Environmental Modelling & Software 21 (2006) 375–405 www.elsevier.com/locate/envsoft A MATLABÒ function for Network Environ Analysis Brian D. Fatha,*, Stuart R. Borrettb,c a Biology Department, Towson University, Towson, MD 21252, USA Institute of Ecology, University of Georgia, Athens, GA 30602, USA c Skiddaway Institute of Oceanography, 10 Ocean Science Circle, Savannah, GA 31411, USA b Received 7 June 2004; received in revised form 28 October 2004; accepted 5 November 2004 Available online 12 March 2005 Abstract Network Environ Analysis is a formal, quantitative methodology to describe an object’s within system ‘‘environ’’ment [Patten, B.C., 1978a. Systems approach to the concept of environment. Ohio Journal of Science 78, 206–222]. It provides a perspective of the environment, based on general system theory and input–output analysis. This approach is one type of a more general conceptual approach called ecological network analysis. Application of Network Environ Analysis on ecosystem models has revealed several important and unexpected results [see e.g., Patten, B.C., 1982. Environs: relativistic elementary particles or ecology. American Naturalist 119, 179–219; Patten, B.C., 1985. Energy cycling in the ecosystem. Ecological Modelling 28, 1–71; Fath, B.D., Patten, B.C., 1999a. Review of the foundations of network environ analysis. Ecosystems 2, 167–179], which have been identified and summarized in the literature as network environ properties. To conduct the analysis one needs ecosystem data including the intercompartmental flows, compartmental storages, and boundary input and output flows. The software presented herein uses these data to perform the main network environ analyses and environ properties including unit environs, indirect effects ratio, network homogenization, network synergism, network mutualism, mode partitioning, and environ control. The software is available from The MathWorks MATLABÒ Central File Exchange website (http://www.mathworks.com/matlabcentral/fileexchange/loadCategory.do). Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Ecological Modelling; Ecological Network Analysis; Network Environ Analysis 1. Introduction Ecological Network Analysis (ENA) is a mathematical methodology to study within system interactions for a given system structure (connectance pattern), function (flow regime), and boundary input (Higashi and Burns, 1991). One could say that ENA is a formal realization of synecology, which is mostly concerned with interrelations of material, energy and information among system components as opposed to autecology, which focuses on the individual organisms and populations themselves. * Corresponding author. Fax: C1 410 704 2405. E-mail address: bfath@towson.edu (B.D. Fath). 1364-8152/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.envsoft.2004.11.007 ENA starts with the assumption that a system can be represented as a network of nodes (vertices, compartments, components, storages, objects, etc.) and connections between them (arcs, links, flows, etc.). In ecological systems the connections are often based on the flow of conservative units such as energy, matter, or nutrients between the system compartments. If such a flow exists, then we say there is a direct transaction between the two connected compartments. These direct transactions give rise to both direct and indirect relations between all the objects in the system. Network analysis provides a system-oriented perspective because it is based on uncovering patterns and influences among all the objects in a system. Therefore, it gives a view on how components are tied to a larger web of interactions. 376 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 The intellectual lineage for Ecological Network Analysis comes from economics, which developed the ability to quantify indirect monetary flows in economic systems. Hannon (1973) first applied economic input– output analysis (Leontief, 1951, 1966) to investigate flow distribution in ecosystems. His models were linked by the energy flow through the food web and he pursued this line of research primarily to determine interdependence of organisms in an ecosystem based on their direct and indirect energy flows. Several formulizations of ENA have arisen including Embodied Energy Analysis (Herendeen, 1981, 1989), Ascendency Analysis (Ulanowicz, 1980, 1986, 1997) and Network Environ Analysis (Patten, 1978a, 1981, 1982, 1985, in preparation). A recent paper by Allesina and Bondavalli (2004) presented a user friendly, Windows-based version of Ascendency Analysis (Ulanowicz, 1982, 1986, 1997). That paper contributes an important role to increase the visibility and usability of that specific network methodology. In a similar vein, we hope to increase the exposure and facilitate the use of Network Environ Analysis by presenting a MATLABÒ function (m-file) to calculate the basic environ parameters and properties. 