Spatial Price Equilibrium and Food Webs: The Economics of Predator-Prey Networks

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1 Spatal Prce Equlbrum and Food Webs: The Economcs of Predator-Prey Networks Proceedngs of the 2011 IEEE Internatonal Conference on Supernetworks and System Management F.-Y. Xu and J. Dong, Edtors, IEEE Press, Bejng, Chna, pp 1-6. Anna Nagurney Department of Fnance and Operatons Management Isenberg School of Management Unversty of Massachusetts Amherst, MA 01003, USA Emal: Ladmer S. Nagurney Department of Electrcal and Computer Engneerng Unversty of Hartford West Hartford, CT 06117, USA Emal: Abstract In ths paper, we prove that the equlbrum of predator-prey networks s, n fact, a spatal prce equlbrum. Ths result demonstrates the underlyng economcs of predatorprey relatonshps and nteractons and provdes a foundaton for the formulaton and analyss of complex food webs, whch are nature s supply chans, through the formalsm of network equlbrum. Moreover, t rgorously lnks the equlbrum condtons of commodty networks n whch a product s produced, transported, and consumed, wth those of ecologcal networks n whch prey are consumed by predators. Index Terms spatal prce equlbrum, supply chans, food webs, predator prey models, food chans, networks, network economcs, economcs of bologcal systems, ecologcal networks, network equlbrum, regonal scence, operatons research, transportaton, supernetworks I. INTRODUCTION Equlbrum s a central concept n numerous dscplnes from economcs and regonal scence to operatons research / management scence and even n ecology and bology. Examples of specfc equlbrum concepts nclude the wellknown Walrasan prce equlbrum n economcs, Wardropan (traffc network) equlbrum n transportaton scence, and Nash equlbrum n game theory [1]. In ecology, equlbrum s n concert wth the balance of nature, n that, snce an ecosystem s a dynamcal system, we can expect there to be some persstence or homeostass n the system [2], [3], [4]. Moreover, equlbrum serves as a valuable paradgm that asssts n the evaluaton of the state of a complex system. Equlbrum, as a concept, mples that there s more than a sngle decson-maker or agent, who, typcally, seeks to optmze, subject to the underlyng resource constrants. Hence, the formulaton, analyss, and soluton of such problems may be challengng. Notable methologes that have been developed over the past several decades that have been successfully appled to the analyss and computaton of solutons to a plethora of equlbrum problems nclude varatonal nequalty theory and the accompanyng theory of projected dynamcal systems ([1], [5] and the references theren). Fascnatngly, t has now been recognzed that numerous equlbrum problems as vared as the classcal Walrasan prce equlbrum problem, the classcal olgopoly problem, the portfolo optmzaton problem, and even mgraton problems [6], whch n ther orgnal formulatons dd not have a network structure dentfed, actually possess a network structure. In addton, such well-recognzed network equlbrum problems as traffc network equlbrum problems wth applcatons to congeston management on urban roads as well as to ar traffc, and even to the Internet [7], as well as spatal prce equlbrum problems, ([8], [9], [10]) also have an underlyng network structure (wth nodes correspondng to locatons n space). Furthermore, t has now been establshed, through the supernetwork [11] formalsm that even supply chan network problems, n whch decson-makers (be they manufacturers, retalers, or consumers at demand markets) compete across a ter, but necessarly cooperate (to varous degrees) between ters, can be reformulated and solved as (transportaton) network equlbrum problems. The same holds for complex fnancal networks wth ntermedares [12]. In addton, the supernetwork framework has even been appled to the ntegraton of socal networks wth supply chans [13] and wth fnancal networks [14]. Hence, t s becomng ncreasngly evdent that seemngly dsparate equlbrum problems, n a varety of dscplnes, can be unformly formulated and studed as network equlbrum problems. Such dentfcatons allow one to: 1. graphcally vsualze the underlyng structure of systems as networks; 2. aval oneself of exstng frameworks and methodologes for analyss and computatons, and 3. gan nsghts nto the commonalty of structure and behavor of dsparate complex systems that underly our economes and socetes. Nevertheless, although deep connectons and equvalences have been made (and contnue to be dscovered) between/among dfferent systems through the (super)network formalsm, the systems studed, to-date, have been exclusvely

