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Quantitative Methods for Ecological Network Analysis Computational Biology and Chemistry 28 (2004) 321–339 Quantitative methods for ecological network analysis Robert E. Ulanowicz∗ Chesapeake Biological Laboratory, University of Maryland Center for Environmental Science, Solomons, MD 20688-0038, USA Received 2 September 2004; accepted 5 September 2004 Abstract The analysis of networks of ecological trophic transfers is a useful complement to simulation modeling in the quest for understanding whole-ecosystem dynamics. Trophic networks can be studied in quantitative and systematic fashion at several levels. Indirect relationships between any two individual taxa in an ecosystem, which often differ in either nature or magnitude from their direct influences, can be assayed using techniques from linear algebra. The same mathematics can also be employed to ascertain where along the trophic continuum any individual taxon is operating, or to map the web of connections into a virtual linear chain that summarizes trophodynamic performance by the system. Backtracking algorithms with pruning have been written which identify pathways for the recycle of materials and energy within the system. The pattern of such cycling often reveals modes of control or types of functions exhibited by various groups of taxa. The performance of the system as a whole at processing material and energy can be quantified using information theory. In particular, the complexity of process interactions can be parsed into separate terms that distinguish organized, efficient performance from the capacity for further development and recovery from disturbance. Finally, the sensitivities of the information-theoretic system indices appear to identify the dynamical bottlenecks in ecosystem functioning. © 2004 Elsevier Ltd. All rights reserved. Keywords: Ascendency; Cycling in ecosystems; Ecosystem status; Flow networks; Information theory; Input–output analysis; Nutrient limitation; Trophic levels 1. Why network analysis? trivial tasks of calibrating the model according to some set of system observations and validating it against another in- Ecologists have precious few tools at their disposal to rep- dependent set of data. resent phenomena that transpire at the level of the whole- Unfortunately, grave problems, both practical and con- ecosystem. Yet, they increasingly are being exhorted to ap- ceptual, beset such whole-ecosystem simulation modeling. proach environmental problems at the level of the whole On a practical level, the results of the endeavor appear to ecosystem (NSF, 1999). To date, the most common tool for leave much to be desired (Sheffer and Beets, 1994). Which quantifying systems-level events is simulation modeling. Be- is not to ignore some successful models consisting of one or fore one can simulate an ecosystem, it is necessary first to a few processes, but as the number of interacting processes identify the relevant taxa that comprise it. Thereupon, the in- increases, problems multiply disproportionately (Platt et al., vestigator must parse out the significant interactions among 1981). Some argue that difficulties in calibrating and validat- those taxa. It is only after these preliminaries that the actual ing simulation models arise from the propagation of errors modeling commences, as the modeler then quantifies each across the nexus of interacting processes. Whence, more ac- such interaction in algorithmic fashion. The aggregated for- curate and precise values for the model parameters should mulae are then executed under an appropriate shell on some ameliorate the problem. Others, however, are convinced that suitable platform. Finally, there follow the arduous and non- interacting nonlinear processes inevitably lose their power to predict (Lorenz, 1963). Furthermore, prediction ability ∗ Tel.: +1 410 326 7266; fax: +1 410 326 7378. wanes as the interacting processes increase in number and/or E-mail address: [email protected]. nonlinearity (Ulanowicz, 1979). Hence, in order to avoid 1476-9271/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compbiolchem.2004.09.001 322 R.E. Ulanowicz / Computational Biology and Chemistry 28 (2004) 321–339 pathologies that inevitably crop up in the behaviors of cou- pled process models, biologically unrealistic assumptions are made out of mathematical convenience (to maintain stabil- ity), thereby further impairing prediction ability. It seems frustrating that nature thwarts attempts to model ecosystems accurately, and thereby control them, but mod- elers seldom draw the conclusion that the conceptual frame- work that supports the simulation process is itself flawed. After all, multi-process models work quite well for phys- ical and chemical systems, and the modeling process is built upon the selfsame assumptions that guide the larger Fig. 1. A directed flow graph. body of science—namely, closure, atomism, reversibility, determinism and universality (Depew and Weber, 1994; Ulanowicz, 1999). Why should matters go suddenly awry 2. Required information with ecosystems? It happens that, for a while now, propo- nents of “deep-ecology” have maintained