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 between things, including organisms, and their environments in terms of four system theoretical abstract 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 creaon and a genon. An environ is a creaon and its corresponding input environment, or a genon and its related output environment. The theory is presented in terms of three propositions that: (1) recognize two distinct environments (input and output) asso- ciated with things, (2) establish things and their environments as units (environs) to be taken together, and (3) partition systems into input and output environs associated with intrasystem creaons and genons, respectively. OHIO J. SCI. 78(4): 206, 1978
Ecology is the biological science of en- organism, loose usage frequently extends vironment. It considers environment as the concept from individuals to groups a derivative of physiology in the sense {our environment), or suggests something that environment contains resources to absolute (the environment). The dic- be mobilized by organisms, and condi- tionary defines environment variously as: tions of life under which this mobilization "the surrounding conditions, influences or must occur. The resource in least sup- forces that influence or modify; the whole ply at any given time is rate limiting (law complex of climatic, edaphic and biotic of the minimum), as is the factor, such as factors that act upon an organism or an temperature, in greatest extreme (law of ecological community and ultimately de- tolerance). Thus, the organism is seen termine its form and survival; the aggre- by ecology to inhabit a physiological life gate of social and cultural conditions that space bounded by conservative and non- influence the life of an individual or com- conservative elements of its environment munity," (Merriam-Webster 1971). The —resources and factors, respectively. significant features of environment in The nature and composition of this life ordinary usage are that some defined sub- space varies according to the character ject (individual or group) is immersed in of the larger system of which the organ- or surrounded by it, and influenced by ism is seen as a part. Population aspects it through a causal relationship. This of environment encompass the intra- causality, as developed below, is the specific reproductive, genetic, demo- basis for the present attempt to express graphic and social worlds of the organ- environment in terms of system theory, ism. A community aspect refers to in- which is the purpose of this paper. terspecific biotic associations. The eco- system aspect takes into account all fea- SYSTEMS DEFINITIONS tures of the organism's biotic and abiotic Systems ecology is a branch of ecology interactions. that applies systems thinking and meth- ods to ecological problems. Several def- Although the strict ecological idea of initions of basic system concepts are use- environment is based on the individual ful in prospect of a systems approach to defining environment. A system is a 1Manuscript received January 19, 1977 and in revised form June 15, 1978 (#77-6). partially interconnected (interacting or 2University of Georgia Contributions in Sys- causally joined) set of components. In- tems Ecology, No. 41. teractions may be mediated by energy- 206 Ohio J. Sci. EIA-SYSTEMS APPROACH 207 matter through transactions, or by infor- an organism. Non-environment consists mation through communications. Trans- of all phenomena (indirect, historical and actions and communications correspond, organism-caused) which never enter into respectively, to transfers of conservative a direct environmental relation with the resources and nonconservative factors in organism. the physiological account of environment Mason and Langenheim (1957) as- described above. serted, "the environment of any organ- In a hierarchical model of nature, any ism is the class ... of those phenomena given system can usefully be abstracted that enter a reaction system of the organ- as three discrete levels separated out of a ism or otherwise directly impinge upon it hierarchical continuum: system, subsys- to affect its mode of life at any time tem and supersystem. Subsystems are throughout its life cycle as ordered by the components of the systems. Supersys- demands of the ontogeny of the organism tems are composed of systems. Koestler's or as ordered by any other condition . . . (1967) term "holon" for a hierarchical that alters its environmental demands." system can be used to refer to any of Only direct factors were considered part these three levels of organization, accord- of environment. "[Indirect and histori- ing to the frame of reference. cal] factors both function to condition a A system is closed if it does not interact phenomenon ... to which an organism with another system, and open if it re- then reacts. Important as this is to the ceives causes from or generates effects to ecosystem the only [organism] reaction another system. A system boundary ... is to an already conditioned phe- provides the interface with other systems nomenon. The state of the phenomenon and is defined by specifying its component prior to its conditioning is outside the set. Input is any movement of energy- scope of operational . . . and . . . po- matter or information from supersystem tential environment. . . . This may seem to system, and output is any similar to rest upon trivial distinctions, but we movement across the system boundary in are convinced that this is the precise the opposite direction. boundary between clarity and confusion in the problems of the environment." ECOLOGICAL CONCEPTS OF Thus, chains and networks of historical ENVIRONMENT causation, which condition direct factors, Environment as a concept has not been are excluded from Mason and Langen- treated very seriously in ecological litera- heim's (1957) concept of environment: ture and only a few explicit works are "... we must reject the implication that available. Mason and Langenheim (1957) . . . [causal] chains