Dynamic Ecological System Analysis

D YNAMIC ECOLOGICAL SYSTEM ANAL YSIS HUSEYIN CO SKUN ∗ A Holistic Analysis of Compartmen tal Systems Abstract. This article develops a new m athematical metho d for holistic analysis of nonlinear dynamic co mpartmenta l systems through the system decomposition theory . The method i s based on the nov el dynamic s ystem and subsystem partitioning metho dologies through which compartment al systems are d ecomposed to the utmost lev el. The dynamic system an d subsystem partitioning e nable trac king the evolution of the initial stocks, environmen tal inputs, and intercompartmen tal s ystem ﬂo ws, as well as the asso ciated storages derived fr om these stocks, inputs, and ﬂows individually and separately within the system. Moreov er, the transient and the dynamic direct, indir ect, acyclic, cycling, a nd transfer ( diact ) ﬂo ws and associated storages transmitted along a giv en ﬂo w path or from one compartment, dir ectly or indirectly , to any other ar e analytically characte rized, systematically classiﬁed, and mathematically formulated. F urther, the article dev elops a dynamic technique based on the diact transactions for the quant itative classi ﬁcation of in tersp eciﬁc in teractions and the determination of their strength within foo d w ebs. Ma j or concepts and quant ities of the curren t static net work analyses are also extended to nonli near dynamic s ettings and integrat ed with the pr oposed dynamic measures and indices within the prop osed unif yi ng mathematical framework. Therefore, the prop osed methodology enables a holistic view and analysis of ecological systems. W e consider that this m ethodology brings a nov el complex system theory to the service of urgent and challenging en vironmenta l problems of the da y and has the p otent ial to lead the wa y to a more f ormalistic ecological s cience. Key words. system decomp osi tion theory , complex systems theory , dynamic ecological netw ork analysis, nonlinear dynamic compartmen tal systems, dynamic system and subsystem partitioning, transien t ﬂo ws and storages, diact ﬂo ws and storages, fo o d webs, interspeciﬁc interact ions, dynamic input-output economics, so cio-economic systems, dynamic input-output analysis, epidemiology , in- fectious diseases, to xicology , pharmacokinetics, neural net works, chemical and biological systems, con trol theory , information theory , i nformation di ﬀusion, so cial netw orks, computer netw orks, mal- wa re propagation, graph theory , traﬃc ﬂow AMS sub ject classiﬁcati ons. 34A34, 35A24, 37C60, 37N25, 37N40, 70G60, 91B74, 92B20, 92C42, 92D30, 92D40, 93C15, 94A15 1. In tro duction. Compartmental systems are mathematica l a bs tractions o f net- works that mo del b ehaviors of contin uous ph ysical systems comp ose d of discrete living and nonliving homoge ne o us co mp o ne nts. Based o n conser v ation principles, sy stem compartments are in ter connected through the ﬂo w o f energ y , matter, or currency betw een them and their en vironment. Ther efore, formulating ﬂows and ass o ciated storage s accur ately and explicitly is critica lly imp ortant in qua nt ifying c o mpartmen- tal s ystem function. V ar io us mathematical asp ects of compartmental systems are studied in the litera ture [ 27 , 2 ]. While man y ﬁelds utilize compar tment al mo del- ing, this approa ch prov es particular ly well-suited for analy sis of ecolo gical systems to address environmental phenomena . Due to the current tec hnologica l adv ancements as w ell a s scientiﬁc understandings of p opulation and industr ial gr owth and reso urce demands, environmen tal issues have assumed center stage in human communities. On the o ther hand, in spite of this increased attention to the en viro nmen t, traditional ecology has an a pplied nature and is still in the empiric a l stage of development. In the mains tream fr a mework of traditional ecolo gy , a ﬁrs t principles-base d for mal theory has yet to e mer ge. This disconnect narrows the scop e of a pplicability of the ﬁeld a nd reduces its ability to deal with complex orga nism-environmen t rela tionships. T o that extent, eco logy and ∗ Departmen t of M athematics, Uni ve rsity of Georgia, Athens, GA 30602 ( hcoskun@uga.edu ). 1 2 HUSEYIN COSKUN environmen tal science are limited in their capacity to r ealistically mo del and analyze complex systems. Ma thematical theories and mo deling hav e s ig niﬁcant p otential to lead the wa y to a mor e formalis tic and theore tical eco s cience devoted to the dis cov ery of basic scientiﬁc laws. More exa c t, pr ecise, and incisive environmen tal applicatio ns can then be materia lized based on this understanding. Sound rationales have b e en oﬀered in the litera ture for ecological netw ork analysis, but these are for sp ecial cases , such a s linear a nd sta tic mo dels. One such s tatic approach ca lled the envir on t he ory has bee n developed b y [ 38 , 3 4 ] bas e d on economic input-output analysis of [ 31 , 3 2 ] introduced into ecolo g y by [ 20 ]. E c ologica l netw orks and co mplexity in living systems are a nalyzed als o in the context of information the ory , thermo dynamics [ 44 , 24 , 45 , 4 6 ], and hier ar chy the ory [ 1 ], yet only for static systems. Several softw are hav e b een developpe d to computerize these static metho ds [ 47 , 8 , 15 , 2 8 , 40 , 5 ]. Although the steady- state analysis is well-established, dy namic analysis of non- linear compa r tment al sy stems has rema ined a lo ng-standing, o pe n proble m. F o r ex- ample, Finn’s cycling index—a celebr ated ecosystem measure that quantiﬁes cycling system ﬂows deﬁned in sta tic ecolog ical netw o rk a nalyses over four deca des ag o—has still not b een made a pplicable to ecosystem mo dels that change ov er time [ 17 ]. The indirect eﬀects in ecos ystems hav e also long b e en a well-established empirica l fact [ 37 , 42 , 49 , 36 , 3 5 , 48 ]. Theoretica l ex plorations of the concept beg an as ear ly a s the 1970s , and it has b een a topic o f scholarly conv e rsation for the pa st ﬁve decades [ 26 , 39 , 16 , 3 3 , 13 ]. Despite the urgent need, the indir ect ﬂow and storag e tr ansfers hav e never b een formulated befor e. There ar e earlier a pproaches in the literature for the analy s is of dynamic ec o systems, but these are either esse nt ially clos ed-form abstract for mu lations [ 19 ], or designed fo r sp ecia l cas es, such as linea r s ystems with time-dep e ndent inputs [ 23 ]. In a ddition, there ar e also ag ent-based techniques for dy- namic compartmental system analysis [ 41 , 30 , 2 9 ]. These are , how ever, c omputational metho ds that rely on netw ork particle tracking simulations. In ec osystem ecolo gy , fo o d webs pr ovide a framework to link communit y struc- ture with ﬂows of energy and material through trophic interactions and, therefore, relate bio diversit y with ecosystem function. T emp oral v ar iation in web architecture and nonlinea rity are dis cussed in the liter ature [ 14 , 51 ]. It is suggested that the dynamic nature of fo o d webs is aﬀecting ecosystem attributes. Nonlinearit y and dy- namic b ehavior, such as extinction in foo d w ebs, how ever, has yet to b e addressed metho dologically . No t only fo o d webs, but to day’s ma jor environmental and eco- logical phenomena and pro blems–human impact, climate c hange, bio div ersity loss, etc.– all inv o lve change, which demonstra tes that the need for dynamic metho ds for nonlinear s ystem analy s is is no t only appr opriate, but also ur gent [ 7 , 22 ]. This is the ﬁr st manuscript in the litera ture that potentially a ddresses the dis- connect b etw een the curr ent static and computationa l metho ds and applied ec o logical needs. W e consider that the metho dolog y prop osed her ein, in eﬀect, bring s a nov el complex sy stem theo ry to the service of pr essing and challenging environmental prob- lems of the day . Due to its theoretical and ma thematical nature , it has the p otential to lead the wa y to a more for malistic ecolog ical s cience. The prop osed metho do logy is a comprehensive approa ch in the sense that the ma jor co ncepts and quantities in the current static ecologic al netw o rk analy ses a re extended from static to nonlinear dy- namic settings, as w ell as integrated eﬀectively with the pr op osed dynamic mea sures in this unifying mathematical framework. This nov el and unifying approach lea ds to a holistic analysis of e c osystems . Aligned with the mathematical theory , called the system de c omp osition the ory , DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 3 int ro duced r ecently by [ 10 ], the prop osed co mprehensive metho d is comp os e d of the nov e l dynamic system a nd subsystem p artitioning metho dolo gies . The system par ti- tioning metho dolo gy yields the subthr oughﬂow and su bstor age ve ctors and matric es that repr esent the ﬂows a nd stor ages generated by the initial sto c ks and individ- ual environmen tal inputs in each compa rtment separ ately . Ther efore, the system partitioning enables dynamically decomp os ing comp osite compar tmen tal ﬂows and storage s into sub compartmental segments ba sed on their c o nstituent sour ces from the initial sto cks a nd environmen tal inputs. In other w ords, this metho dolo gy enables dynamically tr acking the evolution of the initial sto cks and environmen tal inputs, as well as the asso cia ted stora ges der ived fro m these sto cks a nd inputs individually and separately within the s y stem. The t ra nsient ﬂows transmitted along a given ﬂow path and the as so ciated stor- ages genera ted by these ﬂows in ea ch compartment on the path ar e then formulated through the subsystem partitioning metho dolo gy . Therefore, this metho dolo gy allows for the dynamic decomp osition of a rbitrar y comp osite intercompartmental ﬂows and asso ciated stor ages into the tra nsient subﬂow a nd substor age s egments along a given set of subﬂow paths. Consequently , the subsystem partitioning enables dynamically tracking the fate o f a rbitrar y int erco mpartmental ﬂo ws and as so ciated storag e s within the subsystems. Moreover, the sprea d of an arbitra ry ﬂow or stor age se g ment from one compartment to the en tire system can b e determined and monitor ed. F or the quantiﬁcation of intercompartmental ﬂow and sto r age tra nsfer dynamics, the dir e ct , indir e ct , acyclic , cycling , and tr ansfer ( di act ) ﬂows and a sso ciated stor ages tr a nsmit- ted fro m one compartment, directly or indir ectly , to any other a re also analytically characterized, systematically clas s iﬁed, and mathema tica lly fo rmulated. In a nut shell, the s y stem and subsy stem pa r titioning metho dologies dynamically determine the distribution of the initial stocks, environmen tal inputs, and arbitra ry in- tercompartmental ﬂows, as well as the orga nization of the as s o ciated s to rages de r ived from these sto cks, inputs, and ﬂows individually and separately within the system. In other words, the pro p o sed metho d as a whole enables tracking the evolution of the initial sto cks, environmen t inputs, and arbitra ry in tercmopa rtmental system ﬂows, as well as asso ciated storages individually and separately . The dynamic quantities such as the subthroughﬂows, subs to rages, and transient a nd di act ﬂows and sto r- ages are systematica lly introduced through the prop o sed metho d for the ﬁrst time in the literature. E quippe d with these mea sures, the prop osed metho dology ser ves a s a quantitativ e platform for testing empirical hypo thes es, ecological inferences, and, po tent ially , theoretica l developmen ts. The metho d a ls o co nstructs a fo unda tion for the development of new mathematica l system analysis to ols as quantitativ e ecolo gical indicators. Multiple s uch dy na mic dia ct mea sures and indices of matrix , vector, a nd scalar type s which may prove useful for e nvironmental a ssessment and ma na gement were sy stematically introduced by [ 9 ]. The tempo ral v a riations o f tr o phic interactions in fo o d webs is a n imp orta nt to pic in ecolog y as outlined a bove [ 1 4 , 51 ]. The conditions or sta tes of communities in fo o d webs, suc h as extinction, can be dynamically regula ted b y the tempora l v a r iations and seasona l shifts. The present manuscript develops also a novel mathematica l tech- nique based on the diact transactio ns for the dynamic classiﬁca tion of interspeciﬁc int erac tio ns, and notably , for the determination of their strength within fo o d webs. This technique eﬀectively addre sses the no nlinearity in and dynamic ar chitecture of the fo o d chains and webs. The prop os e d metho do logy is applicable to a ny conserv ative co mpartmental sys- tem of naturo genic or anthrop ogenic nature. The metho d can b e used, for exa mple, 4 HUSEYIN COSKUN to ana lyze mo dels desig ned for material ﬂows in industry [ 3 ]. It can als o b e used to analyze mass or ener gy transfers b etw een sp e c ies of diﬀerent tro phic levels in a co m- plex net work or a long a given fo o d chain of a fo o d web in nonlinear dynamic settings [ 21 , 4 , 18 ]. Although the motiv ating applicatio ns ar e ecologica l and environmental for this pap er, the applicability of the pr op osed metho d extends to other rea lms, such as eco nomics, phar macokinetics, chemical r eaction kinetics, epidemiology , bio medical systems, neural netw o rks, so cia l netw orks, and infor mation sc ience—in fact, wherever dynamical co mpartmental mode ls of conserved quantit ies c an be constr ucted. An input-output a nalysis in economics was develop e d several dec a des ago , but o nly for static systems [ 31 , 3 2 ]. The pro po sed metho dology in the co ntext of econo mics, in particular, c a n b e considered a s the mathematical foundatio n of the dynamic input- output e c onomics . The pro po sed method is applied to t wo mo dels in Section 3 to illustrate its eﬃ- ciency and wide a pplicability . In the ﬁrs t ca se study , a linear eco system mo del intro- duced by [ 23 ] is analyzed. The second case study concer ns nutrien t transfer within a nutrien t-pro ducer- consumer ecosystem [ 19 ]. Analy tica l and numerical solutio ns for the substo rages, subthro ughﬂows, tr ansient and di act transactions , and residence times a re pre s ented for b oth mo dels. The in tersp eciﬁc interactions in the no nlin- ear mo del and their strength are also analyzed thro ugh the prop osed mathematical classiﬁcation technique. This pap er is org a nized as follows: the mathematical metho d is introduce d in Section 2.1 , the transient and diact ﬂows and storag es a re formulated in Section 2 .5 , system analysis a nd mea sures are discussed in Section 2.7 , results and case studies are pr esented in Sectio n 3 , and discussion and conclusio ns fo llow in Section 4 and 5 . 