2. Network Environ Analysis There has been much confusion regarding environment as a concept and how one defines it. On the most basic level, environment is all that is external to an object. This has led to the standard object–environment duality where emphasis is placed on the direct flows that come into contact with the object. This has been carried so far as to consider all indirect flows as nonrelevant or historical, suggesting that including them would lead to an ‘‘infinite regress’’ (Mason and Langenheim, 1957). Another interpretation of environment, based on an object embedded in a system, includes the summary contribution of the within-system flows that affect that object. In other words, the environment of an object within a specified system (and all objects are parts of a system) can be refined to recognize the special relationship an object has with the other objects within the system boundary. For distinction, Patten (1978a) termed this second, within-system environment the object’s ‘‘environ.’’ An object’s environ ends at the system boundary. Objects and connections in the external environment, beyond the system boundary, are not distinguished so exchanges between them are not material to the analysis. Exchanges across the boundary of the system with the external environment are deemed inputs or outputs. One important aspect of environ theory is an explicit representation of the two environs, input and output, for each object. Furthermore, it is possible to quantify the environs, and thus, the direct and indirect effects between any two objects in the system. In principle, environ analysis can be applied to a system as simple as a pair of interconnected objects. In practice it has been most readily applied to models of entire ecosystems. Patten (1978a) introduced a systems theory of the environment and put forth three foundational tenets. First, each object has both an input environ, those flows introduced at the system boundary leading up to the object, and an output environ, those flows emanating from the object back to the other system objects before exiting at the system boundary. Second, the purpose of a system boundary is to provide a reference state for the system of study, without which environ analysis is impossible because the analysis collapses back down to the object’s boundary. A system boundary is necessary to distinguish between the system’s environment (the infinite regress) and its component objects’ environs (within system processes). The third key realization is that the individual environs (and the flow carried in each one) are unique such that the system comprises the set union of all environs of each orientation (input or output), which in turn partition the system level of organization. Network Environ Analysis has also been a fruitful way of investigating system level properties of ecosystems. In particular, a series of ‘‘network statistics’’ such as indirect effects ratio, homogenization, synergism, and mutualism have grown up around this analysis that express the role of each entity in a larger system. See Patten (in preparation), Fath and Patten (1999a), and Fath (2004a,b) for further details regarding these properties and the history of Network Environ Analysis. The purpose of this paper is to document software to calculate input and output environs and several of the other basic network properties. We first briefly describe the methodology and network properties. We then introduce the software, which is available from The MathWorks MATLABÒ Central File Exchange website (http://www.mathworks.com/matlabcentral/fileexchange /loadCategory.do) and reproduced in Appendix A. Appendix B gives the program output for a well-studied example of an oyster reef model (Dame and Patten, 1981). In Appendix C, we compile a glossary for the Network Environ Analysis notation and in Appendix D a glossary for common MATLAB notation. 