that ecology is truly A working definition of an ecological network is that it different from other disciplines (Naess, 1988); that it forces is a representation of the answers to two questions: (1) who the investigator to alter profoundly his/her conceptions about eats whom?, and (2) at what rate? As with simulation model- how nature operates. Only recently have some of the fea- ing, preliminary to the first question it becomes necessary to tures been articulated that distinguish ecological phenom- identify the significant taxa or nodes comprising the ecosys- ena from more fundamental physical and chemical events tem. For each of the taxa one then needs to know which of (Ulanowicz, 1999). These matters are beyond the scope of the other nodes are present in its diet. Once this qualitative this exposition, suffice it here to mention: (a) an inversion information is known for all taxa, the result can be repre- in the magnitudes of characteristic lifetimes (and with it the sented in one of two ways: (a) one can present the system direction of causality) as one passes from the individual or- graphically as a directed graph, or digraph. On the digraph ganism to the ecosystem, and (b) the singular nature of and the nodes are usually represented as boxes, and each trans- crucial role played by aleatoric events over the histories of action is represented as an arrow that originates out of the ecosystems. prey taxon and terminates (with an arrowhead) at the preda- Leaving these intriguing academic issues aside, the very tor node (Fig. 1); (b) the connection topology among n nodes practical question arises, “If simulation modeling is less than can likewise be represented as an n × n square adjacency ma- satisfactory, what alternative approach might culminate in a trix, where a one in the entry for row i and column j means 2 deeper and more satisfactory understanding of whole ecosys- that an amount of material flows from predator i to prey j. tems?” Almost a quarter century ago SCOR Working Group By contrast, a zero entry signifies that no palpable transfer 59 (Platt et al., 1981) suggested that systems ecologists sim- occurs between i and j (hence the notion of a binary net- ply forego the latter stages of the modeling process altogether. work). The advantage of using the matrix approach and asso- They suggested that one concentrate instead on what can be ciated linear algebra is that one may then deal with systems of inferred from the identification and parsing tasks alone; that arbitrary dimension. one pay more attention to processes (flows) than to objects Now, ecosystems are necessarily open, meaning that they (stocks). This advice eventually was acted upon by a succes- exchange material and energy with their surroundings. These sor working group (#73) that assembled an inchoate set of exogenous transfers require that one emend both the graph- analytical tools which collectively became known as “ecosys- ical and analytical representations. Without loss of general- tem network analysis” (ENA) (Wulff et al., 1989). ity, one can assume that all inputs to a particular taxon be An assumption underlying ENA has been that the config- lumped as a single entry. (If such is not the case, multiple uration of processes represents the “anatomy” of the ecosys- types of inputs can be treated as will presently be discussed tem; and that, as in medical practice, such anatomy will re- for exogenous outputs.) Actual physical inputs take the form veal much about the history, current status and workings of of primary production, immigration or inbound advection of the ecosystem.1 The remainder of this work will discuss how material or energy. The lumped input is represented graph- to create ecological flow networks and analyze them in sys- ically by an arrow that originates out of no visible taxon tematic fashion. and terminates (with an arrowhead) into the actual receiving node (Fig. 2). For reasons that will become obvious presently, it is use- 1 It should be noted in passing that an independent school of investiga- ful to distinguish between two different types of exogenous tions into the structure of “foodwebs” has evolved in parallel with ENA (e.g., outputs: the first is the export of material or energy still useful Yodzis, 1989; Cohen et al., 1990; Polis and Winemiller, 1995). Foodweb re- search treats unweighted (binary) networks of trophic interactions, whereas ENA addresses the [sometimes large] differences in the magnitudes of the 2 Several other investigators (e.g., Patten, 1985) reverse the order of the connections. subscripts. That is, flow is in the direction from column j to row i. R.E. Ulanowicz / Computational Biology and Chemistry 28 (2004) 321–339 323 Fig. 2. The trophic exchanges of energy (kcal m−2 y−1) in the Cone Spring ecosystem (Tilly, 1968). Arrows not originating from a box represent exogenous inputs. Arrows not terminating in a box portray exogenous outputs. Ground symbols represent dissipations. to other ecosystems of comparable scale. Examples of such for each taxon, i, exports are emigration, harvesting by humans, and advection n n out of the system.
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