2. Metho ds . A new mathematica l theor y for the dynamic deco mpo sition of nonlinear co mpartmental systems has recently be e n introduced by [ 10 ]. In line with this theory , a mathematica l metho d for the dynamic analys is of nonlinear ecolo gical systems is developed in the present pa p e r . The pr op osed theo ry is based on the novel s yst em and subsystem p artitioning metho dolo gies . The system and subsystem partitioning determine the distribution of the initial sto cks, environmen tal inputs, and intercompartmental ﬂows, as well a s the organiza tion of the asso ciated stor ages derived fro m these sto cks, inputs, and ﬂows individually and separately within compartmental systems. The pro p osed metho d, therefore, as a whole, yields the dec omp osition of a ll system ﬂows a nd storag e s to the utmost level. The metho d tog ether with the corres po nding co ncepts and qua nt ities will b e introduced in this section. The terminolo gy and notations us ed in this pap er ar e adopted from [ 10 ] as follows: n nu mber of compa rtments t time [t] x i ( t ) total material (mass ) [m] (or energy , currenc y ) in co mpart- men t i , i = 1 , . . . , n , at time t f ij ( t, x ) nonnegative ﬂow from compartment j to i , at time t [m / t] y i ( t, x ) = f 0 i ( t, x ) environmen tal ( j = 0) output from compartment i at time t z i ( t, x ) = f i 0 ( t, x ) e nvironmental input into co mpartment i at time t The g overning equations for the compartmental dynamics are (2.1) ˙ x i ( t ) = ˇ τ i ( t, x ) − ˆ τ i ( t, x ) DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 5 for i = 1 , . . . , n . The s tate vector x ( t ) = [ x 1 ( t ) , . . . , x n ( t )] T is a diﬀerentiable function of compartmental storag es with the initial conditions of x ( t 0 ) = x 0 = [ x 1 , 0 , . . . , x n, 0 ] T where the sup ers cript T repres e nts the ma trix tra nsp ose. The tota l inﬂow, ˇ τ i ( t, x ), and outﬂow, ˆ τ i ( t, x ), are called the inwar d a nd outwar d thr oughﬂows at compartment i , resp ectively , and for mulated as (2.2) ˇ τ i ( t, x ) = n X j =0 f ij ( t, x ) and ˆ τ i ( t, x ) = n X j =0 f j i ( t, x ) for i = 1 , . . . , n . The nonlinear diﬀer e ntiable function f ij ( t, x ) ≥ 0 repres ents non- negative ﬂow rate fr o m compartment j to i at time t . In gener al, it is assumed that f ii ( t, x ) = 0 , but the following analy s is is also v alid for nonnegative ﬂow fro m a com- partment into itself. Index j = 0 stands for the environment. W e further ass ume that f ij ( t, x ) has the following sp ecial form: (2.3) f ij ( t, x ) = q x ij ( t, x ) x j ( t ) where q x ij ( t, x ) is a nonlinear function of x a nd t , a nd ha s the same pr op erties a s f ij ( t, x ). W e will ca ll q x ij ( t, x ) = f ij ( t, x ) /x j ( t ) the ﬂow intens ity dir ected fro m c om- partment j to i p er unit stora ge or the ﬂow distribution factor for system stora ges in the context of the pro p o sed metho dology [ 11 ]. Combining E qs. 2.1 and 2.2 and sepa rating environmental inputs a nd o utputs, the sys tem of gov erning eq uations takes the following standar d form: (2.4) ˙ x i ( t ) =   z i ( t, x ) + n X j =1 f ij ( t, x )   −   y i ( t, x ) + n X j =1 f j i ( t, x )   with the initial co nditions x i ( t 0 ) = x i, 0 , for i = 1 , . . . , n . There ar e n eq uations; one for each compa r tment . The condition, Eq . 2.3 , guaranties non-negativity of the compartmental stora g es, that is x i ( t ) ≥ 0 for all i . If an environmen tal input or initial condition is p ositive, that is, z i ( t, x ) > 0 o r x i, 0 > 0, the corres po nding stor age v alue is alwa ys strictly p ositive, x i ( t ) > 0 . The prop o sed methodolog y is desig ned for c onservative c omp artmental systems . A dynamical system is called c omp artmental if it ca n b e ex pressed in the form of E q. 2.4 . The compartmental systems will b e calle d c onservative if all internal ﬂow rates add up to zero when the system is closed, that is, when there is neither environmental input nor output: (2.5) n X i =1 ˙ x i ( t ) = 0 when z ( t, x ) = y ( t, x ) = 0 for all t where 0 is used for bo th the n × n zer o matrix and zero vector of size n [ 10 ]. F or notationa l conv enience, we deﬁne a dir e ct ﬂow matrix function F of size n × n as F ( t, x ) = ( f ij ( t, x )) and the inwar d and outwar d thr oughﬂow ve ctor functions as (2.6) ˇ τ ( t, x ) = [ ˇ τ 1 ( t, x ) , . . . , ˇ τ n ( t, x )] T = z ( t, x ) + F ( t, x ) 1 and ˆ τ ( t, x ) = [ ˆ τ 1 ( t, x ) , . . . , ˆ τ n ( t, x )] T = y ( t, x ) + F T ( t, x ) 1 , 6 HUSEYIN COSKUN resp ectively , where z ( t, x ) = [ z 1 ( t, x ) , . . . , z n ( t, x )] T and y ( t, x ) = [ y 1 ( t, x ) , . . . , y n ( t, x )] T are the input and ou t put ve ct or functions, and 1 denotes the column vector of s iz e n whose entries are all one. 2.1. System partitionin g m etho dol ogy . In this section, we in tro duce the dynamic system p artitioning metho dolo gy for ana lytically partitioning the governing system into mutually exclusive and exhaustiv e subsystems , as a simpliﬁed v ersion of the system de c omp osition metho dolo gy recently pr op osed b y [ 10 ]. By mutu al ex- clusiveness , w e mean that tra nsactions are po ssible only a mong co rresp o nding sub- c omp artm ents b elong to the same subsystem. By exhaustiveness , we mean that a ll generated subsys tems s um to the ent ire system so partitioned. The system partition- ing enables dynamically partitio ning co mp o s ite compartmental ﬂo ws and storag es into sub c ompartmental seg ment s based o n their constituent so urces from the initial sto cks and environmen tal inputs of the same conserved qua ntit y . The system partition- ing metho dolog y , cons e q uently , yields the subthr o ughﬂow a nd substora ge matrices representing the distribution of the initial sto cks and environmental inputs, as well as the organiza tion of the asso ciated stora ges derived from these sto cks and inputs individually and se pa rately within the system. In other words, this metho do lo gy en- ables tracking the evolution of the initial sto cks and environmental inputs, as well as asso ciated stor ages individually and s eparately within the sy stem. The system partitioning inv o lves the dynamic sub c omp artm entalization and ﬂow p artitioning co mpo nents, whos e mechanisms ar e expla ined in this s e c tion (Figs. 1 and 2 ). The rela ted concepts a nd notations are summar ized b elow: x i k ( t ) storage in sub compartment k of co mpa rtment i , that is, in sub c ompartment i k , k = 0 , . . . , n , at time t , generated by environmen tal input z k ( t, x ) during [ t 0 , t ] f i k j k ( t, x) nonnegative ﬂow from sub compartment j k to i k at time t y i k ( t, x) = f 0 i k ( t, x) environmen tal ( j = 0) output from sub compar tmen t i k at time t z i k ( t, x) = δ ik z i ( t, x ) environmen tal input into sub compartment i k at time t , where δ ik is the discrete delta function The system is partitioned ex plicitly a nd analytically into mu tually e xclusive and exhaustive subsystems as follows: E ach co mpartment is pa rtitioned into n + 1 sub- compartments; n initially empty sub compar tmen ts for n environmen tal inputs and 1 sub c ompartment for the initia l sto ck of the compar tmen t. The notation i k is used to represent the k th sub c ompartment of the i th compartment for i = 1 , . . . , n and k = 0 , . . . , n . The subs c r ipt index k = 0 represents the initial sub compartment of compartment i (see Fig. 1 ). The s torage in sub compar tment i k will b e ca lled the substor age in i k and denoted by x i k ( t ). More sp eciﬁca lly , the substor age x i k ( t ) is deﬁned as the storage in com- partment i at time t that is ge ne r ated by the environmental input into compar tmen t k 6 = 0, z k ( t ), during the time in terv a l [ t 0 , t ] (see Fig. 2 ). C o nsequently , due to the exhaustiveness o f the system partitioning , we have (2.7) x i ( t ) = n X k =0 x i k ( t ) , i = 1 , . . . , n. DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 7 x 1 1 x 1 0 x 1 3 x 1 2 z 1 y 1 x 1 x 2 1 x 2 0 x 2 3 x 2 2 z 2 y 2 x 2 x 3 1 x 3 0 x 3 3 x 3 2 z 3 y 3 x 3 f 13 f 31 f 23 f 32 f 12 f 21 Fig. 1: Sc hematic repres entation o f the dynamic sub co mpartmentalization in a three - compartment mo del sys tem. E a ch s ubs ystem is color ed diﬀerently; the se c o nd subsys - tem ( k = 2) is blue, for ex a mple. O nly the sub compar tmen ts in the sa me subsystem ( x 1 2 ( t ), x 2 2 ( t ), and x 3 2 ( t ) in the seco nd subsystem, for exa mple) interact with ea ch other. Subsystem k receives environmental input only at s ubco mpartment k k . The initial subsys tem receives no environmental input. The dynamic ﬂow par titioning is not represented in this ﬁgure. Compare this ﬁgur e with Fig. 2 , in whic h the sub- compartmentalization and co rresp onding ﬂow pa rtitioning are illustrated fo r x 1 ( t ) only . W e deﬁne a new vector v a riable for the s ubstorages as x( t ) = [ x 1 0 ( t ) , . . . , x n 0 ( t ) , x 1 1 ( t ) , . . . , x n 1 ( t ) , . . . , x 1 n ( t ) , . . . , x n n ( t )] T . W e assume that environmen tal input z k ( t, x) ent ers the system a t s ubco mpart- men t k k , for all k . Moreover, no other k th sub c ompartment of any other compartment i , that is, sub compar tmen t i k , rece ives en viro nment al input. This input p artitioning can b e expressed as z i k ( t, x) = δ ik z k ( t, x ) = ( z k k ( t, x) = z k ( t, x ) , i = k 0 . i 6 = k The int erco mpartmental ﬂows ar e also partitioned in line with the sub compart- men talizatio n (see Fig. 2 ). The comp osite intercompartment al direct ﬂow, f ij ( t, x ), is partitioned based on the a ssumption that the sub compar tmental ﬂow segments, f i k j k ( t, x), k = 0 , . . . , n , ar e pr op ortiona l to the cor resp onding substorages , x j k ( t ), with the prop ortiona lity factor of the ﬂow intensit y in the ﬂow direc tion, q x ij ( t, x ). The sub compar tmen tal ﬂow f i k j k ( t, x) w ill b e calle d the subﬂow from sub compa rt- men t j k to i k at time t . It can b e formulated as follows: (2.8) f i k j k ( t, x) = x j k ( t ) f ij ( t, x ) x j ( t ) = x j k ( t ) q x ij ( t, x ) = d j k (x) f ij ( t, x ) 8 HUSEYIN COSKUN x 1 1 x 1 0 x 1 3 x 1 2 z 1 f j 0 1 0 f j 1 1 1 f j 2 1 2 f j 3 1 3 x 1 f j 1 Fig. 2: Sc hematic repr esentation of the dynamic ﬂow partitioning in a three- compartment mo del system. The ﬁgure illustra tes sub compa rtmentalization of com- partment i = 1 and the corre sp onding dynamic ﬂow partitioning from this compart- men t to others, j . where the co eﬃcien ts d j k (x) = x j k ( t ) /x j ( t ) will b e ca lled the de c omp osition factors . Consequently , due to the exhaustiveness o f the system partitioning , we have (2.9) f ij ( t, x ) = n X k =0 f i k j k ( t, x) , i, j = 1 , . . . , n. In s ummary , the dynamic system partitioning metho dology explicitly gener ates m utually exclusive and exha ustive subsystems running within the or iginal system. The k th subsystem is comp osed o f all k th sub c ompartments o f each compartment to- gether with the co r resp onding subﬂows and substorag es. These subsystems hav e the same str ucture and dy na mics as the o riginal sy s tem itself, exce pt for their environ- men tal inputs and initial co nditions (s e e Figs. 1 a nd 2 ). E a ch subsys tem, except the initial one—which is driven by the initial sto cks—is ge nerated by a single environ- men tal input. Ther efore, the num b er of non-intersecting subc ompartments in each compartment is equal to the num b er of inputs or co mpartments, plus one for the ini- tial sto cks. If a n input or all initial conditions are z e ro, the cor resp onding subsys tem bec omes null. Conseq uently , for a sy stem with n compar tment s, each c ompartment has n + 1 non-intersecting sub compartments, and therefor e the system has n + 1 mu- tually exclusive subs y stems indexed by k = 0 , . . . , n . The initial s ubsystem ( k = 0) represents the evolution of the initial s to cks, receives no environmen tal input, and has the same initial conditions as the or ig inal s ystem. The initial conditions for a ll the other sub co mpartments ( k 6 = 0) a re zero, since they ar e initially assumed to b e empt y . The g overning equation for each sub compartment i k then b ecomes (2.10) ˙ x i k ( t ) =   z i k ( t, x) + n X j =1 f i k j k ( t, x)   −   y i k ( t, x) + n X j =1 f j k i k ( t, x)   for i = 1 , . . . , n , k = 0 , . . . , n . There are n × ( n + 1) of such gov er ning equations, one fo r each s ub co mpartment. In o rder to tra ck the evolution of environmen tal inputs within the system individually and sepa rately , all except the initial sub compar tment s are assumed to b e initially empty , as mentioned ab ov e. Therefore, the initia l conditions DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 9 bec ome (2.11) x i k ( t 0 ) = ( x i, 0 , k = 0 0 . k 6 = 0 The g ov erning s ystem of equations, Eq. 2.10 , is solved numerically with the initial conditions, Eq . 2.11 . The r esult y ie lds the substorage s a t any time t , that is, x i k ( t ). The total subco mpartmental inﬂows and outﬂows at compar tmen t i at time t generated by the environmental input into compartment k , z k ( t ), during [ t 0 , t ] can then b e deﬁned, r esp ectively , as (2.12) ˇ τ i k ( t, x) = z i k ( t, x) + n X j =1 f i k j k ( t, x) and ˆ τ i k ( t, x) = y i k ( t, x) + n X j =1 f j k i k ( t, x) for k = 0 , 1 , . . . , n . The functions ˇ τ i k ( t, x) a nd ˆ τ i k ( t, x) will resp ectively b e called inwar d and outwar d subthr oughﬂow a t sub compa rtment i k at time t (s e e Fig. 4 ). Therefore, the sy s tem partitio ning enables dynamically partitioning c omp osite c o m- partmental ﬂows a nd storag es into s ubco mpartmental seg ments based on their con- stituen t sources from the initial sto cks and environmental inputs. W e deﬁne the n × n substor age and asso ciated inwar d a nd outwar d su bt hr oughﬂow matrix functions , X ( t ) , ˇ T ( t, x), and ˆ T ( t, x), resp ectively , as follows: (2.13) X ik ( t ) = x i k ( t ) , ˇ T ik ( t, x) = ˇ τ i k ( t, x) , and ˆ T ik ( t, x) = ˆ τ i k ( t, x) , for i, k = 1 , . . . , n . The substor age a nd a sso ciated inwar d and ou t war d subthr oughﬂow ve ctor functions of size n for the initial subsystem, x 0 ( t ), ˇ τ 0 ( t, x), and ˆ τ 0 ( t, x), can also b e deﬁned, resp ectively , as (2.14) x 0 ( t ) = [ x 1 0 ( t ) , . . . , x n 0 ( t )] T , ˇ τ 0 ( t, x) = [ ˇ τ 1 0 ( t, x) , . . . , ˇ τ n 0 ( t, x)] T , a nd ˆ τ 0 ( t, x) = [ ˆ τ 1 0 ( t, x) , . . . , ˆ τ n 0 ( t, x)] T . W e use the constant v ector notation x 0 for the initial conditions and the function notation x 0 ( t ) for the evolution of these initial stoks fo r t > t 0 with x 0 ( t 0 ) = x 0 . The notation diag( x ( t )) will be used to repr esent the diagona l matrix whose diag - onal elements are the elements of vector x ( t ), and diag ( X ( t )) to repre s ent the diagonal matrix w ho se dia gonal elements are the s ame as the diagonal ele ment s of matrix X ( t ). The n × n diagonal st or age , output , a nd input ma trix functions, X ( t ), Y ( t, x ), and Z ( t, x ) will be deﬁned, resp ectively , as X ( t ) = diag( x ( t )) , Y ( t, x ) = diag( y ( t, x )) , and Z ( t, x ) = diag ( z ( t, x )) . Using Eq. 2.8 , the subthroughﬂow ma trices ca n then b e formulated as follows: (2.15) ˇ T ( t, x) = Z ( t, x ) + F ( t, x ) X − 1 ( t ) X ( t ) ˆ T ( t, x) =  Y ( t, x ) + diag  F T ( t, x ) 1  X − 1 ( t ) X ( t ) = T ( t, x ) X − 1 ( t ) X ( t ) where T ( t, x ) = diag ( ˆ τ ( t, x )) = Y ( t, x ) + diag  F T ( t, x ) 1  . Note that, x ( t ) = x 0 ( t ) + X ( t ) 1 , ˇ τ ( t, x ) = ˇ τ 0 ( t, x) + ˇ T ( t, x) 1 , and ˆ τ ( t, x ) = ˆ τ 0 ( t, x) + ˆ T ( t, x) 1 . 10 HUSEYIN COSKUN The gov erning equatio ns for the decomp osed system, Eq . 2.10 , c a n b e expresse d in terms of the vector a nd matrix functions intro duced a b ove as follows: (2.16) ˙ X ( t ) = ˇ T ( t, x) − ˆ T ( t, x) , X ( t 0 ) = 0 , ˙ x 0 ( t ) = ˇ τ 0 ( t, x) − ˆ τ 0 ( t, x) , x 0 ( t 0 ) = x 0 . W e deﬁne a n n × n matrix function A ( t, x ) as (2.17) A ( t, x ) =  F ( t, x ) − Y ( t, x ) − dia g  F T ( t, x ) 1  X − 1 ( t ) = ( F ( t, x ) − T ( t, x )) X − 1 ( t ) = Q x ( t, x ) − R − 1 ( t, x ) where Q x ( t, x ) = F ( t, x ) X − 1 ( t ) and R − 1 ( t, x ) = T ( t, x ) X − 1 ( t ), a ssuming R ( t, x ) is inv er tible. Note that the ﬁrst ter m in the deﬁnition of A ( t, x ), Q x ( t, x ), represe nt s the int erco mpa rtmental ﬂow intensit y deﬁned in Eq. 2.8 , a nd the seco nd term, R − 1 ( t, x ), represents the o ut ward throug hﬂow intensit y . Co nsequently , we will call A ( t, x ) the ﬂow intensity matrix p er unit stor age. It is s ometimes ca lled the c omp artmental matrix [ 27 ]. The ma tr ix measure R ( t, x ) will b e ca lled the r esidenc e t ime matrix , and the matrix measure Q x ( t, x ) will b e called the ﬂ ow intens it y matrix p er unit sto rage or the ﬂow distribut ion matrix for system stor a ges [ 10 , 11 ]. The governing equatio ns , Eq. 2.1 6 , can be expres sed using the ﬂow intensit y matrix in the following from: (2.18) ˙ X ( t ) = Z ( t, x ) + A ( t, x ) X ( t ) , X ( t 0 ) = 0 , ˙ x 0 ( t ) = A ( t, x ) x 0 ( t ) , x 0 ( t 0 ) = x 0 . The dyna mic system pa rtitioning metho dolo gy that yields a decomp osed system of n 2 + n gov erning equations for all sub compar tments, Eq. 2.18 , from the o riginal system of n governing eq ua tions for all c ompartments, Eq. 2 .1 , can alg ebraically be schematized a s follows (see Figs. 1 and 2 for graphica l illus trations): ˙ x ( t ) = τ ( t, x ) =      ˙ x 1 ˙ x 2 . . . ˙ x n      =      τ 1 τ 2 . . . τ n           ˙ x 1 0 ˙ x 2 0 . . . ˙ x n 0      =      τ 1 0 τ 2 0 . . . τ n 0      and      ˙ x 1 1 · · · ˙ x 1 n ˙ x 2 1 · · · ˙ x 2 n . . . . . . . . . ˙ x n 1 · · · ˙ x n n      =      τ 1 1 · · · τ 1 n τ 2 1 · · · τ 2 n . . . . . . . . . τ n 1 · · · τ n n      = = ˙ x 0 ( t ) = τ 0 ( t, x) ˙ X ( t ) = T ( t, x) dynamic system partitioning In the diagram ab ov e, the n et subthr oughﬂow matrix , T ( t, x), as w ell as the net thr oughﬂow a nd initial thr oughﬂow ve ctors , τ ( t, x) and τ 0 ( t, x), are deﬁned as the diﬀerence betw een the corr e sp onding inw a rd and o utw ar d throughﬂows. That is, (2.19) T ( t, x) = ˇ T ( t, x) − ˆ T ( t, x) , τ ( t, x ) = ˇ τ ( t, x ) − ˆ τ ( t, x ) , and τ 0 ( t, x) = ˇ τ 0 ( t, x) − ˆ τ 0 ( t, x) . DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 11 The system partitioning introduced in this section is input-or ient ed. The g ov ern- ing sy s tem can b e partitioned ba s ed on environmental o utputs instead of inputs, by conceptually reversing all system ﬂows. The following condition o n the dir ect ﬂows, instead of Eq. 2.3 , ensures the p ossibility o f the s ystem par titioning and analysis in bo th the input- a nd o utput-o rientations: (2.20) f ij ( t, x ) = q x ij ( t, x ) x i ( t ) x j ( t ) , i, j = 1 . . . . , n. This form of ﬂo w rates makes b o th the origina l and reversed deco mpo sed systems well-deﬁned. 