3. Methodology 3.1. Terms and notations The software introduced in this paper summarizes algorithms for the well-documented methodology of Network Environ Analysis (Patten, 1978a, 1982, 1985, in preparation; Matis and Patten, 1981; Patten et al., 1990; Fath and Patten, 1999a). In Network Environ B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 Analysis, systems are partitioned into compartments and a conservative substance such as energy, nitrogen, or phosphorus is propagated mathematically through the interconnected network. Let i, j Z 1,., n represent n storage compartments (nodes) within an open physical system demarcated from its surroundings by a boundary across which conservative energy–matter is exchanged. The environment is traditionally indexed by 0, however, in the MATLABÒ code environment is indexed as n C 1 because the software does not allow a zero index. Within-system connections are expressed in an adjacency matrix, A Z (aij), corresponding to the model structure, where aij Z 1 if there is an observed flow from compartment j to compartment i, and aij Z 0 if there is no flow. Boundary transfers, zj0 (or just zj) Z input to j, y0i (or yi) Z output from i, and internal exchanges between compartments, fij Z flow directed from j to i, comprise a set of transactive flows, or transactions, meaning the transferred quantities are conserved. Let the ordered pairs (i, j ), ( j, 0) and (0, i) be flow arcs, j / i, 0 / j, and i / 0 carrying the corresponding flows fij, zj, and yi. Networks are a synthesis of such binary flows. Inputs (zj) or flows ( fij) retained in receiving compartments over time become storages (xi), and storage at i, say, released as flow to j or the environment, respectively, becomes flow ( fji) and output ( yi). The sum of flows into or out of the i-th compartment at any point and T(out) , given by: in time is throughflow, T(in) i i ðinÞ Ti Zzi C n X ðoutÞ fij and Ti jZ1 Z n X fji Cyi : jZ1 At steady state, compartmental inflows and outflows are equal such that dxi/dt Z 0, and therefore, incoming and Z T(out) h Ti. outgoing throughflows are equal: T(in) i i This notation is used to develop a structural analysis, and four functional analyses (flow, storage, utility, and control analysis); the later two functional analyses can be derived from either flows or storages. 3.2. Structural analysis Structural analysis provides important insight into the pattern and connectivity of a model. Path analysis is one type of structural analysis in Network Environ Analysis that enumerates pathways of various lengths between components and the rate at which the number of pathways increases as path length increases. The analysis is performed on the adjacency matrix, A Z (aij), of the model. It is a property of matrix multiplication that Am gives the number of paths of length m between two components, i and j in the model. In systems with feedback (i.e., all realistic ecological models), the number of pathways increases as m increases; therefore, 377 PN Am is a divergent series. This phenomenon is termed pathway proliferation in Network Environ Analysis. The proliferation rate is a significant system attribute because it describes the growth rate of indirect pathways, ultimately describing the number of pathways available for transactions and relations (Borrett and Patten, 2003). Like the population growth rate in a Leslie Matrix (Caswell, 2001), the pathway proC 1) (m) liferation rate (a(m /aij as m / N) is given by the ij largest eigenvalue of A (Fath, 1998). The software returns the adjacency matrix, the pathway proliferation rate, the number of network nodes (n), the proportion of direct connections completed or the network connectanceP (L/n2), and the link density (L/n), where LZ ni; jZ1 aij . mZ0 3.3. Functional analysis Throughflow, Storage, Utility, and Control are four functional analyses used in Network Environ Analysis, each providing different system insight. Throughflow analysis is similar to input–output analyses performed by other ENAs, but storage, utility, and control analyses are unique to Network Environ Analysis. Here, we describe storage analysis in detail, while the others follow as analogues. 3.3.1. Storage analysis Nondimensional, storage-specific, output-oriented, intercompartmental flows are given by pij Z cijDt, for i s j, where, cij Z fij/xj; and for i Z j, pii Z 1 C ciiDt, where cii Z
Ti/xi (Matis and Patten, 1981). The dimensional quantities cii and cij are elements of a Jacobian ‘‘community’’ matrix, C Z (cij). This matrix type is typically employed in population–community ecology for stability or food-web analyses. By introducing small enough time steps Dt into the ciiDt and cijDt values, dimensionless pii and pij quantities are obtained that lie in the range 0 % pii, pij % 1, and thus are interpretable as probabilities (Barber, 1978). From the matrix P Z ( pij) of such probabilities, a dimensionless integral storage intensity matrix Q Z (qij) can be computed as the convergent power series: QZP0 CP1 CP2 CP3 C/CPm C/ZðI
PÞ