constitute a unitary defined environmental phenomena as those event playing a significant role in the en- that have or may have an operational re- vironmental relation even though the lation with any organism. The environ- steps are very important to the ecosys- mental relation of an organism is the sum tem. . . . There is also a philosophical of empirical relations between the en- reason for removing indirect factors from vironmental phenomena and any indi- the concept of environment. To intro- vidual organism. The operational en- duce indirect factors into causal relations vironment of an organism consists of those within the environment is to introduce an instantaneous environmental phenomena infinite regress into the system of expla- that actually enter a relation with an or- nation. Every cause has in turn itself ganism; the concept applies to specific in- a cause which becomes an indirect cause dividual organisms. Space and time of the most recent effect. The regress is frames of the operational environment toward the limbo of ultimate cause along are determined by the organism. The an infinitely reticulating path; for this life span of the organism corresponds to we have neither finite description nor the existence time of its operational en- finite explanation. . . . To include such vironment. Potential environment con- relations in environment is to confuse en- sists of the set of environmental phenom- vironment with its history." ena that may enter into an environmental A systems ecology concept of environ- relation at some point in the ontogeny of ment must take issue with the Mason and 208 BERNARD C. PATTEN Vol. 78 Langenehim theory. The whole thrust causes as well, so long as their eventual of a systems understanding of nature is influences can be propagated to a sub- to reconstruct the main patterns of causa- ject, such as an organism, during its tion in models. Within the confines of a existence interval. Systems ecology mod- finite model forming a whole from inter- els that represent complex intrasystem connected parts, an expanded concept of webs of direct and indirect causation environment of the parts is possible, make it possible to implement such an which includes both direct and indirect expanded concept of environment. A factors. The intrasystem causal net- formal approach to such implementation work is never an unknown infinite regress, is described below. but is explicit to the model boundary which constitutes the limit of finite de- HOLONS scription and explanation which were General systems theory defines a sys- lacking in Mason and Langenheim's time. tem to be a partially interconnected set of While the conditioning of direct causes by objects, then proceeds to describe the ob- indirect effects may be temporally ante- jects and various aspects of their inter- cedent, ecosystems and their models are active coupling. Formal details differ persistent or recurrent organizations so with the specific theory, but most general that historical patterns of causation are systems objects have in common that in relevant, with perhaps small corrections some sense they perform a double map- for evolution, to present and future pat- ping of time into state, then state into terns as well. output. Examples are "finite state ma- Such a systems view of environment chines" of Gill (1962), "abstract objects" has precedent in an ecological work by of Zadeh and Desoer (1963), Wymore's Haskell (1940), who focused on events in (1967) "formal systems," the "general the universe that may eventually in- systems" of Klir (1969) given according fluence an organism during its lifetime. to five definitions, "T-processors" of Their influence is limited by how fast Windeknecht (1971), and "general time causality can be propagated, no faster systems" of Mesarovic and Takahara ultimately than the speed of light. Thus, (1975). All such units may be made corresponding to each instant in the life causal, and can be generalized under the of an organism is a light cone. (Haskell nonspecific hierarchical object, holon 1940, fig. 1) bounding the spatiotemporal (Koestler 1967). An extensive theory of extent of possible causes. The cones di- the causal holon as the basis for a systems minish in time as the universe that can concept of environment has been pre- possibly affect the ageing organism sented elsewhere (Patten et al 1976), contracts: "The cones prepresent . . . based on Zadeh's model (Zadeh and a steadily shrinking region . . . within Desoer 1963, Zadeh 1969). This theory which the fastest moving process—light, is outlined below, with notation modified traveling at about 300,000 km a second— according to Mesarovic and Takahara can start, at any point-instant . . . dur- (1975). ing the organism's existence, and effect To model a causal link between two (sic) it before its end. . . . This region is entities requires some kind of process or equal to a geometric hyperbody, denned object whose action converts cause to below as 'habitat', and, constitutes part effect. Such an object, H, is a relation of 'environment.' Habitat is the "im- on attributes, VeA, that are time func- mediate environment" (Haskell 1940, p. tions in a time domain, T. For each aeA 7), taken as Weaver and Clements (1929) a is a behavior V^eT, a(t) is the value of a denned it: "Every part of the environ- at time teT, a,* is the segment of a prior ment that exerts directly or otherwise to /, and at is the behavior segment of a [i.e., indirectly] a specific influence upon beginning at and following /. This ob- the life of the plant is a factor of the ject definition provides latitude in select- habitat." ing the set A of behavioral attributes. Thus, Haskell's concept of environment The holon becomes oriented when its includes not only the direct causes of set of attributes is partitioned into inputs, Mason and Langenheim, but indirect Z, and outputs, Y. The relation H on Ohio J. Sci. EIA-SYSTEMS APPROACH 209 A is then expressed as a set of input- the supersystem/system interface, then output