2.2. Subsystem ﬂo ws and storages. The sys tem par titioning methodolo gy dynamically decomp os es a no nlinear sys tem into mutually exclusive and exhaustive subsystems through the formulation of a set of governing e q uations derived from sub- compartmentalization and ﬂow par titioning comp onents, as in tro duced a bove. This metho dology ena bles the dynamic analy s is of subsys tems generated by the initia l sto cks a nd environmental inputs individually and separately . The subsystem ﬂows and stora ges are for m ulated in matrix form in this sec tion. W e deﬁne the k th dir e ct subﬂow matrix function fo r the k th subsystem as F k ( t, x) = ( f i k j k ( t, x)), k = 0 , . . . , n . Using the relationships formulated in Eq. 2.8 , this matrix can b e expressed as follows: (2.21) F k ( t, x) = F ( t, x ) X − 1 ( t ) X k ( t ) where X k ( t ) = diag ([ x 1 k ( t ) , . . . , x n k ( t )]) is the diagonal matrix of the subs to rage functions in the k th subsystem. The matr ix X k ( t ) will accor dingly be called the k th substor age matrix function. The k th output and input matrix functions then b ecome Y k ( t, x) = Y ( t, x ) X − 1 ( t ) X k ( t ) and Z k ( t, x) = diag ( z k ( t ) e k ) where e k is the element ary unit vector who s e comp onents are all ze ro except the k th element, which is 1, and we set e 0 = 0 . The k th direct subﬂow matrix, F k ( t, x), and the k th input and output ve ctor funct ions , deﬁned as ˇ z k ( t, x) = Z k ( t, x) 1 and ˆ y k ( t, x) = Y k ( t, x) 1 , ar e the counterparts for the k th subsystem of the direct ﬂow matrix, F ( t, x ), and the input a nd output vectors, z ( t, x ) and y ( t, x ), for the origina l system. Altogether , they represent the subﬂow r e g ime of the k th subsystem. Using the no tations a nd deﬁnitions of Eqs . 2.15 and 2.21 , the k th inwar d and out- war d subthr oughﬂow matric es , ˇ T k ( t, x) = diag ([ ˇ τ 1 k ( t, x) , . . . , ˇ τ n k ( t, x)]) and ˆ T k ( t, x) = diag ([ ˆ τ 1 k ( t, x) , . . . , ˆ τ n k ( t, x)]), for the k th subsystem can be expre s sed as follows: (2.22) ˇ T k ( t, x) = Z k ( t, x) + diag  F ( t, x ) X − 1 ( t ) X k ( t ) 1  , ˆ T k ( t, x) =  Y ( t, x ) + diag  F T ( t, x ) 1  X − 1 ( t ) X k ( t ) = T ( t, x ) X − 1 ( t ) X k ( t ) . W e also deﬁne the de c omp osition and k th de c omp osition matric es , D (x) = ( d i k (x)) and D k (x) = diag ([ d 1 k (x) , . . . , d n k (x)]), as (2.23) D (x) = X − 1 ( t ) X ( t ) = T − 1 ( t, x ) ˆ T ( t, x) , D k (x) = X − 1 ( t ) X k ( t ) = T − 1 ( t, x ) ˆ T k ( t, x) . The second equalities in the deﬁnitio ns of D (x) a nd D k (x) are due to Eq. 2.15 and 2.22 , resp ectively . Note that D (x) and D k (x) dec o mp o se the compartmental throughﬂow 12 HUSEYIN COSKUN matrix, T ( t, x ), into the subthroug hﬂow and k th subthroughﬂow matrices as indicated in E qs. 2.15 and 2.22 , similar to the decomp os ition of F k ( t, x) as formulated b elow in Eq. 2.25 . That is , (2.24) ˆ T ( t, x) = T ( t, x ) D (x) and ˆ T k ( t, x) = T ( t, x ) D k (x) . The k th direct subﬂow a nd substorag e matrices, F k ( t, x) and X k ( t ), can then b e written in the following v ar ious for ms: (2.25) F k ( t, x) = F ( t, x ) D k (x) = Q x ( t, x ) X k ( t ) = Q τ ( t, x ) ˆ T k ( t, x) X k ( t ) = R ( t, x ) ˆ T k ( t, x) where Q τ ( t, x ) = F ( t, x ) T − 1 ( t, x ) will b e called the ﬂow intensity matrix p er unit throughﬂow or the ﬂ ow distribution matrix for s ystem ﬂows in the context of the pro- po sed metho dolog y [ 11 ]. Note that the elements of Q τ ( t, x ), q τ ij ( t, x ), a re sometimes called tr ansfer c o eﬃcients , te chnic al c o eﬃcients in economics, o r stoichiometric c o ef- ﬁcients in chemistry . The s ystem level counterpart of Eq . 2.25 can also b e for mulated as follows: (2.26) ˜ T ( t, x) = F ( t, x ) D (x) = Q x ( t, x ) X ( t ) = Q τ ( t, x ) ˆ T ( t, x) , X ( t ) = R ( t, x ) ˆ T ( t, x) , using Eq . 2.15 . The matrix function ˜ T ( t, x) = ˇ T ( t, x) − Z ( t, x ) will b e ca lled the inter c omp artment al subthr oughﬂow matrix . Comp onent wise, ˜ T ( t, x) = ( ˜ τ i k ( t, x)) can be express ed a s ˜ τ i k ( t, x) = ˇ τ i k ( t, x) − z i k ( t ). Consequently , for any g iven storag e orga - nization within the sys tem, X ( t ), the corr esp onding intercompartmen tal ﬂow distri- butions—that is, intercompartmental inw a rd a nd outw a r d subthroughﬂow matric e s , ˜ T ( t, x) and ˆ T ( t, x)—can b e deter mined a t any time t a s follows: (2.27) ˜ T ( t, x) = Q x ( t, x ) X ( t ) and ˆ T ( t, x) = R − 1 ( t, x ) X ( t ) . It is w orth noting that the r e s idence time matrix can be written in the following v ario us fo rms: (2.28) R ( t, x ) = X ( t ) T − 1 ( t, x ) = X k ( t ) ˆ T − 1 k ( t, x) = X ( t ) ˆ T − 1 ( t, x) as for mulated in Eqs. 2.17 , 2.25 , and 2.26 . F o r la ter use, we also deﬁne three dia gonal matrices a s follows: (2.29) ˜ T ( t, x) = diag ( ˜ T ( t, x)) , ˇ T ( t, x) = diag ( ˇ T ( t, x)) , ˆ T ( t, x) = diag ( ˆ T ( t, x)) . 2.3. Analytic solution to line ar systems. T his s ection formulates analytic solutions to linear systems with time-dep endent inputs. The system par titioning metho dology yields a linear sy stem, if the orig inal sy s tem is linear. That is, if E q. 2.4 is linear, Eq. 2.1 8 is also linear. Due to the cancella tions in matr ix A ( t, x ) deﬁned in Eq. 2.17 , the decomp osed linear sy stem, Eq. 2 .18 , can b e ex pressed in the following matrix for m: (2.30) ˙ X ( t ) = Z ( t ) + A ( t ) X ( t ) , X ( t 0 ) = 0 , ˙ x 0 ( t ) = A ( t ) x 0 ( t ) , x 0 ( t 0 ) = x 0 . DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 13 Let V ( t ) be the fundamental matr ix solution to the system E q. 2.30 , as deﬁned by [ 10 ]. That is, let V ( t ) b e the unique solution o f the system ˙ V ( t ) = A ( t ) V ( t ) , V ( t 0 ) = I . The solutions to E q. 2.30 for substo r age matrix, X ( t ), and initial subs to rage vector, x 0 ( t ), in terms of V ( t ), b ecome (2.31) X ( t ) = Z t t 0 V ( t ) V − 1 ( s ) Z ( s ) ds and x 0 ( t ) = V ( t ) x 0 , as formulated by [ 10 ]. Therefore, the solutio n to the or iginal sy stem, Eq. 2.1 can b e written a s x ( t ) = V ( t ) x 0 + Z t t 0 V ( t ) V − 1 ( s ) z ( s ) ds. F or the s p ecia l case of c onstant diago nalizable ﬂow int ensity matrix A , we hav e (2.32) V ( t ) = ex p  Z t t 0 A ds  = e ( t − t 0 ) A = Ω e ( t − t 0 ) Λ Ω − 1 where Ω is the matrix whose columns a re the eigenvectors o f A, and Λ is the diago nal matrix whos e diago nal elements ar e the eig env a lues of A . F or this particular c a se, Eq. 2.31 takes the fo llowing form: (2.33) X ( t ) = Z t t 0 e ( t − s ) A Z ( s ) ds and x 0 ( t ) = e ( t − t 0 ) A x 0 . Consequently , x ( t ) = e ( t − t 0 ) A x 0 + Z t t 0 e ( t − s ) A z ( s ) ds. A subsys tem sca ling argument is pro p o sed to analyze sta tic sy s tem b ehavior p er unit input by [ 11 ]. The scaled substo rage matrix, S ( t ) = X ( t ) Z − 1 , ca n b e ex pressed for cons ta nt inv ertible input matrix, Z ( t ) = Z > 0 , as follows: (2.34) S ( t ) = Z t t 0 V ( t ) V − 1 ( s ) ds = Z t t 0 e ( t − s ) A ds = Ω  Z t t 0 e ( t − s ) Λ ds  Ω − 1 , using Eq . 2.31 and 2.33 . The static version of this meas ure S ( t ) is widely used in static ecolo gical netw ork analys e s a s outlined in the next section [ 11 ]. An exa mple of the a na lytic so lution to a linear ecos y stem mo del with time de- pendent environmental input is prese nted in Sectio n 3 .1 . 2.4. Static ecolo gical system analysis. A t steady state, the time deriv atives of the state v ariables ar e zero, and all system ﬂows a nd storag es ar e constant. That is, ˙ X ( t ) = 0 and ˙ x 0 ( t ) = 0 . The co nstant static qua nt ities will b e denoted by the same symbo ls without the time argument. The consta nt substora g e ma trix, for ex ample, will b e denoted b y X ( t ) = X . 14 HUSEYIN COSKUN Summing up the equations in Eq . 2.10 side by side ov er index k yields E q. 2.4 bec ause of the relationship ˙ x i ( t ) = n X k =0 ˙ x i k ( t ) , i = 1 , . . . , n, deduced from Eq. 2.7 a nd the deﬁnition of the decomp osition factor s, d i k (x), g iven in Eq. 2 .8 . The r efore, if the partitioned system, Eq. 2.10 , is at steady state, the o riginal system, Eq. 2.4 , is also a t steady state. The s tatic version of the pro po sed dynamic metho dology is introduced by [ 1 1 , 12 ], as summarized below in this sec tio n. Since A is a strictly diago nally dominant co nstant matrix, it is invertible. It can be expre ssed as (2.35) A = ( F − T ) X − 1 = Q x − R − 1 . W e then have the following solutions to Eq. 2.18 for the substo r age matr ix , X ( t ), a nd initial subs torage vector, x 0 ( t ), at s teady state: (2.36) X = − A − 1 Z = X ( T − F ) − 1 Z and x 0 = 0 . F ro m E q. 2.15 and the fact that τ = ˆ τ = ˇ τ and T = ˇ T = ˆ T at steady state, the throughﬂow matrix can b e written in terms of s y stem ﬂows only: (2.37) T = Z + F X − 1 X = Z + F T − 1 T ⇒ T =  I − F T − 1  − 1 Z . The r esidence time matrix R can als o b e expr essed as (2.38) R = X T − 1 = X k T − 1 k = X T − 1 similar to Eq. 2.28 . The sca led s ubs torage and subthroug hﬂow ma trices ar e deﬁned for system analy s is per unit input by [ 11 ]. They a re fo rmulated as S = X Z − 1 and N = T Z − 1 , whe r e Z is inv ertible. Using E qs. 2.36 and 2.37 , thes e matrix mea sures can b e expres sed as follows: (2.39) S = − A − 1 and N =  I − F T − 1  − 1 . Note that S ( t ) formulated in E q. 2.34 is equiv alent to S at steady sta te, that is lim t →∞ S ( t ) = S. The matrice s N and S are c alled the cumu lative ﬂow and stor age distribution matric es in the context o f the prop osed metho dolo gy [ 11 ]. Although the deriv a tion ra tio nales are diﬀerent, the prop o s ed matrix mea sures S and N ar e equiv a lent to the ones formulated in the current static ecolo gical netw o rk analyses, as shown by [ 11 ]. These matrices are treated se parately in the c urrent static metho dologies, a lthough they are natura lly related by a fac to r of the residence time matrix. Equatio n 2.38 implies that T = R − 1 X ⇒ S = R N . This r elationship enables the holistic view of static eco logical netw orks: (2.40) x = S z = R N z = R τ and X = S Z = R N Z = R T DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 15 as introduce d by [ 12 ]. The substo r age and subthroughﬂow matrices can b e sca led by output matrix instead, for the ou t put-oriente d syst em analysis . W e use a ba r notation o ver the output-oriented counterparts o f the input oriented quantities. In the output-oriented analysis, we ass ume that all system ﬂows are conce ptually reversed. That is, ¯ F = F T , ¯ Y = Z , and ¯ Z = Y . The co unt erpar ts of the Eq. 2.40 for the output-or iented analy s is then bec o me (2.41) ¯ x = ¯ S y = R ¯ N y = R ¯ τ and ¯ X = ¯ S Y = R ¯ N Y = R ¯ T where ¯ S = ¯ X Y − 1 , ¯ N = ¯ T Y − 1 , and the diagona l o utput matrix, Y , is as sumed to b e inv er tible. Since x = ¯ x and τ = ¯ τ at s tea dy state, ¯ R = R . It is worth noting that the input- and o utput-o riented c umulative ﬂow and stor a ge distribution matrices are similar. This duality can be expre s sed as (2.42) S X = X ¯ S T and N T = T ¯ N T . The holistic input- a nd output-oriented static ecologica l system a nalyses and their duality hav e r ecently b een introduced by [ 11 , 12 ]. 2.5. Subsystem partitioning m etho dol ogy . In this section, we introduce the dynamic su bsystem p artitioning metho dolog y for fur ther partitioning or segmentation of subsys tems along a given set of m utually exclusive and exhaustive subﬂow paths, as a simpliﬁed version of the subsyst em de c omp osition metho dolo gy recently pro p o sed by [ 10 ]. The subsystem partitioning metho dolo gy dynamically app ortions ar bitrary comp osite intercompartmental ﬂows a nd the as so ciated storag es g enerated by these ﬂows into transient subﬂow a nd substo r age segments along given subﬂow pa ths. The subsystem partitioning , therefore, deter mines the distribution of ar bitrary in tercom- partmental ﬂows and the or ganizatio n of the asso cia ted storag es genera ted by thes e ﬂows within the subsystems. In other words, this metho do logy ena bles tra cking the evolution of arbitrar y intercompartmental ﬂows a nd a s so ciated stor ages within a nd monitoring their spread throughout the system. The natur al su bsystem de c omp osition is deﬁned as the set of mut ually exc lus ive and exhaustive subﬂow paths whose lo ca l inputs and outputs, except for the clo sed paths, ar e environmental inputs and outputs, re sp ectively [ 10 , 11 ]. By m u tual ly ex- clusive subﬂow paths, w e mean that no giv en subﬂo w path is a subp ath , that is, completely inside of a nother path in the s ame subsystem. Exha ustiveness in this context means that such mut ually exclus ive subﬂow paths—tog ether with the co r- resp onding transient s ubﬂows and substor ages along the paths—all to gether sum to the entire subsystem so partitioned. The natur al subsystem decomp osition o f each subsystem then r esults in a m utually exclusive a nd exhaus tive deco mp o sition of the ent ire system. W e will ﬁrst introduce the trans ient ﬂows and storag es b elow. Ther eafter, they will then b e us ed for the for mulation of the diact ﬂows and sto rages in the subsequent section. No man ever steps in the same river twic e. – Her aclitus (535-475 BC) 2.5.1. T ransient ﬂows and storages. As indicated in the famous dictum by Heraclitus that “everything ﬂows,” ﬂows a re one o f the most imp orta nt physical phe- nomena o f existence. In this se c tio n, we for mulate the tr ansient ﬂows and the a sso ci- ated stor ages generated by these ﬂows. 16 HUSEYIN COSKUN The tr ansient and cumulative t r ansient subﬂows alo ng a subﬂow path a re deﬁned as follows: Alo ng a given subﬂow path p w n k j k = i k 7→ j k → ℓ k → n k , the tr ansient inﬂow a t subcompartment ℓ k , f w ℓ k j k i k ( t ), gener ated by the lo cal input from i k to j k during [ t 1 , t ], t 1 ≥ t 0 , is the input segment that is transmitted fro m j k to ℓ k at time t . Similarly , the tr ansient outﬂow generated by the transient inﬂow at ℓ k during [ t 1 , t ], f w n k ℓ k j k ( t ), is the inﬂow s e gment that is transmitted from ℓ k to the next sub c ompartment, n k , along the path at time t . The as so ciated tr ans ient su bstor age in s ubco mpartment ℓ k at time t , x w n k ℓ k j k ( t ), is the s ubs to rage s e gment governed by the tra nsient inﬂow and outﬂow balance during [ t 1 , t ] (see Fig. 3 ). The trans ient o utﬂow at sub compar tment ℓ k at time t along s ubﬂow pa th p w n k j k from j k to n k , f w n k ℓ k j k ( t ), can be formulated as follows: (2.43) f w n k ℓ k j k ( t ) = f n k ℓ k ( t, x) x ℓ k ( t ) x w n k ℓ k j k ( t ) , similar to Eq . 