1 ; ð1Þ where P0 Z I is the identity matrix. The m-th order terms, m Z 1, 2,., account for interflows over all pathways in the system of lengths m, cm. In so doing the network has a graph theory property known as transitive closure (Patten et al., 1976). Input-oriented, storage-specific, intercompartmental flows are given by p#ij Z c#ijDt, where for i s j, c#ij Z fij/xi, and for i Z j, p#ii Z 1 C c#iiDt, where c#ii Z
Ti/xi (MATLABÒ uses P# to denote the transpose of P, therefore we use an additional ‘‘P’’ to designate ‘‘prime’’ such that P# 378 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 becomes PP, and Q# becomes QP below, etc. in the code). Note the output-oriented values were normalized by the donating compartment storage, xj, and the inputoriented values by the receiving compartmental storages, xi. From the matrix P# Z ( p#ij), a dimensionless, integral, input-oriented, storage intensity matrix Q# Z (q#ij) can be computed:
0
1
2
3
m Q# Z P# C P# C P# C P# C/C P# C/

1 Z I
P# : ð2Þ The nondimensional output-oriented integral storage matrix Q in Eq. (1) can be redimensionalized by multiplying by the input vector, z and the time step, Dt, such that x Z QzDt; and the nondimensional inputoriented integral storage matrix Q# in Eq. (2) can be redimensionalized by pre-multiplying by the output vector, y and the time step, Dt, such that x Z yQ#Dt (order of multiplication for the scalar time step is irrelevant). In some cases, it is also useful to examine the dimensional form of the integral matrix, which we include as S Z QDt. The software returns C, C#, P, P#, Q, and Q#, S, and S#. 3.3.2. Throughflow analysis Similar to the case for storage analysis, nondimensional, output-oriented, intercompartmental flows are given by, gij Z fij/Tj; and, input-oriented, intercompartmental flows are given by, g#ij Z fij/Ti. The dimensionless gij and g#ij quantities lie in the range 0 % gij, g#ij % 1, and thus are interpretable as probabilities. From the matrices G Z ( gij) and G# Z ( g#ij), dimensionless integral output and input flow intensity matrices N Z (nij) and N# Z (n#ij) can be computed similar to Eq. (1) from the convergent power series: NZG0 CG1 CG2 CG3 C/CGm C/ZðI
GÞ
1 ; ð3Þ and
0
1
2
3
m N# Z G# C G# C G# C G# C/C G# C/

1 Z I
G# ; ð4Þ where G0 Z I is again the identity matrix, and the m-th order terms, m Z 1, 2,., account for interflows over all pathways in the system of lengths m. The nondimensional output-oriented integral flow matrix can be redimensionalized by multiplying by the input vector, z, such that T Z Nz and the nondimensional inputoriented integral flow matrix can be redimensionalized by pre-multiplying by the output vector, y, such that T Z yN#. The software returns G, G#, N, and N# as well as several system-level summary flow parameters. 3.3.3. Utility analysis Intercompartmental flow utilities are given by, dij Z ( fij
fji)/Ti. The dimensionless dii quantities lie in the range
1 % dii % 1, and thus are not interpretable as probabilities. They do, however, conform to the requirements of a convergent series so long as the magnitude of the largest eigenvalue is less than one. Therefore, a test must be executed before the flow utility power series is taken. Networks found not to fulfill the convergence property are excluded from the utility analysis and flagged in the software output as
9999. From the matrix D Z (dij), a dimensionless integral utility intensity matrix U Z (uij) can be computed:
1 UZD0 CD1 CD2 CD3 C/CDm C/ZðI
DÞ : ð5Þ Note for completeness direct storage utilities could be derived from dsij Z ( fij
fji)/xi and integral storage utilities given by the following power series (convergence restrictions apply): USZDS0 CDS1 CDS2 CDS3 C/CDSm C/
1 ZðI
DSÞ ; ð6Þ however, this parameter has not been thoroughly investigated or presented in the NEA literature. The nondimensional integral flow and storage utility matrices can be redimensionalized by multiplying by the diagonalized throughflow vector, Ť, such that Y Z UŤ and YS Z USŤ. The software returns D, DS, U, US, Y, and YS. 3.3.4. Control analysis Patten (1978b) introduced a Network Environ Analysis based measure of control or dominance (see also Patten and Auble, 1981; Patten, 1982; Fath, 2004b). The measure, expressed in a matrix CN Z (cnij), is based on the ratio of integral flow from compartment j to i to the integral flow from i to j. Compartment j is said to dominate i if its output environ effect on i is larger than is input environ effect on j (cnij Z nij/n#ji O 1). This control relationship was further modified such that when nij/n#ji ! 1, cnij Z 1