2 Prey Predators 1... m j n Fg. 1. The bpartte network wth drected lnks representng the predatorprey problem of a soco-techncal-economc varety. In ths paper, we take on the challenge of provng the equvalence between ecologcal food webs and spatal prce equlbrum problems; thereby, provdng a foundaton for the unfcaton of these dsparate systems and, n a sense, we brng the felds of economcs (and operatons research and regonal scence) closer to ecology (and bology). Ths paper s organzed as follows. In Secton II, we brefly recall the predator-prey model of [4], whch serves as the bass for the equvalence. In Secton III, we establsh the equvalence between predator-prey equlbrum and spatal prce equlbrum. In Secton IV, we develop extensons and propose a dynamc adjustment process, along wth stablty analyss results. Secton V presents numercal examples, whereas Secton VI contans a summary and suggestons for future research. II. THE PREDATOR-PREY MODEL In ths Secton, we brefly revew the predator-prey model [4], whose structure s gven n Fgure 1. We consder an ecosystem n whch there are m dstnct types of prey and n dstnct typed of predators wth a typcal prey speces denoted by and a typcal predator speces denoted by j. The bomass of a speces s denoted by B ; = 1,..., m. E denotes the nflow (energy and nutrents) of speces wth the autotroph speces, that s, the prey, n Fgure 1, havng postve values of E ; = 1,..., m, whereas the predators have E j = 0; j = 1,..., n. The parameter γ denotes the trophc assmlaton effcency of speces and the parameter µ denotes the coeffcent that relates bomass to somatc mantenance. The varable X j s the amount of bomass of speces preyed upon by speces j and we are nterested n determnng ther equlbrum values for all prey and predator speces pars (, j). The prey equatons that must hold are gven by: γ E = µ B + X j, 1,..., m. (1) Equaton (1) means that for each prey speces, the assmlated bomass must be equal to ts somatc mantenance plus the amount of ts bomass that s preyed upon. The predator equatons, n turn, are gven by: m γ j X j = B j, j = 1,..., n. (2) Equaton (2) sgnfes that for each predator speces j, ts assmlated bomass s equal to ts somatc mantenance (whch s represented by ts coeffcent tmes ts bomass). Equatons (1) and (2) may be nterpreted as the conservaton of flow equatons, n network parlance, from a bomass perspectve. In addton, there s a parameter φ j ; 1,..., m; j = 1,..., n, whch reflects the dstance (note the spatal component) between dstrbuton areas of prey and predator j, wth ths parameter also capturng the transacton costs assocated wth handlng and ngeston. Accordng to [4], the predaton cost between prey and predator j, denoted by F j, s gven by: F j = φ j κ B + λ j B j, = 1,..., m; j = 1,..., n, (3) where κ B represents the easness of predaton due to the abundance of prey B and λ j B j denotes the ntra-specfc competton of predator speces j. We group the speces bomasses and the bomass flows ntro the respectve m + n and mn dmensonal vectors B and X. Defnton 1: Predator-Prey Equlbrum Condtons A bomass and flow pattern (B, X ), satsfyng constrants (1) and (2), s sad to be n equlbrum f the followng condtons hold for each par of prey and predators (, j); = 1,..., m; j = 1,..., n: F j { = 0, f X j > 0, 0, f X j = 0. (4) These equlbrum condtons reflect that, f there s a bomass flow from to j, then there s an economc balance between the advantages (κ B ) and the nconvenences of predaton (φ j + λ j B j ). Observe that, n vew of (1), (2), and (3), we may wrte F j = F j (X),, j. Clearly, the predator-prey equlbrum condtons (4) may be formulated as a varatonal nequalty problem, as gven below. Theorem 1: Varatonal Inequalty Formulaton of Predator-Prey Equlbrum A bomass flow pattern X R+ mn s an equlbrum accordng to Defnton 1 f and only f t satsfes the varatonal nequalty problem: κ µ X j κ µ γ E + φ j + λ jγ j X j [ X j Xj ] 0, X R mn +. (5)

