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M Scotti. Development Capacity. In Sven Erik Jørgensen and Brian D. Fath (Editor-in-Chief), Ecological Indicators. Vol. [2] of Encyclopedia of Ecology, 5 vols. pp. [911-920] Oxford: Elsevier. Author's personal copy Ecological Indicators | Development Capacity 911 animals that appear on and in dung and the processes they Cross WF, Benstead JP, Frost PC, and Thomas SA (2005) Ecological stoichiometry in freshwater benthic systems: Recent progress and initiate are highly predictable, but in detail depend on the perspectives. Freshwater Biology 50: 1895–1912. habitat and climate under investigation. Specialized copro- Findlay S and Sinsabaugh R (eds.) (2000) Dissolved Organic Matter in philous fungi, similarly, exhibit a clear sequence of Aquatic Ecosystems. San Diego: Academic Press. Flindt MR, Pardal MA, Lillebø AI, Martins I, and Marques JC (1999) utilization of their habitat, spores of early stages already Nutrient cycling and plant dynamics in estuaries: A brief review. Acta being present in the dung when it is deposited by the Oecologica 20: 237–248. herbivore. Apparently, users of dung are highly specialized, Grac¸ a MAS (2001) The role of invertebrates on leaf litter decomposition in streams – A review. International Review of Hydrobiology and their community is significantly different from what 86: 383–393. usually is referred to as decomposers or detritivores. Grac¸ a MAS, Ba¨ rlocher F, and Gessner MO (2005) Methods to Study Depending on climate, dung patches may become comple- Litter Decomposition: A Practical Guide. Dordrecht: Springer. Kirkman H and Kendrick GA (1997) Ecological significance and tely decomposed in less than 2 months but may also last for commercial harvesting of drifting and beach-cast macro-algae and more than 2 years. Thus, as for most types of detritus, seagrasses in Australia: A review. Journal of Applied Phycology 9: 311–326. nutrient release is not sudden but extended over a longer Mackensen J, Bauhus J, and Webber E (2003) Decomposition rates of time interval, so that detritus provides a long-term store coarse woody debris – A review with particular emphasis on and source of nutrients at the basis of ecosystems. Australian tree species. Australian Journal of Botany 51: 27–37. Moore JC, Berlow EL, Coleman DC, et al. (2004) Detritus, trophic dynamics and biodiversity. Ecology Letters 7: 584–600. Orr M, Zimmer M, Jelinski DE, and Mews M (2005) Wrack deposition on See also: Decomposition and Mineralization; Freshwater different beach types: Spatial and temporal variation in the pattern of subsidy. Ecology 86: 1496–1507. Lakes; Freshwater Marshes; Habitat. Schaefer M (1991) Secondary production and decomposition. In: Ro¨ hrig E and Ulrich B (eds.) Ecosystems of the World – 7: Temperate Deciduous Forests, pp. 175–218. Amsterdam: Elsevier. Sinsabaugh RL, Osgood MP, and Findlay S (1994) Enzymatic models Further Reading for estimating decomposition rates of particulate detritus. Journal of the North American Benthological Society 13: 160–171. Attiwill PM and Adams MA (1993) Nutrient cycling in forests. New Swift MJ, Heal OW, and Anderson JM (1979) Decomposition in Phytologist 124: 561–582. Terrestrial Ecosystems. Berkeley: University of California Press. Coleman DC, Crossley DA, and Hendrix PF (2004) Fundamentals in Soil Wetzel RG (1995) Death, detritus, and energy flow in aquatic Ecology. Amsterdam: Elsevier. ecosystems. Freshwater Biology 33: 83–89.
Development Capacity M Scotti, University of Parma, Parma, Italy
ª 2008 Elsevier B.V. All rights reserved.