nij/n#ji otherwise cnij Z 0. The storage version of this measure is CQ, where cqij Z qij/ q#ji. The storage-based dominance relationships are always identical to the throughflow-based measures because intensive throughflows and storages are related by the compartment turnover ratesdwhich cancel out in the ratio measure. NEA.m calculates both CN and CQ. 4. Network and environ properties In this section, we introduce the specific properties the software returns. A listing and brief description of B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 379 Table 1 Network parameters and environ properties returned by MATLABÒ function in ep30 ! 1 vector # Abbreviation Short description [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] ‘# nodes, n’ ‘# links, L’ ‘connectance, L/n2’ ‘link density, L/n’ ‘path proliferation’ ‘TST’ ‘Cycling Index (T )’ ‘MODE_0 boundary’ ‘MODE_1, 1st pass’ ‘MODE_2, cycled’ ‘MODE_3, dissipative’ ‘MODE_4, boundary’ ‘Amp (T, output)’ ‘Amp (T, input)’ ‘I/D (T, output)’ ‘I/D (T, input)’ ‘Homog(T,output)’ ‘Homog(T,input)’ ‘Aggradation’ ‘Cycling Index (S )’ ‘Amp (S, output)’ ‘Amp (S, input)’ ‘I/D (S, output)’ ‘I/D (S, input)’ ‘Homog(S,output)’ ‘Homog(S,input)’ ‘Synergism(T )’ ‘Mutualism(T )’ ‘Synergism(S )’ ‘Mutualism(S )’ number of nodes or compartments number of direct flows or arcs connectance link density l1 ðAÞZrate of pathway proliferation ðdominant eigenvalue of AÞ total system throughflow cycling index for throughflow Z TSTc/TST boundary input first-passage flow cycled flow last passage dissipative flow boundary output network amplification (throughflow, output) network amplification (throughflow, input) indirect-to-direct effects ratio or network nonlocality (throughflow, output) indirect-to-direct effects ratio or network nonlocality (throughflow, input) network homogenization (throughflow, output) network homogenization (throughflow, input) P network aggradation Z TST/ z Z average path length cycling index calculated for storage analysis network amplification (storage, output) network amplification (storage, input) indirect-to-direct ratio or network nonlocality (storage, output) indirect-to-direct ratio or network nonlocality (storage, input) network homogenization (storage, output) network homogenization (storage, input) benefit–cost ratio or network synergism (throughflow) positive to negative interaction ratio or network mutualism (throughflow) benefit–cost ratio or network synergism (storage) positive to negative interaction ratio or network mutualism (storage) each of these is given in Table 1. The formulas can be found in the code in Appendix A. 4.1. Unit environ analysis The first property is the quantitative environ, both in the input and output orientation. Since each compartment has two distinct environs there are 2n environs in total in an n-compartment system. The output environ Ek Z (eijk) (i Z 1, 2,., n C 1, j Z 1, 2,., n C 1, k Z 1, 2,., n) for the k-th compartment is calculated by multiplying G times the diagonalized matrix of the k-th column of N minus the diagonal of the k-th column of N, such that eijk Z gij ! diag(nik)
diag(nik) for i Z 1, 2,., n, j Z 1, 2,., n. In MATLAB code: E(1:n,1:n,k) Z G ! diag(N(:,k))
diag(N(:,k)); when constructed in the manner, elements on the principle diagonal of Ek are the negative throughflows of each compartment generated by the unit input. The column sums of Ek (i Z 1, 2,., n, j Z 1, 2,., n) give the negative output vector which is multiplied by
1 and incorporated into Ek as row n C 1, and row sums the negative input vector which is multiplied by
1 and incorporated into Ek as column n C 1. When assembled, the result is the output oriented flow from each compartment to each other compartment in the system and across the system boundary. In a similar fashion, input environs are calculated by multiplying the diagonalized matrix of the k-th column of N# by G# minus the diagonal of the k-th column of N# (in code: EP (1:n,1:n,i) Z diag(NP(k,:)) ! GP
diag(NP(k,:))). Input (SE#) and output (SE) oriented storage environs are calculated in analogous fashion using P, Q, P#, and Q# (Matis and Patten, 1981). These results comprise the foundation of Network Environ Analysis since they allow for the quantification of all within system interactions, both direct and indirect, on a compartment-by-compartment basis. In the software, E, E#, SE, and SE# are returned as n C 1 ! n C 1 ! n arrays. The n C 1 row is the unit output vector and the n C 1 column is the unit input vector for each n environs. 4.2. Indirect to direct effects measure This property is one of the most important since it compares the strength of indirect (non-touching) flow in the compartment’s environ to the direct flow. Indirect effects are calculated as the integral contributions minus the direct and initial boundary input. The indirect to direct effects ratio is a measure of the relative strength of 380 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 these two factors. Mathematically, this is simply the following ratio  