time segments: (z,y)eH, zeZ and propagated through the interactive net- yeY. The oriented holon associates re- work connecting subsystem holons, and sponse (output) time sequences with finally dissipated as output effects gene- stimulus (input) histories. In develop- rated to the environment across the sys- ing the holon as a causal object, given tem boundary. The key to recognizing output sequences must be uniquely as- the main features of the theory and its sociated with given input segments. This implications for an improved concept of property is incorporated in the notion of environment lay in focusing on intra- a functional holon, where H is construed system environments associated with sub- as a map (function) of inputs, z, into out- system level holons. These environ- puts, y. Such an object is said to be ments may be explicitly identified and determinate, i.e., a time series of inputs measured as a causal reticulum within a from its environment uniquely deter- system model, with consequences that mines a corresponding time series of emerge as three main points of the theory. outputs. These points are developed as specific Dynamic behavior of a determinate ob- propositions in the next three sections. ject occurs in response to the object's en- vironment's behavior, which is received FIRST PROPOSITION as input. This is modeled by introduc- Proposition 1: Every object H defines ing a third set, X, of object variables, two environments: an input environment states. Heuristically, inputs zeZ serve to H', and an output enviroment H". The map time teT into states xeX, and the prerogative of environment definition is states take inputs zeZ into outputs yeY. that of the object. States are generated by a state transition The causal model of subject/environ- function: ment interaction leads to not one, but 0:ZXX->X, two equally plausible and useful concepts and outputs are generated by a response of environment. The first is input en- function: vironment H\ defined by holon H in the act of receiving energy-matter or perceiv- ing information. Behavioral attributes The only other requirement for a de- of the real world that do not impact H as terminate holon to be causal is that it not input during its existence interval cannot respond at time / to inputs received after influence the state of the object. They t. That is, the object cannot anticipate go unrecorded by H and consequently its future environment; it is nonanticipa- are not part of its environment. So tory. If a determinate object were to basic is this environment defining func- generate more than one output sequence tion that this aspect of the holon is given corresponding to a given input sequence, (Patten et al 1976) a special name, creaon, the only way it could do this (since it is to signify an implicit act of environment determinate) would be based on informa- creation. Mason and Langenheim (1957) tion about the future. This possibility restrict the concept of (input) environ- is precluded for the causal object. ment to phenomena that "directly im- The full theory (Patten et at 1976) pinge" upon the organism, whereas Has- should be consulted for details. The kell (1940) includes, in addition, the indi- causal holon may serve at either the rect causes from which direct ones are gen- system or subsystem level. The focus erated. The latter, and the present ap- of the original work was on intrasystem proach, are more consistent with a sys- propagation of causes between subsystem tems view, and in the context of finite level holons. As a result, consequences ecosystem models do not produce the in- of the theory for a system concept of finite causal regress to which Mason and environment were not as clearly per- Langenheim objected. That is, when H ceived as they are now. Environment is is a subsystem level component, W is normally a supersystem level concept. traceable only to the model boundary, be- Causation was considered to be intro- coming beyond this merely undiffer- duced as inputs from an environment at entiated input to the system level. The 210 BERNARD C. PATTEN Vol. 78 within system portion of H} is thus ex- plants are included. For each individual plicit in the concept of input environment. animal, however, its function-circles con- Reciprocally, the second concept of en- stitute a world by themselves, within vironment is that of an output environ- which it leads its existence in complete ment Hn. This begins as a set of potential isolation." environments embodied in the states of Inner world: "The sum of the stimuli H. These states are converted to out- affecting an animal forms a world in itself. puts through interaction of H with other The stimuli, considered in connection objects (creaons). This is, to produce an with the function circle as a whole, form actual output environment from potential certain indications which enable the ani- environment implicit in the state struc- mal to guide its movements. . . . The ture of H requires holon production of animal itself, by the very fact of exercis- potential attributes, then sequential cre- ing such direction, creates a world for it- aon selections to achieve realization of self, which I shall call the inner world." these potentials. Output environment n Surrounding world: "World-of-action H is the resultant causality propagated and world-as-sensed together make a from H as a network of direct and indirect comprehensive whole, which I call the effects. This environment generating surrounding world." property of holons is equally basic to the World-as-sensed and world-of-action creaon function, and to distinguish it the correspond to input and output environ- name genon is given (Patten et al 1976). ments, respectively, and the latter is thus As in the creaon case, an infinite progres- clearly distinguished. Moreover, von sion of effects from // is implied, but at Uexkull's view of the organism/environ- the component level in the context of ment relation is unitary: "The entire finite models, the progression terminates function