10 ULANOWICZ 



species or population level. The enormous analytical difficulties in 

 properly describing most multistable systems, coupled with the high 

 dimensionality of most real ecosystems, has led a number of 

 investigators to explore the possibility that response to stress is best 

 described in terms of macroscopic or emergent variables. Macro- 

 scopic variables are characteristic of the ecosystem as a whole and 

 not just parts of it. They may and often do involve some 

 combination of lesser order variables, however. There is still no 

 consensus as to what consitutes a proper macroscopic variable. 



Several investigators have suggested semiquantitative candidates 

 for macroscopic variables as a consequence of their empirical studies. 

 Kerr (1974) referred to structural transitions as "emergent surprises" 

 and believed that they can be encompassed only by macroscopic 

 theory. He cited the particle-size spectrum of an ecosystem as a 

 convenient indicator of stress in a community. Exogenous stress 

 seems to always affect the larger size organisms disproportionately. 

 Jordan, Kline, and Sasscer (1972) emphasized the ratio of recycling 

 to input as a system variable that characterizes the recovery time of 

 an ecosystem from a temporary stress. Golley (1974) went further; 

 he suggested a temporal hierarchy of three system properties to 

 describe recovery from traumatic stress. First, the system responds to 

 restore its extensive variables (mass); second, the functional options 

 (topological diversity) increase; and, in the final stages of return to 

 undisturbed climax, its response time to disturbance lengthens. 



Presently the reconciliation of microscopic and macroscopic 

 properties of an ecosystem is hampered by the inability of ecological 

 theory to provide appropriate methods for observing community 

 properties (Kerr, 1974). Actually this hierarchical problem has 

 always been extremely important in ecological modeling. There are 

 many opinions on how to aggregate organisms, species, etc., into 

 trophic compartments or functional units (see Halfon, 1978), and it 

 is especially difficult in highly connected or "webbed" ecosystems. 



To address this problem, Kemp and Homer (1977) devised a 

 method for assigning fractions of the energy storage in a given species 

 compartment to various trophic levels. The key to their algorithm is 

 the matrix of partial feeding coefficients, which describes the 

 percentage of the total input to a given species, i, that flows from 

 another species, j. To identify the contents of the fourth trophic 

 level, for example, we identify all pathways three steps removed 

 from a primary producer. The fraction of the end-point species to be 

 assigned to the fourth trophic level is the product of the partial 

 feeding coefficients of the three links along a pathway summed over 

 all existing three-step pathways. Operationally the transformation is 



