhich it interacts only indirectly con-tributes to a unified concept. Considerthe input environ £4' illustrated at thebottom of figure 3. H± takes account ofHs by direct interactive coupling, butcan never have a similar relation toH2 or Hi, with which it is only indirectlyconnected in the model. In the dynamiccase, Hi or H2 may both have gone outof existence by the time i/4 receivescarbon that they processed. What thencan be said of the relation, if any, ofHi to Hi and H2? Similarly, for thegenon case refer to output environ E^depicted at the top of figure 4. Couplingof Hi to Hi is direct but Hx is onlyindirectly related to H3 and H4. Dynam-ically, Hi may no longer exist by thetime its generated effects are propagatedto H3 and H±. What is the environ-mental relation, if any, of Hi to H3 and#4? The denning holon of an input oroutput environ is influenced by or influ-ences all member holons in the environ.The defining holon becomes, in effect, asynthesis of its relations to all direct andindirect phenomena which condition it(creaon) or which it conditions (genon).Thus, a holon and its environments areproperly considered as units, as expressedin the environ concept.

ENVIRON PARTITIONS

A special feature of the present theorynot shared with conventional conceptsof environment is system partition ac-cording to Proposition 3. As indicatedbefore, von Uexkull (1926) held thatorganisms live isolated within the worldof their own function circles. The same


idea appears here in the form of holonsrelating only to things in their ownenvirons. The sense in both cases isnot that entities in nature do not interact,but that the transactions and communi-cations (energy-matter and informationexchanges, respectively) by which theydo so are unique. If environs of differentholons are disjoint, they also may bedissimilar even if the same physicalphenomena are represented. A real en-tity depicted in an environ of Hi mayhave a different character and significancewhen represented in an environ of Hj.An environ is then an abstraction formedby its defining holon—a representationor model of that holon's separate reality.Presumably, it is refined and improvedin some evolutionary synthesis appro-priate to the holon's physical, or biologi-cal nature and level of organization.

How these disjoint models combine toexhaust the concrete reality which isnature is for philosophers, and notecologists, to understand.

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