3 Proof: Note that, by makng use of (1), (2), and (3): F j (X) = κ X j κ γ E + φ j + λ jγ j X j. (6) µ µ We frst establsh necessty. From (4) we have that κ Xj κ γ E + φ j + λ jγ j X j µ µ [ X j X j] 0, Xj 0, (7) snce, ndeed, f Xj > 0, then the left-hand-sde of nequalty (7) pror to the multplcaton sgn s zero, snce the equlbrum condtons (4) are assumed to hold, and, hence, the nequalty n (7) holds; on the other hand, f Xj = 0, then both the expresson before the multplcaton sgn n (7) (due to the equlbrum condtons) s nonnegatve as s the one after the multplcaton sgn (due to the assumpton of the nonnegatvty of the bomass flows), and the result n (7) also follows. Summng now (7) over all prey speces and over all predator speces j yelds the varatonal nequalty (5). In order to prove suffcency, we proceed as follows. Assume that varatonal nequalty (5) holds. Set X kl = Xkl for all kl j and substtute nto (5), whch yelds: κ Xj κ γ E + φ j + λ jγ j X j µ µ [ X j X j] 0, Xj 0, (8) from whch equlbrum condtons (4) follow wth note of (6). III. THE EQUIVALENCE BETWEEN PREDATOR PREY PROBLEMS AND SPATIAL PRICE EQUILIBRIA As noted n [1], the concept of a network n economcs was mplct as early as n the classcal work of Cournot [15], who not only seems to have frst explctly stated that a compettve prce s determned by the ntersecton of supply and demand curves, but had done so n the context of two spatally separated markets n whch the cost of transportng the good between markets was consdered. Samuelson [8] provded a rgorous mathematcal formulaton of the spatal prce equlbrum problem and explctly recognzed and utlzed the network structure, whch was bpartte. In spatal prce equlbrum problems, unlke classcal transportaton problems, the supples and the demands are varables, rather than fxed quanttes. The work was subsequently extended by [9] and others (cf. [16], [17], [10], [1], and the references theren) to nclude, respectvely, multple commodtes, and asymmetrc supply prce and demand functons, as well as other extensons, made possble by such advances as quadratc programmng technques, complementarty theory, as well as varatonal nequalty theory (whch allowed for the formulaton and soluton of equlbrum problems for whch no optmzaton reformulaton of the governng equlbrum condtons was avalable). We now brefly recall the spatal prce equlbrum problem. For a varety of spatal prce equlbrum models, we refer the nterested reader to [1]. There are m supply markets and n demand markets nvolved n the producton / consumpton of a homogeneous commodty. Denote a typcal supply market by and a typcal demand market by j. Let s denote the supply of the commodty assocated wth supply market and let π denote the supply prce of the commodty assocated wth supply market. Let d j denote the demand assocated wth demand market j and let ρ j denote the demand prce assocated wth demand market j. Group the supples nto the vector s R m and the demands nto the vector d R n. Let Q j denote the nonnegatve commodty shpment between the supply and demand market par (, j) and let c j denote the nonnegatve unt transacton cost assocated wth tradng the commodty between (, j). Assume that the transacton cost ncludes the cost of transportaton. Group the commodty shpments nto the vector Q R+ mn. The followng feasblty (conservaton of flow) equatons must hold: for every supply market and each demand market j: and s = d j = Q j, = 1,..., m, (9) Q j, j = 1,..., n. (10) Equatons (9) and (10) reflect that the markets clear and that the supply at the supply market s equal to the sum of the commodty flows to all the demand markets. Also, the demand at each demand market must be satsfed by the sum of the commodty shpments from all the supply markets. Defnton 2: Spatal Prce Equlbrum The spatal prce equlbrum condtons, assumng perfect competton, take the followng form: for all pars of supply and demand markets (, j) : = 1,..., m; j = 1,..., n: π + c j { = ρj, f Q j > 0 ρ j, f Q j = 0. (11) The spatal prce equlbrum condtons (11) state that f there s trade between a market par (, j), then the supply prce at supply market plus the unt transacton cost between the par of markets must be equal to the demand prce at demand market j n equlbrum; f the supply prce plus the transacton cost exceeds the demand prce, then there wll be no shpment between the supply and demand market par. Let K denote the closed convex set where K {(s, Q, d) Q 0, (9) and (10) hold}. The supply prce, demand prce, and transacton cost structures are now dscussed. Assume that, for the sake of generalty, the supply prce assocated wth any supply market may depend upon the supply of the commodty at every supply market, that s, π = π (s), = 1,..., m, (12)