P throughflow
P for P P the
output oriented case: I=DZ niZ1 njZ1 nij
iij
gij = niZ1 njZ1 gij . When the ratio is greater than one, indirect flows are greater than direct flows. This ratio can be calculated for the input and output oriented throughflow and storage analyses (respectively, these are denoted as I/D(T,in), I/D(T,out), I/D(S,in), I/D(S,out)). Analysis of many models has shown that these ratios are often greater than one, indicating the non-intuitive result that indirect effects have greater contribution than direct effects (Higashi and Patten, 1989). This is core evidence for the rationale behind systems modeling and systems perspective because it states that indirect effects are greater, therefore exerting greater dominance, than direct effects in a network. This clearly has implications for understanding feedback and direct versus indirect control in networks. 4.3. Network homogenization The homogenization property yields a comparison of resource distribution between the direct and integral flow intensity matrices. Due to the contribution of indirect pathways, flow in the integral matrix tends to be more evenly distributed than that in the direct matrix. A statistical comparison of the distributions can be made by calculating the coefficient of variation of the direct and integral matrices (Fath and Patten, 1999b). For example, the coefficient of variation of the direct flow intensity matrix G is given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n P P 2 ðgij
gij Þ from j to i, rarely are, but can exceed one when cycling drives more than the equivalent of one unit of input flow over the pathways linking those two compartments. If any off-diagonal element of N (Q) is greater than one, then amplification is said to occur because a virtual input or output of one is implied by the nondimensional analysis. This measure can be applied to both the input and output orientations, and is notated as Amp(T,in), Amp(T,out), Amp(S,in), Amp(S,out). 4.5. Network synergism Synergism implies that positive utility exceeds negative utility in the system. Utility is the throughflow or storage scaled value of net transactions between entity pairs. To determine whether this occurs a comparison is made between positive and negative utilities of the dimensionalized integral utility matrix, Y Z UŤ, which quantifies the magnitude of the positive and negative utilities. Synergism is said to occur when the magnitude of positive utility exceeds the magnitude of negative utility, which is the same as saying the ratio of the positive to negative utility exceeds one (Patten, 1991; Fath and Patten, 1998). For completeness, we include the application of this measure to the storage case, though this parameter has not been thoroughly investigated nor previously presented in the literature. These ratios are denoted as Synergism(T ) and Synergism(S ). 4.6. Network mutualism jZ1 iZ1 CVðGÞZ ðn2
1Þ gij ; ð7Þ where gij is the mean of the elements of G. Network homogenization occurs in the output oriented throughflow case when the coefficient of variation of N is less than the coefficient of variation of G because this indicates that the network flow is more evenly distributed in the integral matrix. The test statistic employed here looks at whether the ratio CV(G)/CV(N) exceeds one. In a similar fashion, this measure can be applied to the input throughflow and the input and output oriented storage cases. These measures are denoted as Homog(T,in), Homog(T,out), Homog(S,in), Homog(S,out). 4.4. Network amplification The amplification property deals explicitly with the values in the integral flow or storage matrices. Diagonal elements of N (Q) are almost always great than one. Off-diagonal elements, representing the integral flow In addition to quantifying the direct and indirect relations the utility matrix can be used to determine qualitative relations between any two components in the network such as predation, mutualism, or competition. Entries in the direct utility matrix, D, or integral utility matrix, U, can be positive or negative (