2.8 , due to the eq uiv alence of ﬂow and subﬂow intensities, whe r e the transient substor age, x w n k ℓ k j k ( t ), is determined by the gov er ning mass balance equa tio n (2.44) ˙ x w n k ℓ k j k ( t ) = f w ℓ k j k i k ( t ) − ˆ τ ℓ k ( t, x) x ℓ k ( t ) x w n k ℓ k j k ( t ) , x w n k ℓ k j k ( t 1 ) = 0 . The equiv a lence o f the throughﬂow a nd subthroug hﬂow intensities, a s well a s the ﬂow and subﬂow int ensities in the same direction, that is (2.45) q x nℓ ( t, x ) = f nℓ ( t, x ) x ℓ ( t ) = f n k ℓ k ( t, x) x ℓ k ( t ) and r − 1 ℓ ( t, x ) = ˆ τ ℓ ( t, x ) x ℓ ( t ) = ˆ τ ℓ k ( t, x) x ℓ k ( t ) are g iven by Eqs. 2.8 a nd 2 .15 , for ℓ, n = 1 , . . . , n , and k = 0 , 1 , . . . , n [ 10 ]. Therefore, since the int ensities in Eq s. 2.43 and 2.44 can b e expressed at b oth the compar tmen tal and subco mpartmental levels, the subsystem partitioning is actually indep endent fro m the system partitioning . That is, the s a me analysis can b e done along ﬂow pa ths within the system, instead of subﬂow paths within the subsystems. This allows the ﬂexibility of tracking a rbitrar y intercompartmental ﬂows and stor ages genera ted by all or individual environmen tal inputs within the system. The governing equations, Eqs. 2.43 and 2.44 , establish the foundation of the dynamic su bsyst em p artitioning . These equations for each sub compartment alo ng a given ﬂow path o f interest will then be coupled with the pa rtitioned system, E q. 2.10 , or the or iginal system, Eq . 2.4 , a nd be solved simultaneously . The equations can a lternatively be solved individually and separately onc e the o riginal or partitioned sy stem is solved. The tra ns ient subﬂo ws and substo rages ar e deﬁned for line a r subﬂow paths ab ov e. The sum o f the tra nsient inﬂows from sub co mpartment j k to ℓ k and the o utﬂows from ℓ k to n k generated at sub c o mpartment ℓ k at time t by the lo cal input into the connection of a given non-self-intersecting closed subﬂow path p w n k j k during [ t 1 , t ], t 1 ≥ t 0 , will resp ectively b e called the inwar d and outwar d cumulative tr ansient subﬂow at sub c ompartment ℓ k at time t . The as so ciated storage generated by the inw a rd cum ulative tra ns ient subﬂow will b e calle d cumulative tr ansient substor age . These inw ar d and out ward cumulative transient s ubﬂows will b e denoted by ˇ τ w ℓ k ( t ) and ˆ τ w ℓ k ( t ), resp ectively , a nd ass o ciated cumu lative transient substorage by x w ℓ k ( t ). They can b e formulated a s (2.46) x w ℓ k ( t ) = m w X m =1 x w, m n k ℓ k j k ( t ) , ˇ τ w ℓ k ( t ) = m w X m =1 f w, m ℓ k j k i k ( t ) , ˆ τ w ℓ k ( t ) = m w X m =1 f w, m n k ℓ k j k ( t ) DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 17 x w n k ℓ k j k x ℓ k f ℓ k j k f w ℓ k j k i k f w n k ℓ k j k f n k ℓ k ˆ τ ℓ k Fig. 3: Schematic representation of the dynamic subsystem decompo sition. The transient inﬂow and outﬂow r ate functions, f w ℓ k j k i k ( t ) and f w n k ℓ k j k ( t ), at and a s so- ciated tra nsient substorage, x w n k ℓ k j k ( t ), in subco mpa rtment ℓ k along subﬂow path p w n k j k = i k 7→ j k → ℓ k → n k . for k = 0 , 1 , . . . , n , where the sup erscript m represents the cycle num b er, and m w is the n umber of cycles, that is, the num ber of times the path p w n k i k pass through sub c ompartment ℓ k . La rge n umber of terms, m w , in computatio n of these summations reduce trunca tio n erro rs and, thus, improve the approximations. Using the equiv alence o f ﬂow intensities as formulated in Eq. 2.45 , it has b ee n recently shown for compartmental systems tha t the parallel subﬂows and the cor re- sp onding subthroughﬂows and substor ages are prop or tional [ 10 ]. By p ar al lel subﬂows , we mean the intercompartmental ﬂows that tr ansit throug h diﬀere nt subco mpart- men ts o f the same compartment along the s ame ﬂow pa th at the s ame time. This prop ortiona lity can b e formulated a s (2.47) ˆ τ k ℓ ( t, x) ˆ τ k k ( t, x) = x k ℓ ( t ) x k k ( t ) = f i ℓ k ℓ ( t, x) f i k k k ( t, x) for k = 1 , . . . , n and ℓ = 0 , . . . , n , where the denomina tors a re nonzero. 2.5.2. The di act ﬂows and storages. In this section, we formulate ﬁve ma in transaction types for nonlinea r sy stems at b oth the sub co mpartmental and compart- men tal levels using t wo approa ches: the dir e ct ( d ), indir e ct ( i ), cycling ( c ), acyclic ( a ), a nd tr ansfer ( t ) ﬂows a nd the asso ciated stor ages genera ted by these diact ﬂows. The ﬁrst approach based on the subsys tem partitioning metho dology will b e called the p ath-b ase d appr o ach , and the se c o nd approach based on the system partitioning metho dology will b e ca lled the dynamic appr o ach . The c omp osite t ra nsfer ﬂow will b e deﬁned a s the total int erco mpartmental tr an- sient ﬂow that is g enerated by all environmental inputs from one co mpartment, di- r e ctly o r indir e ctly through other compar tmen ts, to a nother. The c omp osite dir e ct , indir e ct , acyclic , and cycling ﬂows from the initial compar tmen t to the terminal com- partment are then deﬁned as the dir ect, indirect, no n-cycling, and cyc ling seg ment s at the terminal co mpa rtment o f the comp osite tra ns fer ﬂow (see Fig. 4 ). The cyc ling and acyclic ﬂows can, therefor e , b e in terpreted a s the ﬂows that v isit the terminal compartment m ultiple times and only onc e , resp ectively , after be ing transmitted fro m the initial compar tment. The c omp osite tr ansfer s ubﬂow within the initial subsys tem can also b e deﬁned as the total intercompartmental transient subﬂow that is derived from all initial sto cks from one initial sub compar tment , dir e ctly or indir e ct ly through other initial sub com- partments, to another. The c omp osite dir e ct , indir e ct , acyclic , and cycling subﬂows within the initial subsystem from the initial sub co mpartment to the terminal sub com- 18 HUSEYIN COSKUN x i i x i x j i x j x j 0 τ t ij ˆ τ j ˆ τ j 0 τ c ij τ d ij z i ˆ τ i i ˜ τ j i τ d j i τ i j i τ c j i τ i ij Fig. 4: Schematic representation of the simple and comp os ite d iact ﬂows. Solid arrows repr esent direct ﬂo ws, and da shed a rrows repres ent indirect ﬂo ws through other co mpartments (no t shown). The c o mp o site di act ﬂows (black) genera ted b y outw a rd throughﬂow ˆ τ j ( t, x ) − ˆ τ j 0 ( t, x) (i.e. der ived from all environmental inputs): direct ﬂow, τ d ij ( t ), indirect ﬂow, τ i ij ( t ), acyclic ﬂow, τ a ij ( t ) = τ t ij ( t ) − τ c ij ( t ), cycling ﬂow, τ c ij ( t ), and transfer ﬂow, τ t ij ( t ). The simple dia ct ﬂows (blue) ge nerated by outw ard subthroughﬂow ˆ τ i i ( t, x) (i.e. derived fro m single en vironmental input z i ( t )): direct ﬂow, τ d j i ( t ) = τ d j i i i ( t ), indirect ﬂow, τ i j i ( t ) = τ i j i i i ( t ), acy clic ﬂow, τ a j i ( t ) = τ a j i i i ( t ) = τ t j i ( t ) − τ c j i ( t ), cycling ﬂow, τ c j i ( t ) = τ c j i i i ( t ), and transfer ﬂow, τ t j i ( t ) = ˜ τ j i ( t, x) = ˇ τ j i ( t, x) − z j i ( t ). No te tha t the cycling ﬂows at the ter minal (sub)co mpartment may include the seg ments o f the direct and/or indirect ﬂows at that (sub)compartment, if the cycling ﬂows indirec tly pass through the co rresp onding initial (sub)compa r tment (see Fig. 5 ). Therefore, the acyclic ﬂows a re comp ose d of the seg ment s of the direct and/or indirect ﬂows. partment are then deﬁned as the dir ect, indirect, no n-cycling, and cyc ling seg ment s at the terminal subco mpartment of the c omp osite tra ns fer subﬂow. The simple t r ansfer ﬂow will b e deﬁned as the total intercompartmental transient subﬂow that is generated by the single e nvironmental input fro m an input-receiving sub c ompartment, dir e ctly or indir e ctly thro ugh o ther compartments, to another sub- compartment. The simple dir e ct , indir e ct , acyclic , a nd cycling ﬂows from the initial input-receiving sub compartment to the ter minal sub compartment are then deﬁned as the dir ect, indirect, non-cycling, and cycling segments at the ter minal sub c ompart- men t of the simple transfer ﬂow (see Fig. 4 ). The a sso ciated simple and co mp o site diact stor ages are deﬁned a s the storages g e nerated by the corresp onding di act ﬂows. Let P t i k j k be the set of mutually exclus ive subﬂow paths p w i k j k from sub compart- men t j k directly or indirectly to i k in subsy s tem k . The sets P d i k j k and P i i k j k are also deﬁned as the s e ts of mutually exclusive dir e ct and indir e ct subﬂow p aths p w i k j k from sub c ompartment j k dir e ct ly and indir e ctly to i k , resp ectively . Similar ly , the sets P c i k j k and P a i k j k are deﬁned as the sets o f mutually ex clusive cyclic and acyclic subﬂ ow p aths p w i k j k from j k to i k with a close d and line ar subpath at terminal sub compartment i k , resp ectively (se e Fig. 4 ). The cyclic subﬂow set , P c i k , can a lternatively b e deﬁned a s the set of mutually exclusive subﬂow pa ths p w i k from sub compartment i k indir e ctly back to itself. The num b er o f subﬂow paths in P * i k j k will b e deno ted by w k , where the sup ers cript ( * ) repr esent any o f the diact s ymbols. The c omp osite dia ct subﬂow fro m sub compartment j k to i k , τ * i k j k ( t ), is deﬁned as the sum of the cumulativ e tr ansient subﬂows, ˇ τ w i k ( t ), ge ne r ated by the outw ard sub- throughﬂow at subc ompartment j k , ˆ τ j k ( t, x), during [ t 1 , t ], t 1 ≥ t 0 , and tra ns mitted DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 19 int o i k at time t along all subﬂow paths p w i k j k ∈ P * i k j k . The asso ciated c omp osite dia ct substor age , x * i k j k ( t ), at sub compar tmen t i k at time t is the sum of the cumulativ e tran- sient substorag es, x w i k ( t ), gene r ated by the cumulativ e tr ansient inﬂows, ˇ τ w i k ( t ), during [ t 1 , t ]. Alternatively , x * i k j k ( t ) can b e deﬁned as the stor age segment generated b y the comp osite diac t inﬂow τ * i k j k ( t ) in subcoma prtment i k during [ t 1 , t ]. Note that, for the c y cling case, the ﬁrs t e nt rance of the trans ie nt subﬂows and substor ages into i k are no t considered a s cycling subﬂows and substora g es. The comp osite dia ct subﬂows and substorag es can then b e formulated a s follows: (2.48) τ * i k j k ( t ) = w k X w =1 ˇ τ w i k ( t ) and x * i k j k ( t ) = w k X w =1 x w i k ( t ) . The sum of all comp osite d iact subﬂows a nd substorag e s from sub c ompartment j k to i k within ea ch subsystem k 6 = 0 will b e ca lled the c omp osite diact ﬂow a nd stor age from compar tment j to i a t time t , τ * ij ( t ) and x * ij ( t ), genera ted by all e nvironmen tal inputs during [ t 1 , t ]. They can b e formulated as (2.49) τ * ij ( t ) = n X k =1 τ * i k j k ( t ) and x * ij ( t ) = n X k =1 x * i k j k ( t ) . F or no tational conv enience, we deﬁne n × n ma trix functions T * k ( t ) and X * k ( t ) whose ( i, j ) − e lement s a re τ * i k j k ( t ) and x * i k j k ( t ), resp ectively . That is, (2.50) T * k ( t ) =  τ * i k j k ( t )  and X * k ( t ) =  x * i k j k ( t )  , for k = 0 , . . . , n . These matrix measures T * k ( t ) and X * k ( t ) are calle d the k th c omp osite diact subﬂow and ass o ciated substor age matrix functions . The c orresp o nding c om- p osite diac t ﬂow and a s so ciated stor age matrix functions generated by en viro nment al inputs ar e T * ( t ) =  τ * ij ( t )  and X * ( t ) =  x * ij ( t )  , resp ectively [ 10 ]. The simple dicat ﬂows a nd stor ages generated b y single en viro nment al inputs can b e formulated in terms o f their comp osite counterparts a s follows: (2.51) τ * i k ( t ) = τ * i k k k ( t ) and x * i k ( t ) = x * i k k k ( t ) . T o dis ting uish the comp osite a nd simple diact ﬂow and storage matrices, we use a tilde no tation over the s imple versions. That is, the simple diac t ﬂow and storage matrices, for example, will b e denoted by ˜ T * ( t ) =  τ * i k ( t )  and ˜ X * ( t ) =  x * i k ( t )  . The simple diact thr oughﬂow and c omp artmental stor age matric es and ve ctors can then be formulated as (2.52) ˜ T * ( t ) = dia g ( ˜ T * ( t ) 1 ) ⇒ ˜ τ * ( t ) = ˜ T * ( t ) 1 and ˜ X * ( t ) = dia g ( ˜ X * ( t ) 1 ) ⇒ ˜ x * ( t ) = ˜ X * ( t ) 1 . The comp os ite counterparts of these quantities can similarly b e formulated in par allel. The diﬀerence b etw een the co mpo site a nd s imple dia ct ﬂows, τ * ik ( t ) a nd τ * i k ( t ), and storage s , x * ik ( t ) and x * i k ( t ), is that the comp os ite ﬂow and storage fr o m compar t- men t k to i are gener ated by outw a rd throug hﬂow ˆ τ k ( t, x ) − ˆ τ k 0 ( t, x) de r ived fro m all environmental inputs a nd their simple counterparts from input-receiving sub co m- partment k k to i k are g enerated b y outw a rd subthroug hﬂow ˆ τ k k ( t, x) derived fr om single environmental input z k ( t, x ) (see Fig. 4 ). In that sense , the co mp o site and 20 HUSEYIN COSKUN x i k x k k z k f k k i k direct (or cycling) subﬂow indirect su b ﬂow τ i i k k k Fig. 5: Sc hematic re pr esentation for the complementary nature o f the simple indir ect and cycling ﬂows within the k th subsystem. The co mp o s ite dir ect subﬂow, f k k i k ( t, x), is repr e sented by solid ar row. This s ubﬂow also c ontributes to the simple cycling ﬂow at sub compar tment k k . The simple indirect subﬂow, τ i i k k k ( t ), through other compartments (no t shown) is repr esented by dashed ar row. simple d iact ﬂows and s torages measur e the inﬂuence of one compartment on a n- other induced by a ll and a single environmental input, res pe c tively . The comp osite diact subﬂows and substora ges within the initial subsystem, τ * i 0 k 0 ( t ) and x * i 0 k 0 ( t ), from compartment k to i a re then generated by outw ard throughﬂow ˆ τ k 0 ( t, x) derived from all initial sto cks. The simple and co mp o site dia ct ﬂows hav e b een explicitly for mulated for static systems thro ugh the system partitioning methodo logy in a r ecent study by [ 11 ]. This static appr o ach will b e extended to dynamic systems po int wise in time, that is, at each time step, in what follows. In addition to the p ath-b ase d appr o ach intro duced ab ov e, this alter native appr oach to formulate the dynamic diac t ﬂows and sto r ages will b e called the dynamic appr o ach . The simple transfer ﬂow from an input-receiving sub co mpartment k k to i k can be expre ssed as follows: (2.53) τ t i k ( t ) = τ t i k k k ( t ) = n X j =1 f i k j k ( t, x) = ˇ τ i k ( t, x) − z i k ( t, x) = ˜ τ i k ( t, x) for i, k = 1 , . . . , n . Note that the simple transfer ﬂow at sub compartment i k is equa l to the intercompartmental subthro ughﬂow at that sub c ompartment. The simple direct ﬂow from k k to i k is (2.54) τ d i k ( t ) = f i k k k ( t, x) . The simple indirect ﬂow from k k to i k at time t can then b e formulated as the s imple transfer ﬂow, τ t i k ( t ), diminished by the simple direct ﬂow, τ d i k ( t ). That is, (2.55) τ i i k ( t ) = τ i i k k k ( t ) = n X j =1 j 6 = k f i k j k ( t, x) = ˇ τ i k ( t, x) − z i k ( t, x) − f i k k k ( t, x) . Due to the reﬂex ivity of the cycling ﬂow, the simple cycling ﬂow from an input- receiving sub compartment k k back in to itself can b e formulated in terms of the simple indirect o r transfer ﬂows as follows: (2.56) τ c k k ( t ) = τ c k k k k ( t ) = τ t k k ( t ) = τ i k k ( t ) = n X j =1 f k k j k ( t, x) = ˇ τ k k ( t, x) − z k k ( t, x) . DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 21 That is , the simple cycling ﬂow is the simple transfer o r indirect ﬂow from a sub- compartment ba ck into itse lf. The complement ary na ture of the cy c ling and indir ect ﬂows a re s chematized in Fig. 5 . The pro po rtionality g iven in Eq. 2.60 implies that the simple (comp o s ite) cycling (sub)ﬂow from a n input-r eceiving (arbitr ary) s ubc o m- partment k k ( i k ) into i k is (2.57) τ c i k ( t ) = τ c i k k k ( t ) = τ c i k i k ( t ) = ˇ τ i i ( t, x) − z i i ( t, x) ˆ τ i i ( t, x) ˆ τ i k ( t, x) . The simple acyclic ﬂow can, ther efore, b e formulated as (2.58) τ a i k ( t ) = τ a i k k k ( t ) = τ t i k ( t ) − τ c i k ( t ) = ˇ τ i k ( t, x) − z i k ( t, x) − ˇ τ i i ( t, x) − z i i ( t, x) ˆ τ i i ( t, x) ˆ τ i k ( t, x) . Note that the simple acyclic ﬂow from subco mpartment k k back into itself is zero : (2.59) τ a k k ( t ) = ˜ τ k k ( t ) − τ c k k ( t ) = 0 . This means that there is, obviously , no acyclic ﬂow from a sub compartment back int o itself. The prop o rtionality of the parallel s ubﬂows and the corr e sp onding s ubthrough- ﬂows and s ubs torages formulated in Eq . 