circle formed from inner world at the system level boundary beyond and surrounding world . . . constitutes a which only undifferentiated output is whole which is built in conformity with recognized. The within system portion n plan, for each part belongs to the others, of H is thus explicit in the concept of and nothing is left over to chance . . . output environment. where there is a foot, there is also a path; Neither Haskell (1940) nor Mason and where there is a mouth, there is also food; Langenheim (1957) considered output where there is a weapon, there is also an environment as a proper component of enemy. ... If this circle is interrupted the general concept of environment. at any point whatsoever, the existence of However, an older physiological theory the animal is imperilled. . . . continuity provides explicit justification for the out- of the complete whole must never be lost put environment, von Eexkiill (1926) sight of." Output and input environ- presented a picture of environment as an ments are continuous through the func- organism surrounder in terms of the fol- tion circles of the organism, and that con- lowing set of concepts: tinuity erases, in theory, any distinction World-as-sensed: "Every animal is a between them. However, there is the subject, which, in virtue of the structure matter of practicality to be considered: peculiar to it, selects stimuli from the "All the [function] circles, however far general influences of the outer world, and they lie separated from one another in the to these it responds in a certain way." world-as-sensed, intersect in the steering World-of-action: "These responses, in apparatus of the inner world, and then their turn, consist of certain effects on separate from one another again in the the outer world, and these again influence world-of-action." World-as-sensed (in- the stimuli." put environment) and world-of-action Function-circle: "In this way there (output environment) are, for all practical arises a self-contained periodic cycle, purposes, separate by virtue of the enor- which we may call the function-circle of mity of reality compared to the identifi- the animal. The function circles . . . able sphere of influence of any single connect up ... in the most various organism (holon). ways, and together form the function- Thus, the first proposition. Every in- world of living organisms, within which teracting thing in nature defines two Ohio J. Sci. EIA-SYSTEMS APPROACH 211 separate and distinct environments, both structive. Causality is expressed as car- taken to include the network of causes bon flow (gC m~2 y^1) and system state is and effects as far as these are traceable in represented by carbon storages (gC m~2). any particular model in which the defin- ing object serves as a component. SECOND PROPOSITION Proposition 2: The internal cause prop- agating structure of systems cannot be com- pletely determined, i.e., all causal paths in the interactive network accounted for, with- out input or output reference to an external environment. The prerogative of realiza- tion of internal system structure is that of FIGURE 1. Steady state marine coprography environment. model (Cale and Ramsay 1970). Holon inputs and outputs represent carbon flows in gC This proposition, developed in detail in mT2 y~', and states represent carbon storages Patten et al (1976), can best be presented in gC m~2. here in terms of an example. Figure 1 Hi Callianassa major illustrates a simple steady state model of HiC. major feces marine coprophagy (Cale and Ramsay #3Benthic invertebrates 1970, description in Patten et al 1976, iJ^Benthic invertebrate feces Appendix). The model consists of four Carbon flow: x's and y's in gC m~2 y"1 holons in series, with a feedback loop z's in gC m~2 connecting H3 and H4. Hi is a mud crab, Callianassa major; H2 is the feces of this Environmental inputs are received at Hi animal; H3 includes all other benthic in- and H3, and outputs from the system are vertebrates of the marine community un- generated (respiration) by all four holons. der consideration; and H^ is defined as Table 1 presents the model in tabular the feces of this latter group of animals. (matrix) form. The model is simple in its interactive To account for all possible holon inter- structure, and for that reason, quite in- actions within such a model, a property
TABLE 1 Steady state marine coprography model H, as shown in figure 1.
from
2 1 ''Entries denote carbon flows in gC m y . The state variables for Hi, . . . , H\ are xi, . . . , x4, respectively; zi is input from the system's input environmentH] to holons H\ in rows i (i = l, . . . , 4); 0 n yoj is output to the output environment H 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. 212 BERNARD C. PATTEN Vol. 78 of mathematical graphs, transitive closure Referring to figure 2, let: (1) Hi and (Ore 1962), is required. This property is represent subsystem level components illustrated for the marine coprophagy model by the set of matrices shown in table 2. Let B = (bij) be a binary
TABLE 2 Boolean matrices for the marine coprophagy model. (a)
(b) FIGURE 2. Derivation of transitive closure Row and column headings are state vari- input and output matrices, (I-Q1)"1 and ables xi, . . . , X4 for holons Hi, . . . , HA. Orien- (I-Q11)"1, respectively, (a) Creaon case; (b) tation is such that column elements propagate genon case. causality to row elements. of an n component system H when i, Boolean adjacency matrix denoting di- j = 1, . . . , n; (2) Hj denote system input rect causal coupling (paths of length one) environment H' when j = 0; and (3) Hi from Hj to Hiy i, j = l, . . . , 4. Per- 2 be system output environment H" when forming matrix multiplication, B entries i = 0. Input from W to Hj (j = 1, . . . , n) identify indirect couplings via paths of ]] 3 is ZJO, and output to H from Hi is yOi. length two, B via paths of length three, In figure 2a, if output yoi is received or and in general Bk via paths of length k. 2 3 perceived from Hi by some observer, then The