Introduction System Overhead and Its Constitutive Terms Ecosystem Network Analysis Number of Roles and Development Capacity Total System Throughput and Average Mutual Ecological Applications Information Further Reading Ascendency and Development Capacity
Introduction Ascendency is estimated as the product of average mutual information (AMI) by total system throughput The ever-increasing interest toward ecosystem status and (TST). It measures the ability of a system to prevail performance stimulates the application of tools for whole- over alternative configurations. AMI evaluates how the system assessment; a prominent collection of quantitative system is developed, defining flow articulation, and TST methods to achieve this aim consists of ecosystem net- expresses the size of an ecosystem as the sum of all work analysis (ENA). the flows. The apparatus of ENA comprises indices, derived Since growth and development are natural processes, from information theory, that quantify global attributes, they possess finite limits: the minimum value for ascen- and, in particular, ascendency (A) measures growth (size) dency is 0, while its upper boundary, defined as and development (organization of flows) of ecosystems. development capacity (C), is affected by system topology. Author's personal copy 912 Ecological Indicators | Development Capacity
As a consequence, with a given number of nodes and 255 TST, the highest development capacity is assigned to a wholly connected network (with all flows of equal inten- Bacteria 5205 sity), while a balanced and completely articulated 75 1600 3275 topology (linear and closed chain) shows the minimum 2309 development capacity, corresponding, only in this case, to 11 184 8881 Detritus Plants Detritus the system ascendency. 300 200 feeders
The remaining fraction preventing ascendency 860 2003 635 3109 1814 from reaching its theoretical maximum is called total 167 370 overhead ( ), and it is calculated as the difference Carnivores between development capacity (C) and system ascen- dency (A). 203 The article briefly sketches on information theory Figure 1 Cone Spring ecosystem is composed of five principles and probabilistic approach applied to the compartments (plants, bacteria, detritus feeders, carnivores, and study of ecological networks, introducing an explana- detritus) and it shows: two imports (green), three exports (red), tion of development capacity and associated overhead five dissipations (blue), and eight intercompartmental exchanges (black). Flows are expressed in kcal m 2 yr 1. definitions. Then, it shows an alternative equation to define development capacity, described as a function of the 0 1 2 n n + 1 n + 2 effective number of ecosystem roles (a measure of the 0 0 Imports 00 degree to which the system is differentiated into distinct functions). 1 0 Finally, through the comparison between ascendency, 2 overhead, and development capacity, examples of appli-
Respirat n
Expo
cations to real ecosystems are introduced, strengthening elements
n
+
the important part ascendency plays as a whole-system Intercompartmental 3 eleme 3
rts
index. exchanges ions nts
Ecosystem Network Analysis n elements n 0
Network analysis depicts the ecosystem as a collection of n + 1 0 0 0 0 0 nodes (species or trophospecies) connected by arrowhead n + 2 0 0 0 0 0 arcs portraying trophic relations (exiting the prey items and entering the predator). n + 3 elements Trophic exchanges are measured as energy flows (i.e., kcal m 2 yr 1) or nutrient transfers of different currencies Figure 2 Graphical example of T matrix for a generic (i.e., carbon – mg C m 2 yr 1; nitrogen – mg N m 2 n-compartment ecosystem, with size ¼ (n þ 3) (n þ 3). Imports 1 2 1 to system nodes are summarized in the first row, exports and day ; phosphorus – g P m yr ). dissipations are set in the last two columns, and the internal n n Figure 1 depicts the Cone Spring ecosystem with five matrix shows transfers between system compartments. This compartments (n ¼ 5) and eighteen flows distinguished in model of T matrix is adopted for Cone Spring. two imports from outside (to plants and detritus), three exports to outside (from plants, bacteria, and detritus), 2 3 0 11184 0 0 0 635 0 0 five respirations (a ground symbol on each node), and 6 7 6 7 eight intercompartmental exchanges. 6 0 0 0 0 0 8881 300 2003 7 6 7 A graphical scheme can likewise be represented as an 6 7 6 0 0 0 75 0 1600 255 3275 7 6 7 (n þ 3) (n þ 3) square matrix of transfers [T], with tij 6 7 6 0 0 0 0 370 200 0 1814 7 elements describing outflows from row compartment i to 6 7 T ¼ 6 7 ½1 column node j (see Figure 2). First row stands for imports 6 0 0 0 0 0 167 0 203 7 6 7 6 7 from outside to the column compartments, and the last 6 0 0 5205 2309 0 0 860 3109 7 6 7 two columns list exports and respirations from any row 6 7 compartment. The internal n n submatrix reports inter- 4 00 0 000005 nal exchanges. 00 0 00000 Author's personal copy Ecological Indicators | Development Capacity 913