2.47 can b e expressed for the diact subﬂows as follows: (2.60) τ * i ℓ k ℓ ( t ) = τ * i k k k ( t ) ˆ τ k ℓ ( t, x) ˆ τ k k ( t, x) for i, k = 1 , . . . , n and ℓ = 0 , . . . , n , where the deno minator is no nz e r o, ˆ τ k k ( t, x) 6 = 0. Using this pr op ortiona lity a nd the simple diact ﬂows form ulated ab ov e, the co mpo site diact subﬂows from subco mpartment k ℓ to i ℓ can also b e fo r mulated as follows: (2.61) τ d i ℓ k ℓ ( t ) = f i k k k ( t, x) ˆ τ k k ( t, x) ˆ τ k ℓ ( t, x) = f ik ( t, x ) ˆ τ k ( t, x ) ˆ τ k ℓ ( t, x) τ i i ℓ k ℓ ( t ) = ˇ τ i k ( t, x) − z i k ( t, x) − f i k k k ( t, x) ˆ τ k k ( t, x) ˆ τ k ℓ ( t, x) τ a i ℓ k ℓ ( t ) =  ˇ τ i k ( t, x) − z i k ( t, x) ˆ τ k k ( t, x) − ˇ τ i i ( t, x) − z i i ( t, x) ˆ τ i i ( t, x) ˆ τ i k ( t, x) ˆ τ k k ( t, x)  ˆ τ k ℓ ( t, x) τ c i ℓ k ℓ ( t ) = ˇ τ i i ( t, x) − z i i ( t, x) ˆ τ i i ( t, x) ˆ τ i k ( t, x) ˆ τ k k ( t, x) ˆ τ k ℓ ( t, x) τ t i ℓ k ℓ ( t ) = ˇ τ i k ( t, x) − z i k ( t, x) ˆ τ k k ( t, x) ˆ τ k ℓ ( t, x) for t > t 0 . Note that ˆ τ k k ( t 0 , x) = 0 and we ass ume that ˆ τ k k ( t, x) is nonzer o for all t > t 0 . The second e q uality of the ﬁr st eq ua tion in Eq. 2.61 fo r the comp os ite direct subﬂow is due to the equiv alence o f ﬂow and subﬂow intensities in the same direction, as formulated in Eq. 2.45 [ 10 , 11 ]. The co mpo site dia ct ﬂows gener ated by environmental inputs then b ecome (2.62) τ * ik ( t ) = n X ℓ =1 τ * i ℓ k ℓ ( t ) = τ * i k ( t ) ˆ τ k k ( t, x) n X ℓ =1 ˆ τ k ℓ ( t, x) = n * ik ( t ) ( ˆ τ k ( t, x ) − ˆ τ k 0 ( t, x)) 22 HUSEYIN COSKUN T able 1: The dynamic diac t ﬂow distr ibutio n and the simple and co mp o site d iact (sub)ﬂow matrices . The sup erscript ( * ) in ea ch equatio n r epresents any of the di act symbols. F o r the sake of reada bilit y , the function arguments a r e dro pp ed. diact ﬂow distr ibution matrix ﬂows d N d = F T − 1 T * = N * ( T − ˆ T 0 ) T * ℓ = N * ˆ T ℓ ˜ T * = N * ˆ T i N i = ˜ T ˆ T − 1 − F T − 1 a N a = ˜ T ˆ T − 1 − ˜ T ˆ T − 1 ˆ T ˆ T − 1 c N c = ˜ T ˆ T − 1 ˆ T ˆ T − 1 t N t = ˜ T ˆ T − 1 where the dia ct ﬂow distribution factor , n * ik ( t ), is n * ik ( t ) = τ * i k ( t ) / ˆ τ k k ( t, x). The dynamic diac t ﬂow distribution matrix function can b e deﬁned a s N * ( t ) = ( n * ik ( t )). The dynamic diact ﬂow distribution matrices, as w ell as the dynamic simple and comp osite diac t ﬂow matrices are explicitly formulated in T able 1 based on their comp onent wis e deﬁnitions in Eq. 2.61 , similar to their sta tic counterparts introduced by [ 11 ]. The inv e rted matrices in the table ar e ass umed to b e in vertible. F or the sake of r eadability , the function arguments a r e dropp ed in the table. The comp osite diac t s ubs torages can also b e for mulated using the cor resp onding diact subﬂows as tra nsient inﬂows in E q . 2.44 as follows: (2.63) ˙ x * i ℓ k ℓ ( t ) = τ * i ℓ k ℓ ( t ) − ˆ τ i ( t, x ) x i ( t ) x * i ℓ k ℓ ( t ) , x * i ℓ k ℓ ( t 1 ) = 0 for t 1 > t 0 , i, k = 1 , . . . , n and ℓ = 0 , . . . , n . The solution to this g ov erning equa- tion, x * i ℓ k ℓ ( t ), r e presents the diact substor age in sub c ompartment i ℓ at time t ≥ t 1 generated by the corresp o nding diac t subﬂow, τ * i ℓ k ℓ ( t ), during [ t 1 , t ] (s e e Fig. 3 ). The analytic solutions for linear sy stems are in tro duced in Sec tion 2.3 . In this case, the governing equation fo r the diact s ubs torages , Eq. 2.63 , can b e solved ex- plicitly for x * i ℓ k ℓ ( t ) as well. The solutio n b ecomes (2.64) x * i ℓ k ℓ ( t ) = Z t t 1 e − R t s r − 1 i ( s ′ ,x ) ds ′ τ * i ℓ k ℓ ( s ) ds where r − 1 i ( t, x ) = ˆ τ i ( t, x ) /x i ( t ) is the o utw ar d thr oughﬂow intensit y function. The supplementary r elationship b etw een the simple direct and indirect subﬂows given in E q. 2.55 ca n be ex pressed in matrix form, in terms o f bo th ﬂows a nd storag e s, as follows: (2.65) ˜ T t ( t ) = ˜ T d ( t ) + ˜ T i ( t ) and ˜ X t ( t ) = ˜ X d ( t ) + ˜ X i ( t ) . A similar s upplementary rela tio nship can b e formulated b etw een the simple cycling and acyclic ﬂows a nd storag es: (2.66) ˜ T t ( t ) = ˜ T c ( t ) + ˜ T a ( t ) and ˜ X t ( t ) = ˜ X c ( t ) + ˜ X a ( t ) , due to Eq. 2.58 . The r e ﬂe x ivity of the simple cyc ling ﬂows a nd storag es, fo rmulated in Eq . 2.56 , can also b e written in matrix form, as follows: (2.67) diag ( ˜ T c ( t )) = diag ( ˜ T t ( t )) = diag ( ˜ T i ( t )) , diag ( ˜ X c ( t )) = diag ( ˜ X t ( t )) = diag ( ˜ X i ( t )) . DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 23 T able 2: The input-oriented, ﬂow-based diac t ﬂow and stora ge distribution and the simple and comp osite dia ct (sub)ﬂow and (sub)stor age matrice s. The s upe r script ( * ) in ea ch equation represents any of the diact s y mbols. diact ﬂow a nd storage distribution matr ices ﬂows storag e s d N d = F T − 1 S * = R N * T * = N * T T * ℓ = N * T ℓ ˜ T * = N * T X * = S * T X * ℓ = S * T ℓ ˜ X * = S * T i N i = ( N − I ) N − 1 − F T − 1 a N a = ( N − 1 N − I ) N − 1 c N c = ( N − N − 1 N ) N − 1 t N t = ( N − I ) N − 1 The matrix relations hips for mulated in Eqs. 2.65 , 2 .66 , and 2 .6 7 ca n similarly b e expressed for the comp o s ite di act ﬂows a nd storag e s. 2.6. Static subsystem partiti o ning and diac t transactions. The static version of the dynamic subsystem par titioning methodolog y given in Eqs. 2.43 a nd 2.4 4 has r ecently been for mulated by [ 11 ]. This static partitioning is summariz e d in this section. Since time deriv a tives ar e zero a t steady s tate, we set ˙ x w n k ℓ k j k ( t ) = 0 in E q. 2.44 . The sta tic tra nsient o utﬂow at sub compar tment ℓ k along s ubﬂow path p w n k j k = i k 7→ j k → ℓ k → n k from j k to n k , f w n k ℓ k j k , and the a s so ciated tra nsient substorag e gener- ated in ℓ k , x w n k ℓ k j k , by the tra nsient inﬂow, f w ℓ k j k i k , ar e then for mulated as fo llows: (2.68) x w n k ℓ k j k = x ℓ τ ℓ f w ℓ k j k i k and f w n k ℓ k j k = f nℓ x ℓ x w n k ℓ k j k = f nℓ τ ℓ f w ℓ k j k i k . The second equa lity for f w n k ℓ k j k is obtained by us ing the ﬁrs t equation for x w n k ℓ k j k in the ﬁrs t equality . The rela tionships in Eq. 2.68 e s tablish the foundation of the st atic subsystem p artitioning . Through the system par titioning methodo logy , the static di act ﬂows and storag es are formulated in matrix form by [ 11 ], as pr esented in T able 2 . Note that the ma tr ix N used in the table is deﬁned a s N = diag ( N ). 2.7. System m easures and indices. The dynamic system partitioning metho d- ology yields the subthroughﬂow and substora ge vectors a nd matrice s that measure the inﬂuence of the initial sto cks and environmen tal inputs o n system compartments in terms o f the ﬂow and sto rage gener ation. These vector and ma trix mea sures enable tracking the evolution of the initial sto cks, e nvironmental inputs, as well as the a sso ci- ated stor ages so urced from these sto cks and inputs individually and s eparately within the system. F or the analy sis o f intercompartmental ﬂow and sto rage dynamics, the dynamic subs y stem par titioning metho dology then formulates the tra nsient and dy- namic diact ﬂows and as so ciated storages . The transient and di act transactions enable tracking of a rbitrary int erco mpartmental ﬂows and stora ges a long a particular and all p ossible ﬂow paths within the s ystem and determining the inﬂuence of sys tem compartments dir ectly and indir ectly on one a nother. The prop osed metho dology cons tr ucts a foundation for the development o f new mathematical sy stem analysis to ols a s quantitativ e ecosy stem indicators. In addi- tion to the measures summa r ized ab ov e, m ultiple nov el dynamic a nd static system analysis too ls o f matrix, vector, and scalar t yp es hav e recently b een introduced in separate works, ba s ed on the pr op osed metho dolo g y [ 9 , 1 2 ]. More sp eciﬁcally , these 24 HUSEYIN COSKUN manuscripts int ro duce measures and indices for the d iact eﬀects, utilities, exp osures, and residence times, a s well a s the system eﬃciencies, stres s, and r esilience. 2.8. Quan titativ e classiﬁcation of intersp eciﬁc in teractions. An immedi- ate ecological application of the prop ose d methodo logy is the qua nt itative analysis of fo o d webs and chains. The s ystem co mpartments of a fo o d web eco system repr e- sent s p e cies, the conserved quantit y in question b ecomes nutrient o r energ y , and ﬂow paths corr esp ond to fo o d chains in the web. In this setting, direct ﬂow r ate f ik ( t, x ) represents an interspeciﬁc interaction, such a s preda tion, b etw een sp ecies i a nd k and measures the r ate of nutrien t or ener gy ﬂow fr om sp ecies k in a low er trophic level to i in the next level at time t . Nutrient or ener gy stored in sp ecies i through a ll tr ophic int erac tio ns is r epresented by x i ( t ). Communit y eco logy classiﬁes int ersp eciﬁc int erac tio ns qualitatively using net work top ology without regar d for system ﬂows [ 36 , 3 5 ]. This structural determination, how ever, gets more co mplica ted, if at all p o ssible, with the inc r easing complexity of intricate fo o d webs [ 25 , 5 0 ]. Multiple fo o d chains o f p otentially diﬀerent lengths betw een tw o sp ecies , for example, disallow the cla ssiﬁcation based on the length of the chains [ 6 ]. A mathematical characterization and classiﬁca tion technique for the analysis of the nature and strength of fo o d chains has recently b een prop osed by [ 11 ] for s ta tic systems . This s ection introduces a dynamic version of this technique with slight mo diﬁcatio ns. The prop o s ed metho dology ca n qua ntit atively determine the net b eneﬁt in terms of ﬂow and stora ge transfers received by the inv olved sp ecie s from each other. The sign analysis of the d iact intersp e ciﬁc inter actions determines the neutral and a ntagonis- tic nature of the interactions—whether the interaction is b eneﬁcial or harmful to the sp ecies inv olved. The st re ngth analysis then quantiﬁes the strength of these diact in- teractions. The sign and str ength of the dia ct int er actions induced by environmental inputs b etw een s pe cies i and j will b e deﬁned resp ectively as follows: (2.69) δ * ij ( t ) = s gn ( τ * ij ( t ) − τ * j i ( t )) and µ * ij ( t ) = | τ * ij ( t ) − τ * j i ( t ) | ˇ τ i ( t, x ) + ˇ τ j ( t, x ) where sgn( · ) is the sign function, and the sup er script ( * ) represents a ny of the dia ct symbols. F o llowing the co nv ention of communit y ecology , instead of (+1) and ( − 1 ), (+) and ( − ) notations will b e used for the sig n o f the diact in teractions. The strength, 0 ≤ µ * ij ( t ) ≤ 1 , is deﬁned to b e zer o , if bo th terms in its denominator ar e zero . F or the ana lysis of di act interactions r anging fro m the individual and lo ca l to the system-wide and g lo bal scale, the streng th of the interactions can b e formulated with the no r malization by τ * ij ( t ) + τ * j i ( t ), τ t ij ( t ) + τ t j i ( t ), ˇ τ i ( t, x ) + ˇ τ j ( t, x ), a s in Eq. 2.69 , or ˇ σ τ ( t ) = 1 T ˇ τ ( t, x ) in the given or der , where ˇ σ τ ( t ) will b e called the total inwar d system thr oughﬂow . As an exa mple, fo r the a nalysis of lo cal diact in terac tio ns at the global scale, µ * ij ( t ) can be deﬁned as µ * ij ( t ) = | τ * ij ( t ) − τ * j i ( t ) | / ˇ σ τ ( t ). The d iact neutra l r elationship b etw een sp ecies i and j and “ pr e dation ” o f sp ecies i on j can quantitativ ely b e characterized, r esp ectively , as follows: (2.70) τ * ij ( t ) − τ * j i ( t ) = 0 ⇒ δ * ij ( t ) = (0 ) , τ * ij ( t ) − τ * j i ( t ) > 0 ⇒ δ * ij ( t ) = (+ ) . This classiﬁca tion can also b e extended to the tr ansient inter actions betw een tw o sp ecies a lo ng a given fo o d chain using the tra nsient inﬂows. W e will use the notations δ w ij ( t ) and µ w ij ( t ) for the sign and str ength o f the net ﬂow from compartment j indirectly to i through tr ophic interactions along a given fo o d chain, p w ij . DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 25 The classiﬁcatio n of the d iact interactions induced by the initial sto cks can b e determined by us ing the comp osite di act subﬂows for the initial s ubsystem, τ * i 0 j 0 ( t ), in Eqs. 2.69 and 2.7 0 . The storag e-based qua ntitative deﬁnition of the diact interac- tions ca n be formulated in parallel by substituting the di act storag es for the corr e- sp onding d iact ﬂows in the deﬁnitions ab ov e as well. The stor age-bas ed form ulations represent the history o f the d iact interactions during [ t 1 , t ] w hile the ﬂow-based for- m ulations r epresent simultaneous interactions a t time t . W e will use the sup ers cript x to distinguish the storag e-based meas ures, δ * ,x ij ( t ) and µ * ,x ij ( t ). F or the clas s iﬁcation of the di act interactions induced by individual environmental inputs, the simple diact ﬂows and stor ages can b e us ed instead o f their comp osite counterparts in Eqs. 2.69 and 2.70 . A tilde nota tion will b e used ov er the measures for simple dia ct in tersp eciﬁc int erac tio ns, ˜ δ * ij ( t ) and ˜ µ * ij ( t ). A mathematical technique for the dynamic characteriz a tion and cla ssiﬁcation of the ma in interspeciﬁc interaction types, s uch as neutralism, mutualism, commensa l- ism, co mpe tition, and exploitation, has also been develope d recently in a separate pap er [ 9 ]. 3. Results. The prop os ed dynamic metho dolo gy is applied to a linea r and non- linear dynamic ecosystem mo dels. Numerical r e sults for the s ystem ana ly sis to ols developed in this manuscript, such as the substor age a nd s ubthroughﬂow matrix mea- sures, as well as the tr ansient and dynamic d iact ﬂows a nd s torages , ar e pres ented in this section. The results indica te that the prop osed metho dology pr ecisely quantiﬁes s ystem functions, prop erties , and behaviors, enables tracking the ev olution of the initial sto cks, environmental inputs, and intercompartmen tal ﬂows, a s well as asso cia ted storage s individually and separa tely within the system, is s e nsitive to per turbations due to even a brie f unit impulse , and, thus, can b e used for r igoro us dynamic analys is of nonlinear ecologic a l systems. It is worth no ting , how ever, tha t this pre s ent work prop oses a mathematical metho d —a sys tema tic technique designed for solving and analyzing any no nlinea r dynamic co mpartmental mo del—and it is not itself a mo del . Therefore, w e fo cus more on demonstrating the eﬃciency and wide applicability of the metho d. It is exp ected that once the metho d is a ccessible to a broa der communit y of environmen tal ecolog ists, it can b e used fo r ecological inferences and the holistic analysis of sp eciﬁc mo dels of interest. 