table 2 matrices B, B and B may the input environment Hi] required to be readily verified by reference to figure 1. produce yOi is of interest. Reciprocally, oo in figure 2b the environment Hj" of in- k The matrix 2 B denotes all causal fluence generated in response to Zj0 is the k=l concern. paths of all lengths in the system, includ- In deriving these environments it is ing diverging, converging and feedback convenient to introduce two sets of iden- paths. This is the transitive closure tity constraints. property, meaning that all causality prop- (1) Interaction constraints: Zij = yij agated within the system network is ac- = Fii, i, j = 0, . . . ,n. counted for. B* is a transitive closure (2) Steady state constraints: Zi = yi matrix. This matrix for the marine = Ti, i=l, . . . ,n coprophagy model is the last of the set The first identities allow a direct causal that appears in table 2. flux Fij from Hj to Hi to be recognized Leontief (1936) developed a method without distinguishing whether it is an for steady state analysis of economic input Zjj to Hi from Hj (fig. 2a) or an systems that requires the transitive clos- output yij from Hj to Hi (fig. 2b). The ure property. The procedure, as modi- second constraints make it possible to fied and extended by Finn (1976), in ef- recognize the total throughput Ti of Hi fect defines within system input and out- without considering whether it corre- put environments of each component sponds to total input (fig. 2a) or total out- level holon. The more complicated non- put (fig. 2b) from the holon in question. steady state case is discussed in Patten Intrasystem environments of component et al (1976). level holons may now be derived. Ohio J. Sci. EIA-SYSTEMS APPROACH 213
CREAON CASE In matrix notation this becomes In figure 2a, let total output yj from (8) T = TQ' + ,y Hi be n where T is a 2n-dimensional vector of the (3) yj= 2 yij, j = l, . . . , n, n holon throughputs Tj, y is a 2n-dimen- i = 0 sional vector of holon outputs to W\ and where i = 0 denotes output to H". This Q' is a 2n x 2n matrix of fractional direct latter output yoj to the sysem's environ- causes q'ij from H} to Hi per unit of ment can be isolated: throughput Ti [eq. (4)]. The output n vector y and throughput vector T are in- (4) yj= 2 yii+yoj, j = l, . . . ,n; i=l cated in table 1 for the marine coprophagy 1 it is illustrated as yOi for Hi in figure 2a. model. The Q matrix for this model is Applying constraints (1) and (2), the shown in table 3a. Correspondence of last expression (4) can be rewritten the intrasystem submatrix with the Boolean matrix B in table 2 should be (5) noted. Equation (6) can be solved for T: (9) (') The direct cause Fij from Hj to Hi can be expressed as a fraction of the throughput. Ti of Hi-. (6) Fij = q'ijTi, i, j = l, . . . ,n, where which, substituted into (5), gives (10) (I-QI)ir1 = 0ii/yoi,iJ = l,---,n. Here, ij represents the total causal flux (7) Tj= 2 q'uTi+yoi, j = l, . . . .n. (direct, Fij, plus indirect) from Hj to Hi overall possible pathways of propagation
TABLE 3 (-4) Fractional input -matrix Q] and (B) fractional output matrix Qv for the marine coprophagy model.
(A) from
\ Xi x2 x3 x4 Zio Z20 Z30 Z40 Hi \
Xi 0 0 0 0 1.0 0 0 0 x2 1.0 0 0 0 0 0 0 0 x3 0 0.047 0 0.163 0 0 0.789 0 x4 0 0 1.0 0 0 0 0 0
Zio 0 0 0 0 0 0 0 0 Z20 0 0 0 0 0 0 0 0 Z30 0 0 0 0 0 0 0 0 Z40 0 0 0 0 0 0 0 0
(B) from \ Hi \ Xl x2 X3 x4 yoi yo2 yo3 yo4 Hi \
Xi 0 0 0 0 0 0 0 0 x2 0.176 0 0 0 0 0 0 0 X3 0 0.700 0 0.664 0 0 0 0 X4 0 0 0.246 0 0 0 0 0 in yoi 0.824 0 0 0 0 0 0 0 yo2 0 0.300 0 0 0 0 0 0 yo3 0 0 0.754 0 0 0 0 0 yo4 0 0 0 0.336 0 0 0 0 214 BERNARD C. PATTEN Vol. 78 through the interconnection network of occurs, the convergence is to an inverse H, and (I-QOij"1 represents the amount of matrix of the form (I-Q1)""1- Such this flux normalized to one unit of out- matrices are therefore transitive closure put yOi observed from Hi (fig. 2a). Thus, matrices, provided the limit exists. the matrix (I-Q')~x must be a transitive Existence conditions are well known in closure matrix, and conditions to guaran- linear algebra (e.g., Faddeev and Fad- tee this are to be established. The input deeva 1963). Ortega (1972), cited by environment defining (I-Q1)""1 matrix for Hannon (1973), gives the following con- the marine coprophagy model is depicted vergence theorem. Block diagonalize Q', in table 4a. [x1 0 ... 0 1 Just as entries in Q represent direct 0 Q-. 0 causal links of length 1, (Q1)2 denotes causality propagated indirectly over (13) Q' = paths of length 2, (Q1)3 over length 3 paths, and in general (Q')k over paths of .(0 0 . . . Q, , length k. From the identity forming m irreducible block diagonal 2 submatrices such that det Qi' • det Q21 • (11) (I+Q+Q +...)(I-Q) = I, 1 it follows that . . . • det Qm' = det Q . In each block submatrix sum the state variable entries £ -1 (12) lim S (Q'^HI-Q1)-1 in each state variable row. (I-Q') -^cok = 0 exists if and only if for each block sub- if the limit exists. For the series to con- matrix the sum of state variables in each verge, (Q')k—>0 as k—»oo; that is, all row is strictly <1 for at least one state causal paths of all lengths must be ac- variable row. The significance is that counted for. If this (transitive closure) at 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) from
Xi x2 x3 x4 zio z20 Z30 Z40 Hi \
Xi 1.0 0 0 0 1.0 0 0 0 x2 1.0 1.0 0 0 1.0 0 0 0 x3 0.057 0.057 1.195 0.195 0.057 0 0.943 0 x4 0.057 0.057 1.195 1.195 0.057 0 0.943 0
Zio 0 0 0 0 1.0 0 0 0 Z20 0 0 0 0 0 0 0 0 Z30 0 0 0 0 0 0 1.0 0 Z40 0 0 0 0 0 0 0 0
(B) from \ Hi \ Xi x2 x3 X4 yoi yo2 y03 yo4 Hi \
Xi 1.0 0 0 0 0 0 0 0 x2 0.177 1.0 0 0 0 0 0 0 x3 0.148 0.837 1.195 0.794 0 0 0 0 x4 0.036 0.206 0.294 1.195 0 0 0 0
yoi 0.824 0 0 0 1.0 0 0 0 V02 0.053 0.300 0 0 0 1.0 0 0 yo3 0.111 0.631 0.901 0.598 0 0 1.0 0 yO4 0.012 0.069 0.099 0.402 0 0 0 1.0 Ohio J. Sci. EIA-SYSTEMS APPROACH 215
TABLE 5 Block diagonal forms of (A) creaon matrix Q' {table 3a) and (B) genon matrix Qv (table 3b) for the marine coprophagy model.