4 where each π s a known contnuous functon. Smlarly, the demand prce assocated wth a demand market may depend upon, n general, the demand of the commodty at every demand market, that s, ρ j = ρ j (d), j = 1,..., n, (13) where each ρ j s a known contnuous functon. The unt transacton cost between a par of supply and demand markets may, n general, depend upon the shpments of the commodty between every par of markets, that s, c j = c j (Q), = 1,..., m; j = 1,..., n, (14) where each c j s a known contnuous functon. In the specal case where the number of supply markets m s equal to the number of demand markets n, the transacton cost functons (14) are assumed to be fxed, and the supply prce functons and demand prce functons are symmetrc,.e., π s k = π k s, for all = 1,..., n; k = 1,..., n, and ρj d l = ρ l d j, for all j = 1,..., n; l = 1,..., n, then the above model wth supply prce functons (12) and demand prce functons (13) collapses to a class of sngle commodty models ntroduced n [9] for whch an equvalent optmzaton formulaton exsts. We now present the varatonal nequalty formulaton of the equlbrum condtons (11). Theorem 2: Varatonal Inequalty Formulaton of Spatal Prce Equlbrum A commodty producton, shpment, and consumpton pattern (s, Q, d ) K s n equlbrum accordng to Defnton 2 f and only f t satsfes the varatonal nequalty problem: π (s ) (s s ) + c j (Q ) (Q j Q j) ρ j (d ) (d j d j ) 0, (s, Q, d) K. (15) Proof: See [1]. We now establsh our man result. Theorem 3: Equvalence Between Predator-Prey Equlbra and Spatal Prce Equlbra An equlbrum bomass flow pattern satsfyng equlbrum condtons (4) concdes wth an equlbrum commodty shpment pattern satsfyng equlbrum condtons (11). Proof: We establsh the equvalence by utlzng the respectve varatonal nequaltes (5) and (15). Frst, we note that (5) may be expressed as: determne X R+ mn such that κ Xj κ γ E X j X j µ µ + φ j (X j Xj) + λ j γ j Lettng now: X j [ m X j Q j X j,, j, X j ] 0, X R mn +. (16) t follows then that s = n Q j = n X j and d j = m Q j = X j, for all, j, n whch case we may rewrte (16) as: determne (s, Q, d ) K such that [ κ s κ ] γ E [s s ] + φ j (Q j Q µ µ j) + and [ ] λj γ j d j [ d j d ] j 0, (s, Q, d) K. (17) Lettng now: π (s) κ µ s κ µ γ E, = 1,..., m; (18) c j (Q) φ j, = 1,..., m; j = 1,..., n; (19) ρ j (d) λ jγ j d j, j = 1,..., n, (20) we conclude that, ndeed, a bomass equlbrum pattern concdes wth a spatal prce equlbrum pattern. The above equvalence provdes a novel nterpretaton of the predator-prey equlbrum condtons n that there wll be a postve flow of bomass/commodty from a supply market (prey speces) to a demand market (predator speces) f the supply prce (or value of the bomass/commodty) plus the unt transacton cost s equal to the demand prce that consumers (predators) are wllng to pay. Interestngly, the predator-prey model on a bpartte network proposed by [4] s actually a classcal one n that, from a spatal prce equlbrum perspectve, the supply prce at a supply market depends only upon the supply of the commodty at the market; the same for the demand markets. Moreover, the unt transacton/transportaton cost between a par of supply and demand markets s assumed to be ndependent of the flow. Hence, for ths specfc food web model there s an optmzaton reformulaton of the governng equlbrum condtons. Wth the above connecton, we can now transfer the numerous specal-purpose algorthms that are avalable for the soluton of spatal prce equlbra, and whch effectvely explot the underlyng network structure, for the computaton of predator prey bomass equlbra. Moreover, snce spatal prce equlbrum problems can be transformed nto transportaton network equlbrum problems [18] further theoretcal and practcal results can be expected. For completeness, we now provde an alternatve varatonal nequalty to (15) whch captures product dfferentaton n predator-prey networks. Specfcally, we defne dfferentated demand prce functons ρ j, whch reflect the demand prce