3.1. Case study . A linea r dyna mic ecosys tem mo del intro duced by [ 23 ] is ana- lyzed through the pro po sed metho dolo gy in this case study to demonstrate the capa - bilit y of the metho d to a nalytically solve linear sys tems with time- depe ndent inputs. The g r aphical repr e s entations o f the results a re presented. The mo del ha s tw o compartments, x 1 ( t ) and x 2 ( t ) (see Fig. 6 ). The system ﬂows are desc rib ed as F ( t, x ) =  0 2 3 x 2 ( t ) 4 3 x 1 ( t ) 0  , z ( t, x ) =  z 1 ( t ) z 2 ( t )  , y ( t, x ) =  1 3 x 1 ( t ) 5 3 x 2 ( t )  . The g overning equations take the following for m: (3.1) ˙ x 1 ( t ) = z 1 ( t ) + 2 3 x 2 ( t ) −  4 3 + 1 3  x 1 ( t ) ˙ x 2 ( t ) = z 2 ( t ) + 4 3 x 1 ( t ) −  2 3 + 5 3  x 2 ( t ) 26 HUSEYIN COSKUN x 1 ( t ) x 2 ( t ) p 1 1 1 Fig. 6 : Schematic representation of the model ne tw ork . Subﬂow path p 1 1 1 , along which the c y cling ﬂow and stor age functions ar e computed, is red (subsystems are not shown) (Case study 3.1 ). with the initial c onditions [ x 1 , 0 , x 2 , 0 ] T = [3 , 3] T . The sub compar tmen talization step yields the substate v a riables that repr esent the substo r age v a lues as follows: x 1 k ( t ) and x 2 k ( t ) with x i ( t ) = 2 X k =0 x i k ( t ) . The ﬂow partitioning then yields the subﬂows for the subsys tems: F k ( t, x) =  0 2 3 d 2 k x 2 4 3 d 1 k x 1 0  , ˇ z k ( t, x) =  δ 1 k z 1 δ 2 k z 2  , ˆ y k ( t, x) =  1 3 d 1 k x 1 5 3 d 2 k x 2  , where the decomp o sition factors d i k (x) are deﬁned b y E q. 2.8 . The dy namic sys- tem pa rtitioning metho do logy then y ie lds the following g overning equations for the decomp osed sy stem: ˙ x 1 k ( t ) = z 1 k ( t ) + 2 3 x 2 k ( t ) −  4 3 + 1 3  x 1 k ( t ) ˙ x 2 k ( t ) = z 2 k ( t ) + 4 3 x 1 k ( t ) −  2 3 + 5 3  x 2 k ( t ) with the initial c onditions x i k ( t 0 ) = ( 3 , k = 0 0 , k 6 = 0 for i = 1 , 2 . There ar e n × ( n + 1) = 2 × 3 = 6 equations in the s y stem. The sy stem can b e written in matr ix form a s formulated in E q. 2.30 : (3.2) ˙ X ( t ) = Z ( t ) + A X ( t ) , X ( t 0 ) = 0 ˙ x 0 ( t ) = A x 0 ( t ) , x 0 ( t 0 ) = x 0 where the constant ﬂow intensit y matrix A , as deﬁned in Eq. 2.35 , b eco mes A =  − (4 / 3 + 1 / 3) 2 / 3 4 / 3 − (2 / 3 + 5 / 3)  =  − 5 / 3 2 / 3 4 / 3 − 7 / 3  . The governing decomp osed s y stem, Eq. 3.2 , is linea r. W e ca n, there fore, solve it analytically , as for mulated in Section 2.3 . Since the ﬂow intensit y matrix A is DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 27 constant, we hav e the following fundamental matrix s olution as given in Eq. 2.32 : V ( t ) = " 2 e − t 3 + e − 3 t 3 e − t 3 − e − 3 t 3 2 e − t 3 − 2 e − 3 t 3 e − t 3 + 2 e − 3 t 3 # . F or z = [1 , 1] T , the solutions for the matrix equa tion, Eq. 3.2 , then b ecome (3.3) X ( t ) = " 7 9 − e − 3 t 9 − 2 e − t 3 2 9 + e − 3 t 9 − e − t 3 4 9 + 2 e − 3 t 9 − 2 e − t 3 5 9 − 2 e − 3 t 9 − e − t 3 # , x 0 ( t ) =  3 e − t 3 e − t  , as given in Eq. 2.33 . Therefore, the s olution to the o riginal sys tem, E q . 3.1 , in vector form, is x ( t ) = x 0 ( t ) + X ( t ) 1 =  x 1 ( t ) x 2 ( t )  =  2 e − t + 1 2 e − t + 1  . The subthroughﬂow matr ic es, ˇ T ( t ) and ˆ T ( t ), can a lso be expre s sed a s formulated in Eq. 2.15 , using the solution for the substo rage matrix, X ( t ). The stea dy state s olutions ca n also b e computed as formulated in Eq . 2.36 : (3.4) X = − A − 1 = X ( T − F ) − 1 =  7 / 9 2 / 9 4 / 9 5 / 9  and x 0 = 0 . It ca n eas ily b e seen tha t, this steady-state s olution is the same a s the limit of the dynamic solutio n, Eq. 3.3 , as t tends to inﬁnity . Tha t is, lim t →∞ X ( t ) = X . W e also analyze the sys tem with a time dep endent input z ( t ) = [3 + sin( t ) , 3 + sin(2 t )] T . Similar computatio ns with the s ame fundamental matrix, V ( t ), lead us to the following initial s ubs torage vector, x i 0 ( t ), and substo rage matrix comp onents, x i k ( t ): (3.5) x 1 0 ( t ) = x 2 0 ( t ) = 3 e − t , x 1 1 ( t ) = 7 3 − 11 co s ( t ) 30 + 13 sin ( t ) 30 − 5 e − t 3 − 3 e − 3 t 10 , x 1 2 ( t ) = 2 3 − 16 co s (2 t ) 195 − 2 s in (2 t ) 195 − 13 e − t 15 + 11 e − 3 t 39 , x 2 1 ( t ) = 4 3 − 4 c os ( t ) 15 + 2 s in ( t ) 15 − 5 e − t 3 + 3 e − 3 t 5 , x 2 2 ( t ) = 5 3 − 46 co s (2 t ) 195 + 43 sin (2 t ) 195 − 13 e − t 15 − 22 e − 3 t 39 . Using these s olutions, we can e x press the solutions to the original s y stem, Eq. 3.1 , as x 1 ( t ) = 2 X k =0 x 1 k ( t ) and x 2 ( t ) = 2 X k =0 x 2 k ( t ) . The prop osed dynamic metho d solves linear systems analytica lly . Explic it s o lutions can b e used to compute the quantities in ques tion at any time t . The elements o f the inw ard initial throughﬂow vector, ˇ τ 0 ( t ), and subthroughﬂow 28 HUSEYIN COSKUN 0 5 10 15 20 25 time 0 0.5 1 1.5 2 2.5 3 substorage matrix (a) X ( t ) 0 5 10 15 20 25 time 0 1 2 3 4 subthroughflow matrix (b) ˇ T ( t, x) Fig. 7: The graphical r epresentation of the substorag e and inw ard subthroughﬂow matrices, X ( t ) and ˇ T ( t, x), and the initial substorag e and in w ard subthroughﬂow vectors, x 0 ( t ) and ˇ τ 0 ( t, x), for time-dep endent input z ( t ) = [3 + sin( t ) , 3 + sin(2 t )] T (Case study 3.1 ). matrix, ˇ T ( t ), can b e computed using Eq. 2.15 as follows: (3.6) ˇ τ 1 0 ( t ) = 2 e − t , ˇ τ 2 0 ( t ) = 4 e − t , ˇ τ 1 1 ( t ) = 35 9 − 8 cos ( t ) 45 + 49 sin ( t ) 45 − 10 e − t 9 + 2 e − 3 t 5 , ˇ τ 1 2 ( t ) = 742 585 − 184 cos 2 ( t ) 585 + 86 sin (2 t ) 585 − 26 e − t 45 − 44 e − 3 t 117 , ˇ τ 2 1 ( t ) = 28 9 − 22 cos ( t ) 45 + 26 sin ( t ) 45 − 20 e − t 9 − 2 e − 3 t 5 , ˇ τ 2 2 ( t ) = 2339 585 − 128 cos 2 ( t ) 585 + 577 sin (2 t ) 585 − 52 e − t 45 + 44 e − 3 t 117 . The outw ard initial thr o ughﬂows and throughﬂows can also b e obtained similar ly , using Eq . 2.15 . The substo rage and inw ard subthroughﬂow ma trix functions, X ( t ) and ˇ T ( t ), given in Eqs. 3.5 and 3.6 , deter mine the dynamic distribution of the envi- ronmental inputs a nd the o rganiza tion of the ass o ciated storag es generated by these inputs individually and separ ately within the system. In other w ords , using these functions, the evolution of the environmen tal inputs and ass o ciated storage s can b e track ed individua lly and separ ately throughout the system. The gra phica l represe n- tation of X ( t ) and ˇ T ( t ) for time dep e ndent input z ( t ) = [3 + s in( t ) , 3 + sin(2 t )] T are depicted in Fig. 7 . The evolution of initial sto cks and inw ard thro ughﬂows, x i 0 ( t ) and ˇ τ i 0 ( t ), are also presented in Fig. 7 . The r esidence time matrix for this mo del, deﬁned in Eq. 2.2 8 , b ecomes R ( t, x ) = diag ([0 . 6 , 0 . 4 3 ]) . The residence time of compartment 2 is cons tantly smaller than that of compar tmen t 1. That is, r 2 ( t, x ) = 0 . 4 3 < 0 . 6 = r 1 ( t, x ). This result e c ologica lly indicates that compartment 2 is more active tha n 1. DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 29 The subsystem partitioning metho dology allows for the further analysis of the system and brings out additiona l insights that are not av ailable through the state- of-the-art techniques. The comp os ite transfer ﬂow and stor a ge from compartment 2 to 1, τ t 12 ( t ) and x t 12 ( t ), and comp os ite tra nsfer subﬂow and substo rage from initial sub c ompartment 2 0 to 1 0 , τ t 1 0 2 0 ( t ) and x t 1 0 2 0 ( t ), are computed b elow as an a pplication of the prop osed subsystem partitioning metho dolog y . They c an be ex pressed using Eq. 2.49 as (3.7) τ t 12 ( t ) = 2 X k =1 τ t 1 k 2 k ( t ) and x t 12 ( t ) = 2 X k =1 x t 1 k 2 k ( t ) . The s e ts o f mutually exclus ive subﬂow pa ths fro m 2 k to 1 k , P 1 k 2 k , for k = 0 , 1 , 2, can be for mulated as follows: P 1 0 2 0 = { p 1 1 0 2 0 , p 2 1 0 2 0 } , P 1 1 2 1 = { p 1 1 1 2 1 } , P 1 2 2 2 = { p 1 1 2 2 2 } where p 1 1 0 2 0 = 0 0 7→ 1 0 2 0 → 1 0 , p 2 1 0 2 0 = 0 0 7→ 2 0 → 1 0 2 0 , p 1 1 1 2 1 = 0 1 7→ 1 1 2 1 → 1 1 , and p 1 1 2 2 2 = 0 2 7→ 2 2 → 1 2 2 2 . There ar e t wo subﬂow paths in the initial subsy stem, p 1 1 0 2 0 and p 2 1 0 2 0 , and there- fore, w 0 = 2. The corr esp onding transfer s ubﬂow and ass o ciated substorag e functions, as formulated in Eq. 2.48 , b ecome (3.8) τ t 1 0 2 0 ( t ) = 2 X w =1 ˇ τ w 1 0 ( t ) = ˇ τ 1 1 0 ( t ) + ˇ τ 2 1 0 ( t ) , x t 1 0 2 0 ( t ) = 2 X w =1 x w 1 0 ( t ) = x 1 1 0 ( t ) + x 2 1 0 ( t ) . Similarly , we hav e (3.9) τ t 1 k 2 k ( t ) = 1 X w =1 ˇ τ w 1 k ( t ) = ˇ τ 1 1 k ( t ) and x t 1 k 2 k ( t ) = 1 X w =1 x w 1 k ( t ) = x 1 1 k ( t ) for k = 1 , 2, since there is only one subﬂow path in these subsy stems ( w k = 1). The links tha t dire ctly contribute to the cumulative transie nt inﬂow and sub- storage , ˇ τ 1 1 1 ( t ) a nd x 1 1 1 ( t ), at sub compar tmen t 1 1 along p 1 1 1 2 1 are marked with cycle nu mbers, m , in the e x tended subﬂow pa th diag ram b elow: p 1 1 1 2 1 = 0 1 7→ 1 1 2 1 1 − → 1 1 2 1 2 − → 1 1 2 1 3 − → 1 1 · · · The cumulativ e transient inﬂow and substo r age will b e approximated b y tw o terms ( m 1 = 2) using E q. 2.46 : (3.10) x 1 1 1 ( t ) ≈ 2 X m =1 x 1 ,m 2 1 1 1 2 1 ( t ) = x 1 , 1 2 1 1 1 2 1 ( t ) + x 1 , 2 2 1 1 1 2 1 ( t ) , ˇ τ 1 1 1 ( t ) ≈ 2 X m =1 f 1 ,m 1 1 2 1 1 1 ( t ) = f 1 , 1 1 1 2 1 1 1 ( t ) + f 1 , 2 1 1 2 1 1 1 ( t ) . The g ov erning equations, Eqs. 2.43 and 2.44 , for the transie nt subﬂows and asso- ciated substorage s , f 1 ,m 1 1 2 1 1 1 ( t ) and x 1 ,m 2 1 1 1 2 1 ( t ), and the other transient subﬂows and substorag es in volv ed in Eqs. 3.8 a nd 3.9 , are solved s im ultaneously , to gether with the decompo sed system, Eq. 2.10 . Numerical results for the tra nsfer subﬂows and asso ciated substo r ages ar e presented in Fig. 8 . 30 HUSEYIN COSKUN 0 5 10 15 20 25 time 0 0.5 1 1.5 2 2.5 transfer subflows (a) 0 5 10 15 20 2 5 time 0 0.5 1 1.5 2 2.5 3 3.5 4 transfer substorages (b) 0 5 10 15 20 25 time 0 0.5 1 1.5 2 cycling flows and storages (c) Fig. 8: The gra phical representation of (a ) the comp osite transfer ﬂow and (b) storag e , τ t 12 ( t ) and x t 12 ( t ), together with the contributing comp osite transfer subﬂows a nd substorag es, τ t 1 k 2 k ( t ) a nd x t 1 k 2 k ( t ), as w ell as (c) the composite cycling ﬂows and storage s, τ c i 0 i 0 ( t ) + τ c ii ( t ) and x c i 0 i 0 ( t ) + x c ii ( t ) (Case study 3.1 ). The subﬂow paths in P 1 k 2 k for e a ch subsys tem k are mutually exclusive and exhaustive. Therefore , x 1 ( t ) and x 1 0 ( t ) + x 2 1 1 1 0 1 ( t ) + x t 1 0 2 0 ( t ) + x t 12 ( t ) must b e the same, as well as f 12 ( t ) and τ t 1 0 2 0 ( t ) + τ t 12 ( t ). The terms added to x t 1 0 2 0 ( t ) + x t 12 ( t ) for a compariso n, x 2 1 1 1 0 1 ( t ) and x 1 0 ( t ), a re the transient substorage gener ated by environmen tal input in 1 1 (0 1 7→ 1 1 ) and the initial substor age in compartment 1 (see Fig. 7a ). Therefor e , they are not included in the tra nsfer stor age and initial substorag e, x t 12 ( t ) and x t 1 0 2 0 ( t ). These quantities, how ever, ar e appr oximately eq ual as pres ent ed in Fig. 8 : x 2 1 1 1 0 1 ( t ) + x t 1 0 2 0 ( t ) + x t 12 ( t ) ≈ x 1 ( t ) − x 1 0 ( t ) and τ t 1 0 2 0 ( t ) + τ t 12 ( t ) ≈ f 12 ( t ) . The small diﬀerence s are caused by the truncation err ors in the computation o f cumu- lative transient subﬂows, and larg e r m w v alues further improv e the appr oximations. These close approximations demonstrate the accura cy and co ns istency of b oth the system and subsystem pa rtitioning metho dolo gies. Instead of the pa th-based a pproach us e d in the numerical c omputations a b ov e, the diac t ﬂows can also b e obtained a nalytically and explicitly using the dynamic approach a s introduce d in Section 2.5.2 . The comp osite tra nsfer subﬂows τ t 1 k 2 k ( t ) for k = 0 , 1 , 2, and tra nsfer ﬂow τ t 12 ( t ) b ecome: (3.11) τ t 1 0 2 0 ( t ) = 2 e − t , τ t 1 1 2 1 ( t ) = 8 9 − 8 cos ( t ) 45 + 4 sin ( t ) 45 − 10 e − t 9 + 2 e − 3 t 5 , τ t 1 2 2 2 ( t ) = − 1013 585 − 184 c o s 2 ( t ) 585 − 499 sin (2 t ) 585 − 26 e − t 45 − 44 e − 3 t 117 , τ t 12 ( t ) = − 493 585 − 8 cos ( t ) 45 + 4 sin ( t ) 45 − 998 cos ( t ) sin ( t ) 585 − 184 c o s 2 ( t ) 585 + 14 e − t 45 + 14 e − 3 t 585 , as formulated in Eq . 2.61 . The comp osite tra nsfer substorages can then be obta ined b y coupling Eq. 2.63 for the tra nsfer subﬂows with the decomp osed system, Eq. 2.10 , a nd solving them sim ultaneously . Alternativ ely , the corres p o nding trans fer substorag es DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 31 can b e obtained analytically as formulated in Eq. 2.64 : (3.12) x t 1 0 2 0 ( t ) = 3 e − t − 3 e − 5 t 3 x t 1 1 2 1 ( t ) = 8 15 − 26 co s ( t ) 255 − 2 s in ( t ) 255 − 5 e − t 3 + 261 e − 5 t 3 170 − 3 e − 3 t 10 x t 1 2 2 2 ( t ) = − 17 15 + 2534 co s (2 t ) 11895 − 3047 sin (2 t ) 11895 − 13 e − t 15 + 459 e − 5 t 3 305 + 11 e − 3 t 39 x t 12 ( t ) = − 9671 11895 − 26 co s ( t ) 255 − 2 sin ( t ) 255 − 6094 co s ( t ) sin ( t ) 11895 + 5068 cos 2 ( t ) 11895 + 7 e − t 15 + 417 e − 5 t 3 10370 − 7 e − 3 t 390 . The gra phs of these explicit transfer subﬂow and substora ge functions in Eqs. 3.11 and 3.12 obtained through the dynamic a pproach a re exactly the s ame a s the ones obtained b y numerical co mputation thr ough the pa th-based appr oach, E q. 3.7 , as depicted in Fig. 8 . The cycling ﬂows and the asso ciated stora ges generated by these ﬂows are also calculated below for both co mpartments. The sets of m utua lly exclusive subﬂow paths from s ubco mpartment k k to 1 k with a closed subpath at 1 k , P c 1 k k k , a re given as P c 1 0 0 0 = { p 1 1 0 1 0 , p 2 1 0 2 0 } , P c 1 1 1 1 = { p 1 1 1 1 1 } , P c 1 2 2 2 = { p 1 1 2 2 2 } , w her e p 1 1 0 1 0 = 0 0 7→ 1 0 2 0 → 1 0 , p 2 1 0 2 0 = 0 0 7→ 2 0 1 0 2 0 → 1 0 , p 1 1 1 1 1 = 0 1 7→ 1 1 2 1 → 1 1 , a nd p 1 1 2 2 2 = 0 2 7→ 2 2 1 2 2 2 → 1 2 . F or the subﬂo w paths in P c 1 0 0 0 , the co mpo site cycling subﬂows ar e der ived from the initial sto cks, and for the ones in P c 1 1 1 1 and P c 1 2 2 2 , the simple cycling ﬂows are genera ted by the resp ective environmen tal inputs of z 1 ( t ) and z 2 ( t ). The sets of subﬂow paths P c 2 k k k for k = 0 , 1 , 2 can similarly be deﬁned. The simple cycling subﬂow at sub compar tmen t 1 2 along the o nly subﬂow path ( w 2 = 1) in subsys tem 2, p 1 1 2 2 2 ∈ P c 1 2 2 2 , and ass o ciated substo rage ar e τ c 1 2 ( t ) = 1 X w =1 ˇ τ w 1 2 ( t ) = ˇ τ 1 1 2 ( t ) and x c 1 2 ( t ) = 1 X w =1 x w 1 2 ( t ) = x 1 1 2 ( t ) , as for m ulated in Eq. 2.48 . The links contributing to the cycling subﬂow along the path a r e marked with cycle n umbers in the extended s ubﬂow diag ram b elow: p 1 1 2 2 2 = 0 2 7→ 2 2 1 2 2 2 1 − → 1 2 2 2 2 − → 1 2 · · · Note tha t the ﬁr s t ﬂow e ntrance into 1 2 is not co nsidered a s cycling ﬂow. The c umu- lative tr a nsient inﬂow ˇ τ 1 1 2 ( t ) and substora ge x 1 1 2 ( t ) ca n b e approximated by tw o terms ( m 1 = 2) along the closed subpath a s formulated in Eq. 2.46 : x 1 1 2 ( t ) = 2 X m =1 x 1 ,m 2 2 1 2 2 2 ( t ) ≈ x 1 , 1 2 2 1 2 2 2 ( t ) + x 1 , 2 2 2 1 2 2 2 ( t ) , ˇ τ 1 1 2 ( t ) = 2 X m =1 f 1 ,m 1 2 2 2 1 2 ( t ) ≈ f 1 , 1 1 2 2 2 1 2 ( t ) + f 1 , 2 1 2 2 2 1 2 ( t ) . The gov erning equatio ns for the transient subﬂows and asso ciated substora ge func- tions, f w, m 1 2 2 2 1 2 ( t ) and x w, m 2 2 1 2 2 2 ( t ), as well as the o ther transient subﬂows a nd s ubstorage s 32 HUSEYIN COSKUN inv olved in Eq. 3.13 , as for m ulated in Eqs. 2.43 and 2.44 , ar e coupled and so lved si- m ultaneously together with the deco mp o s ed system, E q. 2.1 0 . The numerical results for the co mpo site cycling ﬂows and as so ciated stora ges induced bo th by the environ- men tal inputs a nd initial sto cks, (3.13) τ c i 0 i 0 ( t ) + τ c ii ( t ) = 2 X k =0 τ c i k i k ( t ) and x c i 0 i 0 ( t ) + x c ii ( t ) = 2 X k =0 x c i k i k ( t ) for i = 1 , 2 , are presented in Fig. 8c . Note that, due to the r eﬂexivity of cy c ling ﬂows, the same co mputations can b e done mor e practically in o nly tw o steps using the sets of closed subﬂow paths, P c i k , instead, with the lo cal inputs being the corr esp onding outw ards subthro ughﬂows. The subﬂow path p 1 1 1 = 1 1 7→ 1 1 2 1 → 1 1 in sub compar tment 1 1 with lo cal input ˆ τ 1 1 ( t, x ), fo r example, is depicted in Fig . 6 . The cycling ﬂows can also b e computed along closed pa ths at the compa rtmental le vel, where the lo ca l inputs ar e the outw ards throughﬂows. The comp osite cycling subﬂows can also be computed analytica lly thro ugh the dynamic approach as fo r mulated in E q. 2.57 . As examples, τ c 1 0 1 0 ( t ) and τ c 2 2 ( t ) = τ c 2 2 2 2 ( t ) beco me (3.14) τ c 1 0 1 0 ( t ) = − 36 e − t + 8 0 e 2 t − 1 00 e 1 t − 1 6 e 2 t cos ( t ) + 8 e 2 t sin ( t ) 9 + 50 e 2 t − 7 0 e 3 t + 1 1 e 3 t cos ( t ) − 13 e 3 t sin ( t ) τ c 2 2 2 2 ( t ) = 584 585 − 52 e − t 45 + 44 e − 3 t 117 − 8 sin (2 t ) 585 − 128 cos 2 ( t ) 585 . The comp osite cycling storages can then b e obtained by coupling Eq. 2.63 for the cycling ﬂows a nd storag es with the decomp osed system, E q. 2 .10 , and solving them simult aneous ly . Alternatively , they can b e obtained ana lytically b y using Eq. 2.64 , similar to the transfer storag e s presented above in this example. Because of the lengthy ana lytical for mu lations o f the o ther cycling subﬂows a nd substora ges, o nly τ c 1 0 1 0 ( t ) and τ c 2 2 ( t ) ar e presented in Eq . 3.14 as examples. 