(A) from \ Hi state \ zio z2o Xl Z30 Z40 x2 x4 X3 variables Hi \ row 2 x, 1.0* 0 0 0 0 0 0 0 0 x2 0 0 1.0 0 0 0 0 0 1.0 Zio 0 0 0 0 0 0 0 0 to X;j 0 0 0 0.789* 0 0.047 0.163 0 0.2 x4 0 0 0 0 0 0 0 1.0 1.0 Z2o 0 0 0 0 0 0 0 0 — Z30 0 0 0 0 0 0 0 0 — Z40 0 0 0 0 0 0 0 0 —
(B) from.
\ Xl yoi x2 x4 yo2 X3 yos yo4 Hi \
yoi 0.824* 0 0 0 0 0 0 0 x2 0.176 0 0 0 0 0 0 0
y02 0 0 0.300* 0 0 0 0 0 to x3 0 0 0.700 0.664 0 0 0 0 y4 0 0 0 0.336* 0 0 0 0
X4 0 0 0 0 0 0.246 0 0 yos 0 0 0 0 0 0.754* 0 0 Xl 0 0 0 0 0 0 0 0 state variables 0.176 — 0.700 0.664 — 0.246 — — column 2 *See text below. each submatrix must have input contact . . . , Hi] that it defines will be clarified with the system's input environment H, later. and that this must be true for all of the m subsystems formed by the matrix di- GENON CASE agonalization procedure. Thus, to ac- A parallel development is required to count for all causal propagation within a establish Proposition 2 with respect to system H, it is necessary to refer to an output environment. In figure 2b, let environment H outside of H. This is the total input z\ to H. be Proposition 2, for the creaon case. x 1 n Block diagonalization of Q for the (14) zi= 2 zij+Zio, i=l, . . . , n, marine coprophagy model is illustrated = 1 in table 5a. Row sums appear in the i right hand column. For both Qi' and where z i0 (fig. 2b) is input from the sys- Q2' the sum of state variable rows is < 1 tem level environment W. Applying for at least one such row, namely the row equations (1) and (2) as before gives for xi in Qi' due to input zw to Hi (in- n dicated by an asterisk), and the row for (15) Ti= 2 Fij+Zio, i = l, . . . ,n. X3 in Q2' due to input z30 to Hz (asterisk). 3 = 1 Existence of the transitive closure matrix 1 1 Fij can be expressed as a fraction of the (I-Q )" for this model is thus established, throughput Tj of H}: and the matrix in fact is illustrated in } (16) Fij^q^ijTj, i, j = l, ...,n, table 4a. The input environments Hi , which, substituted into (15), results in 216 BERNARD C. PATTEN Vol. 78 fects within a system H it is necessary to (17) reference, as output, an environmental system Hn external to H. This is Prop- osition 2 expressed for the genon case. In matrix notation this becomes Block diagonalization of Q" for the (18) T = Q"T+z, marine coprophagy model is shown in where T is a 2n-dimensional throughput table 5b. Column sums appear in the vector, z is a 2n-dimensional vector rep- bottom row. For submatrix Qx" the sum of the only state variable column, xi, is resenting inputs from H\ and Q" is a < 1 due to out put yOi form Hi (shown by 2n x 2n matrix of fractional direct effects) *). In Q2" both state variable column 1 sums are < 1 because H and H both gen- qij" from Hj to Hi per unit of Tj [eq. 16c]. n 2 4 Input z and throughput T vectors for the erate output to H (asterisks). And in Q3", column x3 sums to < 1 because of out- marine coprophagy model are indicated put y03 from H3 (asterisk). The existence in table 1. Table 3b shows the Q" ma- condition for (I-Q")-1 is met for this trix. Solving eq. (18) for T: model, and the matrix is shown in table 4b. The output environments H\\ . . . , Hi] defined by this matrix will be demon- strated in the next section. The second proposition has been estab- where lished. The internal interactive struc- (20) (I-Q")ir^ = 0ii/Zio,i,j = l,.:.,n. ture of systems cannot be fully specified, 0ij is the total (direct, Fij, plus indirect) with all causal pathways of all lengths effect of H} on Hi transmitted over all accounted for, without reference to an possible paths interconnecting the com- 1 exogenous input or output environment, ponents of H. (I-Q") ij"" is the same total or both. The systems must be open effect normalized to a unit of input z to 1 j0 systems. The causal pattern within Hj (fig. 2b). Therefore, (I-Q")- re- closed systems cannot be completely quires the transitive closure property, specified, from which it may be con- for which conditions must be established. cluded that it is a function of environ- This output environment denning ma- ments to validate the internal nature of trix for the marine coprophagy model their defining systems. appears in table 4b. As Patten et al (1976) indicate, Propo- As before, Q" denotes direct effects k sition 2 can also be realized from Markov and (Q") indirect effects over paths of chain theory. Its ultimate generality, length k. From identity (11), series con- however, is probably conferred by the vergence is to an inverse matrix, fact that it may be a manifestation of £ Godel's famous theorem (e.g., Nagel and (21) lim 2 (Q")k=(I-Q")r1 Newman 1956) on incompleteness of ^cok = 0 logical systems. Godel in what is con- if the limit exists. Again, diagonalize sidered one of the mathematical land- Q" into m irreducible block sub matrices marks of this century, showed that the satisfying det Qi" • det Q2" . . . • det Qm". consistency of any deductive system can- In each block submatrix sum the state not be established without reference to variable entries in each state variable some external system of logic whose own column. (I-Q")-1 exists if and only if consistency is in question without refer- for each submatrix the sum of state vari- ence to a further external system, etc. ables in each column is strictly <1 for If logical systems have logical "environ- at least one state variable column. That ments" which must be consulted to is, at least one holon in each subsystem demonstrate internal consistency of the represented by a block diagonal matrix former, then it should be no surprise that must have output contact with the out- nature as comprehended by the same put environment Hn of H; no subsystem mind that created logic should possess so defined may lack such contact. Hence, the same characteristic inherent in the to account for all propagation of ef- object/environment relationship. Ohio J. Sci. EIA-SYSTEMS APPROACH 217