1 % dij, uij % 1). The elements of D represent the direct relation between that (i,j ) pairing and the elements of U the integral relations, respectively (Patten, 1991; Fath and Patten, 1998). The direct matrix D, being zero-sum between complementary pairs dij and dji, always has the same number of positive and negative signs. Signs in the integral matrix, U, are determined by the entire web of system interactions. If there are more positive signs than negative signs in the integral utility matrix, then network mutualism is said to occur. Network mutualism reveals the preponderance of positive mutualistic relations in the system. Again for completeness, we include the application of this measure to the storage case, though it has not been thoroughly investigated nor previously presented in the NEA literature. These ratios are denoted as Mutualism(T ) and Mutualism(S ). B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 4.7. Mode partitioning Flow (and flow derived storage) into and out of a specific compartment can be partitioned into five categories or modes: (0) boundary input, (1) first passage, (2) cycled, (3) dissipative, and (4) boundary output, depending on its position relative to the focal compartment. Because this is a partition the modes are mutually exclusive and exhaustive (Fath et al., 2001). Boundary input is flow that starts from the environment and crosses the system boundary into a compartment within the system. It is calculated as Iz, where I is the identity matrix. First passage flow, or mode 1, is flow from any compartment that reaches another (focal) compartment for the very first time. Note, since this is compartment specific, flow cycled between other compartments en route to the focal compartment for the first time is still considered first passage. For example, firstpassage flow to compartment k from i could have traveled the following path: i / j / i / j / k. Cycled flow, or mode 2, is calculated using a derivation of the Finn (1976, 1978) cycling index and represents the amount of flow that has exited a compartment but will return again to that same compartment before being lost from the system such that the compartment in question is both the originating and terminating node for that pathway. Dissipative, mode 3 flow has left the focal compartment never to return again, although it passes through other compartments before crossing the system boundary. Boundary output, mode 4, is flow that exits the system boundary directly from the focal compartment in question. Mode partitioning is described more fully by Higashi et al. (1993) and Fath et al. (2001). 5. Software NEA.m is a MATLABÒ function created to rapidly perform NEA on flow-storage network models (available from http://www.mathworks.com/matlabcentral/ fileexchange/loadCategory.do and in Appendix A). The function implements algorithms for all analyses described in Sections 3 and 4. It requires one input variable, DnC1!nC2 , that summarizes the flow and storage information for the system of interest. The function returns a vector of the system-level environ properties (ep30!1 ) to the workspace (Table 1), displays the comprehensive list of analytical results in the command window, and saves all results as a MATLABÒ data file called NEA_output.mat. 5.1. Input data The input argument DnC1!nC2 is an (n C 1) ! (n C 2) composite matrix that summarizes the system information of the flow–storage network to be analyzed, 381 where n is the number of compartments or nodes of the network. The composite input matrix is:   Fn!n zn!1 xn!1 ; DZ y1!n 0 0 nC1!nC2 where Fn!n is the steady-state intercompartmental flow matrix, zn!1 is the steady-state boundary input, y1!n is the steady-state boundary output, and xn!1 is the storage value. D is oriented such that flows are from columns to rows. In its present form, Network Environ Analysis makes two critical assumptions about the input data. First, flows and storages must be measured in a consistent conservative energy–matter unit. For example, all fluxes could have units of g C m
2 y
1 or mg P cm
3 d
1, while the corresponding storages would be g C m
2 or mg P cm
3. Second, the data must represent a static, steady-state system (T(in) Z T(out)). When implemented, NEA.m checks whether the model meets a steady-state requirement (defined as within a certain tolerance of 0.05% of throughflow at each node). If the model does not meet this requirement, then a warning is given and the analysis does not proceed. It would be necessary to more accurately balance the network flows; algorithms for this are available (Savenkoff et al., 2001; Allesina and Bondavalli, 2003). The static, steady-state assumption is a limitation of the methodology because few ecological systems exist in this condition. Despite this limitation, important insights emerge that appear to challenge conventional ecological theorydsuch as energy cycling (Patten, 1985) and the dominance of indirect effects (Higashi and Patten, 1989; Patten, in preparation). Nonetheless, further work, like that initiated by Hippe (1983), to develop a dynamic Network Environ Analysis, is