3.2. Case study . In this section, a no nlinear re source-pr o ducer-co ns umer ecos y s- tem mo del in tr o duced by [ 1 9 ] is analyzed thro ugh the pr op osed metho dolo gy . A compariso n of the results is no t p o ssible, as the authors did no t pr ovide a ny compu- tational or explicit results in the article. They only provided some results at steady state. Besides a constant environmen tal input, the system is a ls o exa mined for a time dep endent, symmetric Gaussian impulse to illustrate the eﬃcie nc y of the pro- po sed metho d in capturing the sy s tem r esp onse to disturbances. Suc h analysis can be used to q uantif y the system resista nce and res ilience in the face of disturbances and p erturbatio ns. The reso urce-pro duce r-consumer mo del by [ 19 ] consists o f the dynamics for three comp onents: x 1 ( t ) = r ( t ) is the nutrient sto rage (such as phosphor us or nitrogen) present at time t; x 2 ( t ) = s ( t ) r epresents the nutrien t stora ge in the pr o ducer (such as phytoplankton) p opulation; and x 3 ( t ) = c ( t ) denotes the n utrient s torage in the consumer (such as zo opla nkton) p opulation (see Fig. 9 ). The conser v ation of nutrien t is the basic mo del assumption. The system ﬂows are describ ed as follows: F ( t, x ) =    0 d 1 s ( t ) d 2 c ( t ) α 1 s ( t ) r ( t ) α 2 + r ( t ) 0 0 0 β 1 s ( t ) c ( t ) β 2 + s ( t ) 0    , z ( t ) =   z 1 ( t ) z 2 ( t ) z 3 ( t )   , y ( t ) =   r ( t ) s ( t ) c ( t )   , DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 33 x 1 ( t ) x 2 ( t ) x 3 ( t ) p 1 0 1 1 1 Fig. 9: Sc hema tic repre s entation of the mo de l netw o r k. Subﬂow pa th p 1 0 1 1 1 along which the tra nsient subﬂows a nd subs to rages a re co mputed is red (subsystems are not shown) (Case s tudy 3.2 ). where the constant input is z ( t ) = [1 , 1 , 1] T , and the par ameters are given as d 1 = 2 . 7 , d 2 = 2 . 025 , α 2 = 0 . 098 , β 1 = 2 , β 2 = 20 , and α 1 = 1 . The v alue for α 1 was no t pr ovided in [ 19 ] and was chosen arbitra rily for this example. The g overning equations take the following for m: (3.15) ˙ r ( t ) = − r ( t ) + d 1 s ( t ) + d 2 c ( t ) − α 1 s ( t ) r ( t ) α 2 + r ( t ) + z 1 ( t ) ˙ s ( t ) = − (1 + d 1 ) s ( t ) + α 1 s ( t ) r ( t ) α 2 + r ( t ) − β 1 c ( t ) s ( t ) β 2 + s ( t ) + z 2 ( t ) ˙ c ( t ) = − (1 + d 2 ) c ( t ) + β 1 c ( t ) s ( t ) β 2 + s ( t ) + z 3 ( t ) with the initial c onditions of [ r 0 , s 0 , c 0 ] = [1 , 1 , 1]. The system pa rtitioning metho dology is comp ose d of the sub compar tment aliza- tion a nd ﬂow pa r titioning comp o nents. The sub compa rtmantalization yields x 1 k ( t ) = r k ( t ) , x 2 k ( t ) = s k ( t ) , and x 3 k ( t ) = c k ( t ) with x i ( t ) = 3 X k =0 x i k ( t ) . The ﬂow partitioning then gives the ﬂow regime for each subsystem a s follows: F k ( t, x) =   0 d 2 k d 1 s d 3 k d 2 c d 1 k α 1 s r α 2 + r 0 0 0 d 2 k β 1 s c β 2 + s 0   , ˇ z k ( t, x) =   δ 1 k z 1 δ 2 k z 2 δ 3 k z 3   , ˆ y k ( t, x) =   d 1 k r d 2 k s d 3 k c   , where F k , ˇ z k , and ˆ y k describ e the k th direct ﬂow matrix, input, and o utput vectors for the k th subsystem, and the deco mpo sition factors d i k (x) are deﬁned by E q. 2.8 . Therefore, the dynamic system partitioning metho dolo g y yields the following govern- 34 HUSEYIN COSKUN 0 5 10 15 20 25 time 0 0.2 0.4 0.6 0.8 1 1.2 1.4 substorage matrix elements (a) substorages of x 1 ( t ) 0 5 10 15 20 25 time 0 1 2 3 4 subthroughflow matrix elements (b) subth roughﬂows of ˇ τ 1 ( t, x) and ˆ τ 1 ( t, x ) Fig. 1 0: The numerical results for s elected elemen ts (ﬁrst rows) of the substorage, X ( t ), and subthroug hﬂow matrices, ˇ T ( t, x) a nd ˆ T ( t, x), and the initial substora ge, x 0 ( t ), and subthroughﬂow vectors, ˇ τ 0 ( t, x) and ˆ τ 0 ( t, x), for the sys tem with co nstant input z ( t ) = [1 , 1 , 1] T (Case study 3.2 ). ing equations for the decomp osed system: (3.16) ˙ r k ( t ) = δ 1 k z 1 ( t ) + d 1 s k ( t ) + d 2 c k ( t ) − r k ( t ) − α 1 s ( t ) r k ( t ) α 2 + r ( t ) ˙ s k ( t ) = δ 2 k z 2 ( t ) + α 1 s ( t ) r k ( t ) α 2 + r ( t ) − s k ( t ) − d 1 s k ( t ) − β 1 c ( t ) s k ( t ) β 2 + s ( t ) ˙ c k ( t ) = δ 3 k z 3 ( t ) + β 1 c ( t ) s k ( t ) β 2 + s ( t ) − c k ( t ) − d 2 c k ( t ) with the initial c onditions x i k ( t 0 ) = ( 1 , k = 0 0 , k 6 = 0 for i = 1 , . . . , 3. There are n × ( n + 1) = 3 × 4 = 1 2 equatio ns in this system. The system is s o lved numerically and the g r aphs for selected elemen ts of the substorag e and subthroughﬂow matrices a re depicted in Fig. 10 . As seen from the graphs, the system co nv e r ges to a steady-state at ab o ut t ≈ 6. The results show, for example, that the n utrient s to rage in the resource compartment ( i = 1) der ived from nu trient input into the co nsumer compartment ( i = 3 ), x 1 3 ( t ), increases from 0 to 0 . 64 units until the system r e aches the stea dy state, while the initia l nutrien t s torage, x 1 0 , ﬁrst increases from 1 to 1 . 38 units and then v anishes. The throughﬂow into the resource co mpartment g e ne r ated by nutrien t input into the pr o ducer co mpartment ( i = 2), ˇ τ 1 2 ( t, x), incr e a ses until ab out t ≈ 2 . The outw ard throughﬂow at the same sub c ompartment, ˆ τ 1 2 ( t, x), is slightly smaller than inw a rd throughﬂow, ˇ τ 1 2 ( t, x), but has a s imilar b ehavior. As s e en fro m these results, the distribution of environmen tal nu trient inputs and the o r ganizatio n o f the asso ciated nut rient stor ages gener ated by the inputs can b e analyzed individually and separa tely within the system. In general terms, the sta te v aria ble x i ( t ) of the orig inal system fo r the reso urce- pro ducer-co nsumer dynamics, Eq. 3.15 , gives the nutrient stor age in compa rtment i DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 35 0 5 10 15 20 25 time 0 0.2 0.4 0.6 0.8 1 1.2 1.4 substorage matrix elements (a) substorages of x 1 ( t ) 0 5 10 15 20 25 time 0 1 2 3 4 subthroughflow matrix elements (b) subth roughﬂows of ˇ τ 1 ( t, x) and ˆ τ 1 ( t, x ) Fig. 11: The numerical r esults for the selected elements (ﬁrst rows) of the substora ge, X ( t ), and subthroughﬂow matrix functions, ˇ T ( t, x) and ˆ T ( t, x), and the initial subs to r- age, x 0 ( t ), and subthroughﬂow vectors, ˇ τ 0 ( t, x) and ˆ τ 0 ( t, x), for the system with time- depe ndent environmen tal input (Gaussian impulse function) z 2 ( t ) = e − ( t − 15) 2 2 + 0 . 1 , and constant inputs z 1 ( t ) = 1 a nd z 3 ( t ) = 1 (Case study 3.2 ). at time t base d on its initial sto ck, x i ( t 0 ). It cannot be used to distinguis h the n utrient storage derived fr om individual environmen tal nutrien t inputs. On the other ha nd, the state v ariable x i k ( t ) of the deco mp o s ed system, Eq . 3.16 , repr esents the nutrien t storage in compartment i that is derived from the sp eciﬁc environmen tal nutrien t in- put int o compartment k , z k ( t ). Similarly , the state v a riable x i 0 ( t ) of the deco mpo sed system represents the dynamics of the initial nut rient sto cks in co mpartment i . Par- allel interpretations are p ossible for the inw ard and o utw ar d throughﬂow functions of the origina l system, ˇ τ i ( t, x ) and ˆ τ i ( t, x ), and the inw ard and outw ard subthroug hﬂow functions o f the deco mpo sed system, ˇ τ i k ( t, x) a nd ˆ τ i k ( t, x), a s well. The pr op osed dynamic system par titioning metho dolo gy , co nsequently , enables partitioning the compartmental comp osite nutrien t ﬂows and sto rages int o sub com- partmental segments ba sed on their co nstituen t sources from the initial sto cks and environmen tal inputs. In o ther words, the system partitioning enables tracking the evolution o f the initial nutrien t sto cks and en viro nment al nutrien t inputs a well as the as so ciated s torages g enerated b y the s to c ks a nd inputs individually and sepa- rately within the s ystem. This partitioning also allows for co mpiling a histor y of compartments v isited by individual nutrien t inputs separ ately . The s ystem is als o p ertur be d with a Gaussian input z 2 ( t ) = e − ( t − 15) 2 2 + 0 . 1 , which represents a brief, unit lo cal impulse a t ab out t = 15 to demonstr ate the capability of the prop osed metho d to analyze the inﬂuence of time dep endent inputs on the system. The other tw o environmen tal n utrient inputs are kept constant a s b efore for a co mparison, that is , z 1 ( t ) = z 3 ( t ) = 1. The graphical representations for the selected elements of the substora ge a nd subthroug hﬂow matrices ar e given in Fig. 11 . It is clear from the g raphs that the dynamic substora ge and subthroughﬂow matrix measures r eﬂect the impact of the unit impulse at ab out t = 15. Note that, the system completely recovers after the distur ba nce in a bo ut 10 time units. This time interv al can b e taken a s a quantitativ e measure for the re stor ation t ime or system r esilienc e . 36 HUSEYIN COSKUN 0 5 10 15 20 2 5 time 0 0.2 0.4 0.6 0.8 1 transient substorages (a) 0 5 10 15 20 25 time 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 transient subflows (b) 0 5 10 15 20 25 time 0 1 2 3 4 5 6 7 transient substorages 10 -4 (c) 0 5 10 15 20 25 time 0 0.2 0.4 0.6 0.8 1 transient subflows 10 -3 (d) Fig. 12: The numerical r esults for the transient subﬂows and substorag es at each step along subﬂow paths p 1 3 1 1 1 (a,b) and p 1 0 1 1 1 (a.b,c,d). In ﬁgures (a) and (b), functions f 1 3 1 2 1 1 1 ( t ) and x 1 3 1 2 1 1 1 ( t ) are scaled up by a fa ctor of 10 fo r clarity of the presentation (Case study 3.2 ). Therefore, the pr op osed measur es can be used as quantit ative ecologic al indicato rs for v a rious ecos y stem characteristics and b ehaviors. The subsys tem partitioning metho dolo gy is also applied to this mo del to tra ck the fate of ar bitrary intercompartmental ﬂows and the asso cia ted stor ages genera ted by these ﬂows within the subsystems. Along the subﬂow path p 1 3 1 1 1 = 0 1 7→ 1 1 → 2 1 3 1 from sub compartment 1 1 to 3 1 in subsystem 1, the trans ient s ubﬂows and as s o ciated substorag es ar e computed as formulated in Eqs. 2.43 and 2.44 . The n umerical re- sults for the transient subﬂows, f 1 2 1 1 1 0 1 ( t ) and f 1 3 1 2 1 1 1 ( t ), and asso cia ted substor age functions, x 1 2 1 1 1 0 1 ( t ) and x 1 3 1 2 1 1 1 ( t ), ar e presented in Fig. 12b and 12a . The subﬂow path p 1 3 1 1 1 is extended to pa th p 1 0 1 1 1 = 0 1 7→ 1 1 → 2 1 → 3 1 → 1 1 → 2 1 → 1 1 → 0 1 to compute the local output f 1 0 1 1 1 2 1 ( t ) (a segment o f environmen tal output y 1 ( t )) derived fr om the lo cal (and environment al) input z 1 ( t ) = 1 along that particula r pa th DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 37 0 5 10 15 20 25 time 0 1 2 3 4 5 indirect flows 10 -3 (a) indirect (sub)ﬂows 0 5 10 15 20 25 time 0 0.5 1 1.5 2 2.5 3 acyclic flows (b) acyclic (sub )ﬂow s 0 5 10 15 20 25 time 0 0.05 0.1 0.15 0.2 0.25 0.3 cycling flows (c) cycling (sub )ﬂow s 0 5 10 15 20 25 time 0 0.2 0.4 0.6 0.8 1 1.2 indirect storages 10 -3 (d) ind irect (sub) storages 0 5 10 15 20 25 time 0 0.002 0.004 0.006 0.008 0.01 acyclic storages (e) acy clic (sub)storages 0 5 10 15 20 25 time 0 0.05 0.1 0.15 0.2 cycling storages (f ) cycling ( su b)storages Fig. 13: The gra phica l representation of s ome indirect, acyclic, and cy cling ﬂows and sto rages, as w e ll as the corr esp onding initial subﬂo ws and substorag es (Case study 3.2 ). (see Fig. 9 ). That is, the fate of z 1 ( t ) along path p 1 0 1 1 1 within the system is determined. The c orresp o nding transient subﬂow a nd asso ciated substora ge functions a t e ach step (subco mpartment) along the path are a ls o pr esented in Fig . 12d and 12c . Since f 1 0 1 1 1 2 1 ( t ) ≤ 6 . 28 × 1 0 − 5 , at most ab out %0 . 00 6 of z 1 ( t ) exits the system through the given subﬂow path p 1 0 1 1 1 at a ny time t . These results indica te that the pro p o sed dynamic subsys tem partitioning metho d- ology enables dyna mically tr acking the fate of an arbitra ry amount o f nutrient ﬂow and a sso ciated nutrien t storage alo ng a given ﬂow path. Consequently , the sprea d of an arbitrar y a mount of nu trient from o ne compar tment to the entire system ca n be monitored. Mo r eov er, the eﬀect of one compartment o n any other in terms o f the nutrien t tra nsfer, through not only dire ct but also indirect in teractions, ca n be determined. The di act ﬂows and s torage s are intro duced in Section 2.5.2 . The indirect ﬂow and stora ge from co mpartment 1 to 3, τ i 31 ( t ) and x i 31 ( t ), the acyclic ﬂow and sto rage from compa rtment 2 to 1, τ a 12 ( t ) and x a 12 ( t ), and the cycling ﬂow and stor age at com- partment 1, τ c 11 ( t ) and x c 11 ( t ), genera ted by the environmental inputs, as well as the corres p o nding initial subﬂows a nd substor ages der ived from the initial sto cks tra ns - mitted in the s ame dire c tio ns are depicted in Fig. 13 . As see n from the gra phs, all initial diact s ubﬂows and substorages v anish as the system conv erge s to a steady-s tate and, then, the s y stem b ehavior is even tually dominated by the environmental inputs. Ecolog ic ally , the ac y clic ﬂow and stor age, τ a 12 ( t ) and x a 12 ( t ), repr esent the n utrient ﬂow at time t and the asso cia ted nutrient s torage g enerated by this ﬂow during [ t 1 , t ] that visit the reso urce c o mpartment only once—do not return to this compar tmen t 38 HUSEYIN COSKUN 0 5 10 15 20 25 time 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 residence time (a) constant inp ut 0 5 10 15 20 25 time 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 residence time (b) time-d ep endent inpu t Fig. 14: The gra phica l representation for the res ide nce times of the sys tem com- partments with bo th the constant and time-dep endent environmental inputs, z ( t ) = [1 , 1 , 1 ] T and z ( t ) = [1 , e − ( t − 15) 2 2 + 0 . 1 , 1] T (Case study 3.2 ). for a seco nd time la ter—after b eing dir ectly or indirec tly transmitted from the pro- ducer co mpa rtment. The initial a cyclic subﬂow a nd substor age, τ a 1 0 2 0 ( t ) and x a 1 0 2 0 ( t ), represent the same pheno mena within the initial subsys tem. Similar ly , the indirect ﬂow and storage, τ i 31 ( t ) and x i 31 ( t ), repr esent the nutrien t ﬂow and s torage tra nsmit- ted indirectly from the r esource compa rtment thr ough the pr o ducer to the consumer compartment. The cycling ﬂow a nd stor age, τ c 11 ( t ) and x c 11 ( t ), repr esent the nutrien t ﬂow and s torage transmitted indirectly fro m the r esource compar tmen t through other compartments ba ck into itself. The other di act (sub)ﬂows and (sub)stora g es can be int erpre ted similarly , for b o th the subsys tems and initial subsystem to a nalyze the in- tercompartmental dynamics gene r ated resp ectively by the environmen tal inputs and initial sto c ks, individually and separately . The reside nc e time ma trix is a nother nov el mathematical measur e pro p o sed for quantitativ e system a nalysis [ 10 , 9 ]. The i th diagonal element of R ( t, x ) at time t 1 , r i ( t 1 , x ), can be interpreted as the time required for the outw ard throug hﬂow, at the constant rate of ˆ τ i ( t 1 , x ), to completely empty compartment i with the stor age of x i ( t 1 ). The diagonal structure of the res idence time matrix indicates that all sub- compartments of compar tmen t i v anish simultaneously . The r esidence times measure compartmental activity le vels [ 11 ]. The sma ller the residence time the more active the co rresp onding compartment. The deriv a tive of the reside nc e time matrix is called the r everse activity r ate matrix [ 9 ]. The residence time functions for this mo del with b oth the c o nstant and time- depe ndent environmental inputs a re depicted in Fig . 14 , for a compariso n. The res- idence times of b oth the consumer a nd pro ducer compartments are almost c onstant and the s ame in bo th cases . In terestingly , the Gaussia n impulse at the pr o ducer c om- partment, z 2 ( t ), has no signiﬁcant impact on the activity level of the consumer and even that of the pro ducer compartment itself. How ever, the decr e ase in the input into the pro ducer co mpartment from co ns tant z 2 ( t ) = 1 to z 2 ( t ) = e − ( t − 15) 2 2 + 0 . 