5 assocated wth demand (predator) market j for supply (prey) market, such that ρ j (Q) λ jγ j Q j + κ γ E,, j. (21) µ The followng result s mmedate, wth notce to (5), (21), and that Q j X j,, j, and wth π (s) κ µ s, : Corollary 1: Alternatve Varatonal Inequalty Formulaton of Predator-Prey Equlbrum as a Network Equlbrum wth Product Dfferentaton An equlbrum bomass flow pattern satsfyng equlbrum condtons (4) concdes wth an equlbrum commodty shpment pattern wth dfferentated product prces wth the varatonal nequalty formulaton: determne (s, Q ) wth Q R+ mn and (9) satsfed, such that π (s ) [s s ] + [φ j ρ j (Q )] [ Q j Q j] 0, (s, Q) such that Q R mn + and (9) holds. (22) IV. MODEL EXTENSIONS Through the equvalences establshed n Secton III, many possbltes exst for extendng the fundamental network economcs model(s) of food webs (ncludng the predator-prey model recalled n Secton II) presented n [4]. Specfcally, we propose that the unt transacton costs, the φ j s, need no longer be fxed, but can be flow-dependent, and monotone ncreasng, so that competton assocated wth foragng can also be captured. Of course, one may also generalze the correspondng bomass functons to correspond to nonlnear supply prce and demand prce functons and to also generalze the unt transacton cost functons to be nonlnear. Such general spatal prce equlbrum models [1] already exst and the methodologes can then be appled to ecologcal predator prey network systems. In addton, we beleve that general food web models can be reformulated and solved as spatal prce equlbrum problems on more general networks as n [10]. Fnally, we note that, due to the varatonal nequalty formulaton (15), we may explot the connecton between sets of solutons to varatonal nequalty problems and sets of statonary ponts of projected dynamcal systems. In so dong, a natural dynamc adjustment process becomes: Q j = max{0, ρ j (d) c j (Q) π (s))},, j. (23) Lettng ˆF j =π (s) + c j (Q) ρ j (d),, j, we can wrte the followng pertnent ordnary dfferental equaton (ODE) for the adjustment process of commodty (bomass) shpments n vector form as [5]: Q = Π K (Q, ˆF (Q)), (24) where ˆF s the vector wth components ˆF j ; = 1,..., m; j = 1,..., n and where (P K (x + δv) x) Π K (x, v) = lm, (25) δ 0 δ P K (x) = arg mn x z. (26) z K We now present a stablty result (see [5]) snce, due to the equvalence establshed between the two network systems, ts relevance to predator-prey problems s notable. Theorem 4 Suppose that (s, Q, d ) s a spatal prce equlbrum accordng to Defnton 2 and that the supply prce functons π, the transacton cost functons c, and the negatve demand prce functons ρ are (locally) monotone, respectvely, at s, Q, and d. Then (s, Q, d ) s a globally monotone attractor (monotone attractor) for the adjustment process solvng ODE (24). Stronger results, ncludng stablty analyss results, can be obtaned under strct as well as strong monotoncty of these functons, wth the latter guaranteeng both exstence and unqueness of the soluton (s, Q, d ) to (15). We explot the above connecton through our numercal procedure n the next secton where we provde numercal examples. Of course, a dynamc adjustment process, analogous to (23), can be constructed for varatonal nequalty (22). V. NUMERICAL EXAMPLES In ths Secton, we present several numercal examples. We used the Euler method, whch s nduced by the general teratve scheme of [19] and whch has been appled to solve spatal prce equlbrum problem as projected dynamcal systems ([20], [5], where convergence results may also be found). Specfcally, one ntalzes the Euler method wth an ntal nonnegatve commodty shpment pattern and then, at each teraton τ, one computes the commodty shpments for all pars of supply and demand markets accordng to the formula: Q τ+1 j = max{0, a τ (ρ j (d τ ) c j (Q τ ) π (s τ )) + Q τ j}., j (27) The algorthm was consdered to have converged to a soluton when the absolute value of each of the successve commodty shpment terates dffered by no more than ɛ = We utlzed the sequence a τ =.1{1, 1 2, 1 2,...}, whch satsfes the requrements for convergence of the Euler method. The Euler method was mplemented n FORTRAN on a Lnuxbased computer system at the Unversty of Massachusetts Amherst. In order to approprately depct the realty of predator-prey ecosystems, we utlzed parameters, n ranges, as outlned n [4]. The computed equlbrum bomass flows for all the numercal examples are gven n Table 1. Example 1

