## Quantifying Directionality In Ecosystems

It is one thing to describe the workings of autocatalytic selection verbally, but science demands at least an effort at describing how one might go about quantifying and measuring key concepts. At the outset of such an attempt, we should emphasize again the nature of the directionality with which we are dealing. The directionality associated with autocatalysis does not appear in either physical space or, for that matter, in phase space. It is rather more like the directionality associated with time. There direction, or sense, is indicated by changes in a systems-level index—the system's entropy. Increasing entropy identifies the direction of increasing time.

The hypothesis in question is that augmented autocatalytic selection and centripetality are the agencies behind increasing self-organization. Here we note that as autocatalytic configurations displace more scattered interactions, material and energy become increasingly constrained to follow only those pathways that result in greater autocatalytic activities. This tendency is depicted in cartoon fashion in Figure 4.6. At the top is an arbitrary system of four components with an inchoate set of connections between them. In the lower figure one particular autocatalytic feedback loop has come to dominate the system,

2This emergence differs from Prigogine's "order through fluctuations" scenario in that the system is not constrained to toggle into one of two pre-determined states. Rather, complex chance can carry a system into entirely new modes of behavior (Tiezzi, 2006b). The only criterion for persistence is that the new state be more effective, autocatalytically speaking, than the original.

Figure 4.6 Cartoon showing the generic effects of autocatalysis. (a) Inchoate system. (b) Same system after autocatalytic loop has developed.

Figure 4.6 Cartoon showing the generic effects of autocatalysis. (a) Inchoate system. (b) Same system after autocatalytic loop has developed.

resulting in fewer effective flows and greater overall activity (as indicated by the thicker surviving arrows). Thus we conclude that quantifying the degree of constraint in an ecosystem must reflect these changes in both the magnitude and intensity of autocatalytic activities. Looked at in obverse fashion, ecosystems with high autocatalytic constraints will offer fewer choices of pathways along which resources can flow.

The appearance of the word "choice" in the last sentence suggests that information theory might be of some help in quantifying the results of greater autocatalysis, and so it is. Box 4.1 details the derivation of a measure called the System Ascendency, which quantifies both the total activity of the system as well as the degree of overall constraint extant in the system network. A change in the system pattern as represented in Figure 4.6 will result in a higher value of the ascendency.

In his seminal paper, "The strategy of ecosystem development", Eugene Odum (1969) identified 24 attributes that characterize more mature ecosystems that indicate the direction of ecological succession. These can be grouped into categories labeled species richness, dietary specificity, recycling, and containment. All other things being equal, a rise in any of these four attributes also serves to augment the system ascendency (Ulanowicz, 1986a). It follows as a phenomenological principle "in the absence of major perturbations, ecosystems have a propensity to increase in ascendency." This statement can be rephrased to read that ecosystems exhibit a preferred direction during development: that of increasing ascendency.

Box 4.1 Ascendency: a measure of organization

In order to quantify the degree of constraint, we begin by denoting the transfer of material or energy from prey (or donor) i to predator (or receptor) j as Tij, where i and j range over all members of a system with n elements. The total activity of the system then can be measured simply as the sum of all system processes, TST=£nj=i Tj, or what is called the "total system throughput" (TST). With a greater intensity of auto-catalysis, we expect the overall level of system activity to increase, so that T appears to be an appropriate measure. For example, growth in economic communities is reckoned by any increase in gross domestic product, an index closely related to the TST.

In Figure B4.1 is depicted the energy exchanges (kcal/m2/year) among the five major compartments of the Cone Spring ecosystem (Tilly, 1968). The TST of Cone Spring is simply the sum of all the arrows appearing in the diagram. Systematically, this is calculated as follows:

T01 + T02 + T12 + T16 + T17 + T23 + T24 + T26 + T27 ^ T32

= 11,184 + 635 + 8881 + 300 + 2003 + 5205 + 2309 + 860 + 3109 + 1600 + 75 + 255 + 3275 + 200 + 370 + 1814 +167 + 203 = 42,445 kcal/m2 /year where the subscript 0 represents the external environment as a source, 6 denotes the external environment as a receiver of useful exports, and 7 signifies the external environment as a sink for dissipation.

Figure B4.1 Schematic of the network of energy exchanges (kcal/m2/year) in the Cone Spring ecosystem (Tilly, 1968). Arrows not originating from a box represent inputs from outside the system. Arrows not terminating in a compartment represent exports of useable energy out of the system. Ground symbols represent dissipations.

Figure B4.1 Schematic of the network of energy exchanges (kcal/m2/year) in the Cone Spring ecosystem (Tilly, 1968). Arrows not originating from a box represent inputs from outside the system. Arrows not terminating in a compartment represent exports of useable energy out of the system. Ground symbols represent dissipations.

Again, the increasing constraints that autocatalysis imposes on the system channel flows ever more narrowly along fewer, but more efficient pathways—"efficient" here meaning those pathways that most effectively contribute to the autocatalytic process. Another way of looking such "pruning" is to consider that constraints cause certain flow events to occur more frequently than others. Following the lead offered by information theory (Abramson, 1963; Ulanowicz and Norden, 1990), we estimate the joint probability that a quantum of medium is constrained both to leave i and enter j by the quotient TjIT. We then note that the unconstrained probability that a quantum has left i can be acquired from the joint probability merely by summing the joint probability over all possible destinations. The estimator of this unconstrained probability thus becomes Zq TiqIT. Similarly, the unconstrained probability that a quantum enters j becomes Zk TkjIT. Finally, we remark how the probability that the quantum could make its way by pure chance from i to j, without the action of any constraint, would vary jointly as the product of the latter two frequencies, or Zq Tiq Zk TkjIT2. This last probability obviously is not equal to the constrained joint probability, Tij IT.

Information theory uses as its starting point a measure of the rareness of an event, first defined by Boltzmann (1872) as (—k log p), where p is the probability (0 <p < 1) of the given event happening and k is a scalar constant that imparts dimensions to the measure. One notices that for rare events (p ~ 0), this measure is very large and for very common events (p ~ 1), it is diminishingly small. For example, if p = 0.0137, the rareness would be 6.19 k-bits, whereas if p = 0.9781, it would be only 0.032 k-bits.

Because constraint usually acts to make things happen more frequently in a particular way (e.g., flow along certain pathways), one expects that, on average, an unconstrained probability would be more rare than a corresponding constrained event. The more rare (unconstrained) circumstance that a quantum leaves i and accidentally makes its way to j can be quantified by applying the Boltzmann formula to the joint probability defined above, i.e., — klog(Zk Tk] Zq TiqIT2), and the correspondingly less rare condition that the quantum is constrained both to leave i and enter j becomes — k log (TjIT). Subtracting the latter from the former and combining the logarithms yields a measure of the hidden constraints that channel the flow from i to j as k log (TjT/£k Tkj Zq Tqq ).

Finally, to estimate the average constraint at work in the system as a whole, one weights each individual constraint by the joint probability of constrained flow from i to j and sums over all combinations of i and j. That is,

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