Total System Throughput and Average yielding to Mutual Information Xnþ2 Xnþ2 pai ; bj AaðÞ¼; b K pai ; bj log ½9 Besides the graphical and mathematical representation of i¼0 j ¼0 paðÞi pbj ‘who eats whom’ and ‘at what rate’, ENA performs calcu- since lus focusing both on single-compartment and whole- system levels. Xnþ2 TST and AMI are basal indices advocating a systemic paðÞ¼i pai ; bj ½10 approach. j ¼0 The total amount of flows occurring in the system Xnþ2 is called TST, and for Cone Spring is equal to pbj ¼ pai ; bj ½11 42 445 kcal m 2 yr 1: i¼0 To measure transfer uncertainty with network analysis Xnþ2 Xnþ2 notation, the probability that a quantum of matter (or TST ¼ tij ½2 i¼0 j ¼0 energy) would flow from compartment i to j is computed as tij TST defines ecosystem activity (size) and it conceptually pai ; bj ffi ½12 corresponds to what in economy is known as gross TST national product (GDP), an indicator of economic com- with output and input probabilities that can be written as munity size. marginal sums of joint probabilities: Ecosystem development is connected to flow organi- Xnþ2 zation and it increases when uncertainty is diminishing. tij paðÞffii ½13 Mathematically, uncertainty (H), related to a distribution j ¼0 TST of probability over n categories (with scalar constant K), is Xnþ2 t equivalent to pb ffi ij ½14 j TST Xn i¼0 H ¼ – K pi log pi ½3 AMI is the index measuring system organization (eco- i¼1 system development intended as flow articulation). Using System flow disorganization is measured by uncer- the expressions provided by formulas [12]–[14] into [9], tainty and its amount can be distinguished into output and simplifying, one obtains for this quantity the follow- H(a) and input H(b) contributions: ing representation: "# Xnþ2 Xnþ2 Xnþ2 K tij ?TST HaðÞ¼– K paðÞlog paðÞ ½4 AMI ¼ tij ? log P P ½15 i i TST nþ2 t nþ2 t i¼0 i¼0 j ¼0 q¼0 qj v¼0 iv
Xnþ2 HbðÞ¼– K pbðÞi log pbðÞi ½5 j ¼0 Ascendency and Development Capacity With completely independent events, total system uncer- Ascendency, being the product of TST and AMI, takes tainty becomes H(a) þ H(b). However, inputs and outputs the following mathematical form: in ecosystems are not always independent and the asso- "# Xnþ2 Xnþ2 ciated uncertainty can be computed adopting joint t ? TST P ij P probabilities p(ai,bj): A ¼ AMI ? TST ¼ tij ? log nþ2 nþ2 ½16 i¼0 j ¼0 q¼0 tqj v¼0 tiv Xnþ2 Xnþ2 HaðÞ¼; b – K pai ; bj log pai ; bj ½6 Assessing ecosystem growth and development can be done i¼0 j ¼0 comparing ascendency with its maximum and minimum limits. Ascendency, deriving from TST and AMI, shows a Therefore, with inputs and outputs that are not comple- minimum value of 0 (when output and input flow prob- tely independent, abilities are completely independent; see Figure 3a), while HaðÞ; b < HaðÞþHbðÞ ½7 the upper boundary is defined as development capacity. For each ecosystem, the development capacity depends and, in this case, the degree of system organization is on the constraints established by real network topology. defined as When number of compartments (n) and TST are assigned, the highest development capacity is associated to a wholly AaðÞ¼; b HaðÞþHbðÞ– HaðÞ; b ½8 connected and balanced network, decreasing when flows Author's personal copy 914 Ecological Indicators | Development Capacity
(a) (b)
3 3 3 6 AB AB 3 3 3 6 6 3 3 3 3 6 6 33 TST = 48 6 6 TST = 48 AMI = 0 AMI = 1 3 D C A = 0 D C A = 48 3 C = 192 6 C = 144 3 3
(c) (d)
4 12 A B AB
12 4 4 8 8 12 12 TST = 48 TST = 48 AMI = 1.144 611 AMI = 2 D C A = 54.931 35 D C A = 96 4 C = 137.058 65 12 C = 96
Figure 3 Hypothetical networks with four compartments and TST ¼ 48 energy units: (a) the more unarticulated topology with minimum AMI and ascendency values (both equal to 0) and maximum development capacity (192); (b) and (c) intermediate configurations showing increasing AMI and A; C is lower than that observed for the first network; (d) a closed and linear chain (maximally articulated flows) with highest AMI (2) and ascendency equal to development capacity (96).