Propositions 1 and 2 together signify one unit of output yoi from each com- that the object (organism)/environment ponent holon Hi (i=l, . . . , 4). These pair is an inseparable, mutually defining normalized input environs are depicted unit. In the next section, a system is in figure 3. Each environ is relative to a formulated as a composition of such sub- unit output (heavy arrows) from the system level units. component holons. Numbers within the holon symbols denote throughputs re- THIRD PROPOSITION quired to generate the unit outputs; Proposition 3: A system can be con- numbers associated with arrows represent structed as a set union of mutually dis- propagated causes that sum to the joint and exhaustive object/ environment throughputs. Correspondences between elements {environs). The within system figure 3 and table 4a are obvious. To express the normalized environs as car- object/environment units of Propositions 1 2 1 and 2 form a partition at the system level of bon flows (gC m~ y" ) numbers in the organization. figure and table must be multiplied by the corresponding output flux as given in This final proposition can be illustrated figure 1. The normalized versions (fig. advantageously with the marine co— 3) will be used for interpretation. prophagy model. First, the formal state- Consider £4' in figure 3. Observation ment. Let H^ i=l, . . . , n, be a sub- (measurement) of one unit of carbon out- system level component of an n-com- put from H^ specifies the indicated causal ponent system H, with input environ- n network as input environment H±. Cau- ment W and output environment H at sation is traced back through the network the supersystem level. The within sys- to its origins at the system boundary. tem input environment of Hi is Hi\ and Most of the output from Hi derives from the corresponding output environment is n input to Hs (94.3%), and only a small Hi . The creaon/input environment and amount (5.7%) originates with Hi input. genon/output environment units have The relations shown for the remaining been well enough established by Propo- three input environs are self evident. If sitions 1 and 2 that they can be regarded these four normalized environs E\\ . . . , as entities in their own right. They will ? £4 are scaled to actual carbon flows and be termed input and output environs summed, the original figure 1 system is (within system object/environment ] reconstructed. That is, units), Ei and £;" respectively, i=l, 4 . . . , n. This is consistent with normal usage in which the word environ refers (25) H = S E{\ ot nearby surroundings. Here, nearby i=l means within the boundary of the de- Thus, the input environs of figure 3 are fined system. Proposition 3 can be for- nonintersecting [eq. (22)] and also exh- mulated in terms of these units: input haustive [eq. (24)], establishing Proposi- environs do not overlap, tion 3 for the creaon case. Table 4b presents the (I-Q")"1 matrix (22) £iV\Ej' =
1.0 '
.057
• 057
• 195 • 9^3 FIGURE 3. Normalized input environs £/, . . . , £4' which partition the steady state marine coprophagy model. figure 1. The normalized environs (fig. indicating that the output environs E^, 4) will again be interpreted. . . . , £4" are mutually exclusive [eq. (23)] In the upper diagram of figure 4 de- and exhaustive [eq. (24)]. Proposition 3 picting £1", 82.4% of Hi input exits the is therefore established for the genon case. system at Hh 5.3% at H2, 11.1% at H3 Thus, for general systems, but espe- and 1.2% at H4. The within system cially for ecosystems which motivate this propagated effects leading to these out- theory, within system object (organism)/ puts are shown. The other environs pro- environment units (environs) form set vide similar information about the fate partitions at the system hierarchical of other inputs. If these environs are level. Two such partitions are possible, dimensionalized to actual carbon flows one by input environs and the other by (gC m~2 y"1) and the results summed, the output environs. Both are distinct and original figure 1 system is again recom- different as the input and output en- posed. That is, virons defined by a given holon are dis- 4 tinct and different (EiM-Ei", i=l, . . . , (26) #=2 £/, n). von Uexkull (1926) apparently ap- preciated the disjoint property of such Ohio J. Sci. EIA-SYSTEMS APPROACH 219 partitions when he wrote, "For each in- (e.g., Haskell 1940) have been offered. dividual animal, ... its function-circles The systems