needed. The input variable can be coded as a MATLABÒ function so that system data do not need to be reentered multiple times. An example data function for an oyster reef model (Dame and Patten, 1981) is included with NEA.m. 5.2. Implementing NEA.m and data output Once NEA.m is installed in the MATLABÒ operating directory, the function can be implemented on D by typing ‘‘ep Z NEA(D);’’ in the command window. Resultant matrices from structural, throughflow, utility, unit environ analyses as well as a table of the system-level environ indices will be displayed in the command window. The system-level indices are also returned to the workspace as the vector ‘ep’ (Table 1). All results are stored in the MATLABÒ data file ‘NEA_output.mat’. This file can be loaded into the workspace, giving the user access to all resultant matrices for additional investigation and manipulation. Typing ‘‘ep Z NEA(D,0);’’ will 382 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 prevent the results from displaying in the command window. While all results displayed in the command window can by cut and pasted into word processing or spreadsheet programs, there is a simple way to capture the results in an ASCII text file using the diary function in MATLABÒ. To use this function, type diary (‘file_name.txt’) in the command window, where ‘file_name’ is the name of the output file you are creating. Then, run NEA.m by typing ‘‘NEA(D);’’ or ‘‘ep Z NEA(D);’’. When the computations are complete, type ‘‘diary off’’ to turn off the diary function. This file can then be opened in any text editor. Example NEA_output.mat and diary files for the Oyster Reef Model (Dame and Patten, 1981) are included with the software and in Appendix B. 6. Conclusion Network Environ Analysis is one branch of Ecological Network Analysis. It is a powerful tool for investigating the within-system transactions and relations in ecological systems. The software presented herein can be used to calculate the primary parameters and properties of Network Environ Analysis. The analysis itself is not computationally challenging, but does require some familiarity with matrix algebra and graph theory concepts. The software compiles the algorithms and should facilitate use of the methodology. Network Environ Analysis is an active area of research such that not all of its facets could be included here. Also, while the software provides the quantitative results, the challenging task of interpretation is left to the user. Previous applications of Network Environ Analysis (Matis and Patten, 1981; Patten and Matis, 1982; Flebbe, 1983; Patten, 1983, in preparation) may be useful guides for interpretation. It is our intention that dissemination of this software will encourage others to look more closely at the environ methods and be aided in applying them in their own research. Acknowledgements The computational methods and algorithms for the mathematical system theory of environment that we summarize and code here were developed through collaboration between several people over many years. We wish to acknowledge Bernard C. Patten who has served as the chief orchestrator and the following collaborators: M. Craig Barber, Robert W. Bosserman, Thomas P. Burns, John T. Finn, Masahiko Higashi, James Hill, IV, James H. Matis, and Stuart J. Whipple. This manuscript benefited from comments by Bernard C. Patten and anonymous reviewers. SRB was supported in part by a grant from the National Science Foundation (OPP-00-83381). B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 Appendix A. A function for Network Environ Analysis expressed in MATLAB notation 383 384 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 385 386 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 387 388 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 389 390 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 391 392 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 Appendix B. Results from NEA.m analysis of Oyster Reef Model B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 393 394 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 395 396 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 397 398 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 399 400 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 Appendix C. Glossary of Network Environ Analysis notation 401 402 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 Appendix D. A glossary of primary MATLAB notation used in Appendix A 403 404 B.D. Fath, S.R. Borrett / Environmental Modelling & Software 21 (2006) 375–405 References Allesina, S., Bondavalli, C., 2003. 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