Table 1 Development capacity calculated for the hypothetical outflow probabilities as mutually determined (to the out- networks depicted in Figure 3; the values of C decrease when put ai coincides, exclusively, the input bj): the network topology becomes more articulated pai ; bj ¼ paðÞ¼i pbj ½19 Network topology Development capacity (C) and the consequent development capacity is Fully connected (A) 192 Intermediate I (B) 144 Xnþ2 Xnþ2 paðÞi Intermediate II (C) 137 C ¼ K paðÞi log ¼ – K paðÞi log paðÞi ½20 Linear and closed chain (D) 96 i¼0 paðÞi paðÞi i¼0 Xnþ2 Xnþ2 pbj become more articulated (its minimum value is achieved C ¼ K pbj log ¼ – K pbj log pbj ½21 pb pb with closed linear chain topology, when it corresponds to j ¼0 j j j ¼0 system ascendency) (see Figure 3 and Table 1). or, in terms of energy (or matter) transfers (with K ¼ TST), In what follows, minimum and maximum ascendency Xnþ2 Xnþ2 limits are explained through a probabilistic approach. tij C ¼ – tij log ½22 The lower ascendency value rises with fully connected i¼0 j ¼0 TST topology, that is when input and output flow probabilities are completely independent: The more articulated topology (Figure 3d), with receiv- ing node always determined when the compartment from pai ; bj ¼ paðÞi ? pbj ½17 which the flow is exiting is known, implies that C ¼ A. Substituting this relation into eqn [9] yields In general, Xnþ2 Xnþ2 C A 0 ½23 paðÞi pb j AaðÞ¼; b K paðÞi pbj log paðÞpb i¼0 j ¼0 i j From the strictly positive difference between develop- Xnþ2 Xnþ2 ment capacity and ascendency is estimated the system ¼ K paðÞi pbj logðÞ¼ 1 0 ½18 i¼0 j ¼0 overhead ( ), measuring the degree of freedom in flow organization preserved by an ecosystem: Conversely, under minimum uncertainty conditions, ascendency can be inferred setting each inflow and ¼ C – A ½24 Author's personal copy Ecological Indicators | Development Capacity 915
Ascendency and system overhead are usually scaled case of external source collapse (with all the inputs to the with development capacity to define them as percentage system occurring via one single arrow). of the theoretical upper bound on organization: Overhead on exports quantifies pathway multiplicity for medium exiting the system in a usable form. Like the 100 ? A A ðÞ¼% ½25 overhead on inputs, it ranges from a minimum of 0, C when all the matter (or energy) leaving the system is 100 ? ðÞ¼% ½26 concentrated on a single node, to a maximum value C achieved in case of exports evenly distributed on each single compartment. When the topology of exports and their relative importance are assigned, this overhead System Overhead and Its Constitutive component tends to increase with higher amount of Terms matter exported. The dissipative overhead is related to the fraction of medium that is modified by internal processes (i.e., Path multiplicity and low level of flow organization, giv- respiration in ecosystems) and exiting the system in an ing rise to overhead, can be interpreted as system unusable form (flows that do not connect boxes). It inefficiency in processing material and energy, but, in increases with dissipation intensity and because of ther- case of stress and perturbations, they represent an advan- modynamic and ecological constraints must always be tage in terms of system adaptability to new threats. greater than 0. The system overhead can be divided into four separate The fourth component of overhead is related to the contributions, each related to a certain form of multi- redundancy of pathways within the system. It is a con- plicity of pathways: input from outside (overhead on tribution to disorder (inefficiency – disorganization) imports, ), exports to other systems (overhead on I because sending medium over diverse routes costs more exports, ), respirations (dissipative overhead, ), and E D in terms of dissipation than channeling it over few effi- internal transfers (redundancy, ): R cient pathways; nevertheless, it becomes absolutely