concept outlined above constitute a world by themselves, within differs from the normal one in four which it leads its existence in complete particular ways: two environments are isolation." recognized instead of one; indirect cau- sality is included; the object (organism)/ DISCUSSION environment complex is unitary; and Ecology was stated previously in this the units (environs) partition reality. paper to take a fundamentally physio- logical view of environment. This is TWO ENVIRONMENTS consistent with ordinary usage in which The causal holon H is a general systems living or nonliving systems are influenced object that originates not one, but two, by external surroundings. The physio- environments, H (input) and H" (out- logical concept is manifested in Mason put). If H is a system level object, H and Langenheim's (1957) theory, which and Hn are supersystem concepts and limits environment to direct causes only. cannot be further described. If H is a This is the normal ecological view of subsystem, then its within system environment, although other viewpoints environments can be specified to the
.012
1.0
.069
.099
.402
FIGURE 4. Normalized output environs Ei", . . . , £4" which partition the steady state marine coprophagy model. 220 BERNARD C. PATTEN Vol. 78 system boundary as input and output system as input, / is present time, and tn environs, £' and E", respectively. The is the time at which a corresponding environ is a new class of object in system effect is generated as system output. theory. What it may contribute to the Dynamically, H defines its input and understanding of ecological or general output environments Hl and H" instan- systems remains to be seen. For example, taneously at time t through direct inter- where a holon is a biological object, active coupling to other holons of the inheritance and evolution of its environs system. In input environs E\ indirect may be reasonable to consider as outward causality, which conditions the direct projections of known genetic mechanisms. coupling events at /, has already occurred The necessity for objects to interact during the past [t\ i\. Thus, an instan- consistently within environs provides taneous input environ defined at / constraints that almost certainly guar- encompasses a historical network of antee coevolution to be an ecosystem causation extending backward to the level phenomenon (Patten et at 1976; system boundary at /'. Similarly, in- Patten 1977). Prospects for an organ- direct effects in an instantaneous output ismic representation of environment are environ E" are propagated from the direct coupling events at time t during quite real in this theory. n The normal one-environment concept the future, [t, t ]. The instantaneous includes only input environment, von output environ contains the succession of indirect causes and effects extending Uexkull (1926) provided a precedent for } output environment in the notion of forward to the system boundary at t \ function circles that fail of closure (out- Note that the system exists with respect to a cause introduced at t] only during put affecting input) due to complexity ]l of the external world. Propagated effects the interval [/', t ] required for it to generate a corresponding effect at tn, and become lost in the general flux of causa- n tion before they can return as identifiable this is true "vA t e T. The role of holon inputs to the original generating organ- H in the system relative to the same ism. By explicitly recognizing two en- cause is similarly restricted to the same vironments, an analytical potential is interval. Without a temporally finite gained that is absent in a one-sided the- model Mason and Langenheim's (1957) ory. Creaon and genon partitions (eg., objection of infinite regress, and a figs. 3 and 4) are never the same, and pat- counterpart infinite future progression of terns of how they differ are foreseeable the two-environment theory, would be system properties of interest. For ex- valid. So long as a holon's memory of ample, Patten (1978) has analyzed con- the past and horizon to the future are trol relationships in ecosystem models by relatively small, so that its system comparing input and output environs of appears relatively permanent compared component holons. to itself, this permanent organization should be represented in its environs. INDIRECT CAUSALITY Environment as a concept is not instan- taneous. It is natural history, a window Mason and Langenheim (1957) wrote on the relatively near past and future, that to include indirect factors in environ- and to make it so, indirect causality must ment is to confuse environment with be included. Therefore, instantaneous history. In the two-environment ap- input environs E] defined at time t proach the future enters as a similar properly span intervals [t\ t], and cor- objection. How should time be regarded responding output environs En span in a concept of environment? Two intervals [t, tn]. aspects of the question are dynamic and static. The static case reflects this. Static Let H be a component of a system models depict, usually, steady state that exists with respect to a cause during characteristics of systems over time [t\ t\ t}