6 TABLE I EQUILIBRIUM SOLUTIONS FOR THE EXAMPLES (, j) Q j Example 1 Example 2 Example 3 (1, 1) Q (1, 2) Q (1, 3) Q (2, 1) Q (2, 2) Q (2, 3) Q Ths example conssted of two prey speces and three predator speces. The parameters for prey speces 1 were: κ 1 =.10, µ 1 =.50, and γ 1 = 1.00, wth E 1 = 1, 000. The parameters for prey speces 2 were: κ 2 =.10, µ 2 = 1.00, and γ 2 = 1.00, wth E 2 = 1, 000. These values resulted n supply prce functons gven by: π 1 =.2s 1 100, π 2 =.1s The unt transacton costs were: φ 11 =.10, φ 12 =.20, φ 13 =.30, φ 21 =.15, φ 22 =.10, φ 23 =.20. The parameters for the predators were: for predator 1: λ 1 =.02, µ 1 =.20, and γ 1 =.10; for predator 2: λ 2 =.04, µ 2 =.20, and γ 2 =.20. The parameters for predator 3 were: λ 3 =.02, µ 3 =.2, and γ 3 =.1. These parameters resulted n demand prce functons gven by: ρ 1 =.01d 1, ρ 2 =.04d 2, ρ 3 =.01d 3. The computed equlbrum commodty/bomass flow pattern s gven n Table 1. Example 2 The second example had the same data as Example 1 except that now we consdered unt transacton cost functons that captured congeston (as n the model extenson n Secton IV). The unt transacton cost functons were now: φ 11 =.01Q , φ 12 =.02Q , φ 13 =.01Q , φ 21 =.03Q , φ 22 =.04Q , φ 23 =.01Q The computed soluton s gven n Table 1. Example 3 Example 3 had the same supply prce functon and unt transacton cost data as Example 1 but here we consdered the nterestng scenaro dentfed n [4] where λ j = 0.00 for all predators j. Ths scenaro results n all the demand prce functons to be dentcally equal to The computed soluton for ths example s also reported n Table 1. VI. SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH In ths paper we establshed the equvalence between two network systems occurrng n entrely dfferent dscplnes n ecology (and bology) wth economcs (and operatons research and regonal scence). In partcular, we proved the equvalence of the governng equlbrum condtons of predator-prey systems wth spatal prce equlbrum problems through ther correspondng varatonal nequalty formulatons. Through ths connecton, we then unveled natural extensons of the basc bpartte predator-prey network model along wth a dynamc adjustment process. We also presented an alternatve varatonal nequalty formulaton usng a product dfferentaton concept. We provded both theoretcal results as well as numercal examples. We can expect contnung research n network equlbrum models of complex food webs, nature s supply chans, n the future. ACKNOWLEDGMENTS The authors are grateful to the Engneerng Computer Servces at the Unversty of Massachusetts Amherst for settng up Professor Anna Nagurney s Lnux system, whch was used for the numercal experments. Support from the John F. Smth Memoral Fund s acknowledged for the purchase of ths computer system. REFERENCES [1] A. Nagurney, Network Economcs: A Varatonal Inequalty Approach, second and revsed edton, Kluwer Academc Publshers, Norwell, Massachusetts, [2] E. N. Egerton, Changng concepts of the balance of nature, The Quarterly Revew of Bology , [3] K. Cuddngton, The balance of nature metaphor and equlbrum n populaton ecology, Bologcal Phlosophy , [4] C. Mullon, Y. Shn, and P. Cury, NEATS: A Network Economc Approach to Trophc Systems, Ecologcal Modellng , [5] A. Nagurney and D. Zhang, Projected Dynamcal Systems and Varatonal Inequaltes wth Applcatons, Norwell, Massachusetts, [6] A. Nagurney, edtor, Innovatons n Fnancal and Economc Networks, Edward Elgar Publshng, Cheltenham, England, [7] A. Nagurney and Q. Qang, Fragle Networks: Identfyng Vulnerabltes and Synerges n an Uncertan World, John Wley & Sons, Hoboken, New Jersey, [8] P. A. Samuelson, A spatal prce equlbrum and lnear programmng, Amercan Economc Revew , [9] T. Takayama and G. G. Judge, Spatal and Temporal Prce and Allocaton Models, North-Holland, Amsterdam, The Netherlands, [10] S. Dafermos and A. Nagurney, Senstvty analyss for the general spatal economcs equlbrum problem, Operatons Research , [11] A. Nagurney and J. Dong, Supernetworks: Decson-Makng for the Informaton Age, Edward Elgar Publshng, Cheltenham, England, [12] Z. Lu, and A. Nagurney, Fnancal networks wth ntermedaton and transportaton network equlbra: A supernetwork equvalence and renterpretaton of the equlbrum condtons wth computatons, Computatonal Management Scence , [13] J.M. Cruz, A. Nagurney, and T. Wakolbnger, Fnancal engneerng of the ntegraton of global supply chan networks and socal networks wth rsk management, Naval Research Logstcs , 2006.

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