1 results in a n ov erall increa se in the residence time o f the resour c e compartment (and all of its sub c ompartments) from the steady state v alue o f r 1 = 0 . 87 days to r 1 = 0 . 98 days. DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 39 0 5 10 15 20 2 5 time -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 net diact storage transactions (a) sign analysis 0 5 10 15 20 25 time 0 0.01 0.02 0.03 0.04 0.05 diact interaction strengths (b) strength analysis Fig. 15: The gra phical representation for the net di act stor a ge tr a nsactions and the strengths of the di act interspeciﬁc interactions b etw een the pro duce r ( i = 2) and consumer (3 ) compartments fo r the time-dependent input case (Case study 3.2 ). Moreov er, the maximum impulse at t = 1 5 decrea s es this re sidence time, r 1 ( t ), lo cally in time. Tha t is, R (10) = R (25) = diag ([0 . 98 , 0 . 27 , 0 . 33 ]) but R (15) = diag ([0 . 85 , 0 . 27 , 0 . 3 3]) . Consequently , the r esidence time of the r esource co mpartment adversely impacted by the environmental input into the pro ducer compartment. The mathematical classiﬁcatio n of the dia ct interspeciﬁc interactions is a lso in- tro duced in Section 2.8 . The sign a nd stre ngth o f the dia ct interactions, induced by environmen tal inputs, b etw een the pro ducer and co nsumer compa r tments bec ome (3.17) δ * ,x 32 ( t ) = s gn ( x * 32 ( t ) − x * 23 ( t )) and µ * ,x 32 ( t ) = | x * 32 ( t ) − x * 23 ( t ) | x 2 ( t ) + x 3 ( t ) for the storage - based analys is. The n umerical r esults for the net dia ct sto rage trans- actions a nd the stre ngths of the d iact int era ctions ar e pr esented in Fig. 15 . As see n from the sign ana lysis in Fig. 15a , (3.18) δ d ,x 32 ( t ) = (+ ) , δ i ,x 32 ( t ) = δ a ,x 32 ( t ) = δ c ,x 32 ( t ) = δ t ,x 32 ( t ) = ( − ) . These r esults indicate that the diact in teractions induced b y the environment al inputs betw een the pro ducer and cons umer compartments a re all a ntagonistic. Although the consumer compartment directly bene ﬁts fr o m the pro ducer compartment as exp ected, int eresting ly , their indirect, cycling, acy clic, and total interactions are detrimental to the consumer compartment. The strengths of the dia ct iterations are o r dered as follows: (3.19) µ c ,x 32 ( t ) < µ d ,x 32 ( t ) < µ a ,x 32 ( t ) < µ t ,x 32 ( t ) < µ i ,x 32 ( t ) for t ≥ 0 . 39. Ther efore, the indirect interaction b etw een the pro ducer and co nsumer compartments is the strong est of all di act interactions. Since the indirect interac- tion dominates the direct interaction, their ov erall interactions is co unt erintuitiv ely detrimental to the consumer compartment after t = 0 . 3 9. 40 HUSEYIN COSKUN The deta ile d infor mation and inferences enabled by the prop osed metho do logy cannot b e obtained thro ugh the analysis of the or ig inal sy stem by the state-of-the- art techn iques, as demonstra ted in these ca se studies. 4. Discussion. Environmen t is not a n easy concept to deﬁne in genera l and, in particular, to a nalyze mathematically . One reaso n for this is that nature is always on the mov e and ecolo g ical systems strug gle to ada pt to cons tantly changing circum- stances. Althoug h sound r ationales a re oﬀer ed in the literature for the analy sis of natural system dynamics, they ar e only for sp ecia l cases, such a s linear mo dels and static systems. In recent deca des, there ha s b een several attempts to ana lyze dyna mic ecologica l netw orks, but each of them b ears dis a dv antages. The need for dynamic and nonlinear metho do logies has alwa ys existed. This manuscript prop oses a nov el math- ematical metho dolog y for the analysis of nonlinear dyna mic compartmental systems to comprehens ively addr ess these shor tcomings. Considering a h yp othetical ecosystem with several interacting sp ecies for which the eﬀect o f a sp eciﬁc p ollutant needs to b e inv estigated, monitoring the evolution of that p ollutant within this fo o d web w ould be critica l to a ddr essing the p otential harm. Given the initial po llutant sto cks in each of the sp ecies, curr ent deterministic math- ematical metho ds can a nalyze the compo site throug hﬂow and storage of the toxin in the s p ecie s. The evolution of each environmen tal p ollutant input se pa rately within the web, how e ver, cannot b e deter mined throug h the curre nt metho dolo gies. In the case that m ultiple sp ecies exp osed to the same p ollutant from the environmen t, the prop osed sys tem partitioning metho do logy enable s dynamica lly partitioning the com- po site p ollutant ﬂow a t and storag e in any spe c ies into subc ompartmental s e gments based on their constituent so urces from the initial p ollutant sto cks and environmen- tal po llutant inputs. In other words, the sys tem partitioning enables tracking the evolution of the initial p ollutant sto c ks a nd environmental p ollutant inputs in each sp ecies, as w ell as the a s so ciated po llutant storages derived from these inputs and sto cks individua lly and separa tely w ithin the foo d web. The pr op osed s ubs ystem partitioning metho dology can then dynamically track the fate of arbitrar y intercompartmental p ollutant ﬂows a nd asso ciated sto rages in each sp ecies along a given fo o d chain in the web as well. Therefore , the spread of an arbitrar y amount of toxin fro m o ne sp ecies to the ent ire web o r along a sp eciﬁc fo o d chain can b e monitored. Such infor mation can help, for example, to identif y critical pathw ays in which interven tion helps to alleviate the eﬀects of the p ollutant. The direct, indirect, acyclic, cycling , a nd transfer ( dia ct ) ﬂows and as so ciated s torages of the pollutant fro m o ne sp ecies directly or indire c tly to any other—including itself—can also b e determined. More tec hnically , the pro p o sed sy stem partitio ning metho do logy explicitly gen- erates mutually exclusive and exhaustive subsystems. Except the initial s ubs ys- tem—whic h is driven by the initia l sto cks—each subs ystem is driven by a single envi- ronmental input. The subsy stems ar e running within the o riginal s ystem and hav e the same structure and dynamics as the system itself, except fo r their initial sto cks and environmen tal inputs. The system partitioning yields the subthro ughﬂows and sub- storage s that repr e s ent ﬂo ws and storag e s derived from the initial sto cks a nd generated by e nvironmental inputs in each compartment. That is, the comp osite compar tmen- tal stora ges and throug hﬂows, x i ( t ) and τ i ( t ), are dynamica lly partitioned into the sub c ompartmental substor age a nd subthroughﬂow seg men ts, x i k ( t ) and τ i k ( t ), based on their cons tituen t so ur ces fro m the initial sto cks and environmental inputs, x i 0 and z k ( t ). Equipp ed with these mea sures, the s ystem partitio ning ascer tains the dynamic DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 41 distribution of the initial sto cks and environmental inputs and the o r ganizatio n of the asso ciated stor ages derived from these sto cks and inputs within the sy stem. In other words, the system par titioning enables dynamically tracking the evolution of the ini- tial sto cks and environmental inputs individually a nd separately within the sy s tem. The s ystem par titioning metho dolo gy , therefore, re ﬁnes system analysis from the cur- rent static, linear, co mpa rtmental to the dynamic, nonlinear, s ub co mpartmental level to explor e the full complexity of the ecolog ical sy s tems. The subsystems a re then further decomp ose d into subﬂows and substorag es along a set of mutually ex clusive a nd exhaustive directed subﬂow paths. The subsystem partitioning metho dology yields the transient subﬂows a nd substor ages in each sub- compartment a long a given subﬂow pa th within a subsy stem, g enerated by or derived from an arbitrar y intercompartmental ﬂow or storag e. Therefore, arbitra ry co mpo s- ite intercompartmental ﬂows and asso c ia ted storag es ca n b e app ortio ne d dynamically int o tr ansient subﬂow and s ubstorag e segments along a given set of subﬂow pa ths. That is, the tr ansient subﬂows and substorag es deter mine the dynamic dis tribution of arbitr ary int erco mpartmental ﬂows and the or ganization o f the a sso ciated storag es generated by these ﬂows within the subsystems. As a result, the spread of an ar- bitrary ﬂow or stora ge segment from o ne compa r tment to the entire sys tem ca n b e monitored. Moreov er, an archive of compartments vis ited by a rbitrary sys tem ﬂows and s torages can be also compiled. In brief, the subsys tem partitioning methodo l- ogy ena bles tra cking the fate o f a r bitrary intercompartmental ﬂows a nd asso c iated storage s within the subsystems. The pro po sed mathematical metho d, a s a whole, enables the decomp osition to the utmost level, or “atomization,” of the system ﬂows and stora ges. In additio n to the subthroughﬂows, substora ges, transient ﬂows and storages , the dy namic direct, indi- rect, acyc lic , cy cling, and tr ansfer ( diac t ) ﬂows and stor ages fro m one compar tment directly or indirectly to any other —including itse lf—are als o sys tema tically for mu - lated for the quantiﬁcation of intercompartmental ﬂow and stor age dynamics. The diact ﬂows and storag es are der ived explicitly throug h the dy namic a nd pa th-ba sed approaches, whic h a re based on the sys tem and subsystem par titioning methodo logies, resp ectively . As an immedia te application, a mathematical technique that character - izes a nd class iﬁes the neutr a l and a ntagonistic nature of di act interspeciﬁc interac- tions in fo o d webs and determines the streng th of these interactions is also developed based on the diact ﬂows and stora ges. The illustr ative case studies in Section 3 demonstrate the rigor and eﬃciency of these mathema tica l system a nalysis to ols as ecologica l s ystem indicator s. F or a co mparison of the pro p o sed metho dology with the state-o f-the-art tech- niques, we ﬁrst note that, at a steady s tate, the pr op osed dynamic methodolo gy agrees with the current techniques for static ecolo gical netw ork ana lysis, a s shown by [ 11 ]. In recent decades , there hav e been several attempts to ana lyze dynamic ecolog- ical netw orks . The ﬁrs t actual dyna mic analy s is was limited to linear systems with time-dep e ndent input [ 23 ]. The prop o sed metho d is a pplied to the linear ecosys tem mo del intro duce d b y [ 23 ] as a n illustrative case study in Section 3 . It is shown that the analytic solutions obtained by the pro p o sed metho dology agree with tho se obta ined by Hipp e’s appr oach. F urther results that are no t av ailable through Hipp e’s appr oach, such as the di act ﬂows and stora ges, a re als o presented for this linear sys tem. The dy na mic appro ach is ex tended fro m linear to nonlinear sys tems by [ 19 ]. The authors provided, how ever, only closed-for m, abstract fo r mulations that are diﬃcult to apply to rea l cases. The prop o sed metho dology is also applied to the no nlinear ecosystem mo del analyzed by [ 19 ], and the res ults, together w ith their eco logical 42 HUSEYIN COSKUN int erpre ta tions, are pres ented in Section 3 . A compar ison o f our results with the o nes provided by the a uthors was not p o ssible b eca use, unlike the c o mprehensive dynamic analysis enabled by the pr op osed metho do logy for nonlinear sys tems , the authors could only pr ovide asympto tic solutions to the mo del at steady state through their metho dology . Individual-based algorithms that rely on particle tracking sim ulations are also prop osed for dynamic no nlinear ecologica l mo dels in the liter ature. A trunca ted in- ﬁnite series formulation, for example, was prop osed by [ 41 ]. How ever, the authors ’ approach was approximate, computationally resour ce-intensiv e, and oﬀered no g uar- antees of series conv ergence . While a guarantee of c onv ergence is added by [ 29 ], the c o mputation r emained resour ce-intensiv e due to the individual-based simulation techn ique [ 30 , 43 ]. This is the ﬁrst man uscript in the literature that comprehensively a ddresses all the previously identiﬁed problems and shortcomings . The metho d’s prima ry limitation is that it is designed for the ana ly sis of co nserv a tive mo dels deﬁned in Eq. 2.5 . Since the co nserv a tio n pr inciples ar e fundamental laws of na ture, a lar ge class o f real- world problems ar e formulated based on co nserv a tion principles in many ﬁelds. On the other hand, ther e are still v ar ious no n-conserv ative systems that cannot b e analyzed by the prop osed methodo logy in its cur r ent form. The prop osed metho dolog y can easily be extended for similar analy ses to sys- tems o f hig her o rder or dinary and par tial diﬀerential equations whose source ter ms gov erning the intercompartmental in teractio ns are in the form of the conser v ative compartmental sys tems , as deﬁned in Eq . 2.5 . Suc h an extension of the sy s tem parti- tioning metho dolog y to pa rtial diﬀerential e q uations enable s s patiotemp oral ana lysis of eco logical systems. 5. Conclusions . In the pres ent pap er, we develope d a comprehensive mathe- matical metho d for the ana lysis of nonlinea r dynamic compartmental systems throug h the sys tem decomp osition theory . The pr op osed metho d is bas ed on the no vel an- alytical and explicit, m utually exclusive and exha ustive system and subsystem pa r - titioning metho dolog ies. While the prop ose d dy na mic system partitioning provides the subthro ughﬂow and substo rage matric e s to deter mine the distribution of the ini- tial sto cks and environmen tal inputs, as well as the organiz a tion o f asso c iated stor - ages individually and separ ately within the system, the su bsyst em pa rtitioning yields the transie nt ﬂows and stora ges to determine the distribution of arbitrar y intercom- partmental ﬂows and the org anization of a sso ciated sto r ages within the subsy stems. Consequently , the evolution of the initial sto cks, e nvironmental inputs, and arbitra ry int erco mpa rtmental system ﬂows, as well as the asso c ia ted storages derived from these sto cks, inputs, a nd ﬂows can b e track ed individually and separately within the sys- tem. Moreover, the trans ient and the dynamic direct, indir e c t, acyclic, cycling, a nd transfer ( d iact ) ﬂows a nd a sso ciated stor ages transmitted along a g iven ﬂow path or from one compa r tment dir ectly or indirectly to any o ther within the sys tem a re systematically formulated to ascertain the intercompartmen tal dynamics. T ra ditional ecology is s till largely a de s criptive empirical science. T his nar rows the ﬁeld’s scop e of a pplica bility and compromises its ca pacity to dea l with complex eco - logical netw o rks. The pr op osed dynamic metho d enhances the strength and extends the applicability o f the sta te-of-the-ar t techniques and pr ovides signiﬁcant adv ance- men ts in theory , metho dology , and practica lity . It ser ves as a quantitativ e platform for testing empirica l hypotheses, ecological inferences, and, p otentially , theor etical developmen ts. Therefore, this metho d has the p o ten tial to lead the wa y to a mo re DYNAMIC EC O LOGICAL SYSTEM ANAL YSIS 43 formalistic ecolo gical science . W e consider tha t the prop osed metho do logy bring s a nov e l co mplex sys tem theory to the service o f urgent and challenging environmen tal problems o f the day . Several case s tudies from e cosystem ecolo gy are pres ent ed to demonstrate the accur a cy and eﬃciency of the metho d. The pro p o sed metho do logy a lso lays gr oundwork for the developmen t o f new mathematical sy s tem analys is to ols as q uantitativ e ecologica l indica tors. The time depe ndent nature of these quan tities enables also their time deriv atives and in tegrals to be formulated as novel system measur es. Multiple s uch dynamic di act measures and indices o f matr ix , vector, and scalar types which may pr ov e useful for environmental assessment and mana gement a re sy stematically int ro duced in a sepa rate pap er by [ 9 ]. Ac knowledgmen ts. The author would like to thank the members o f the eco sys- tem ecology gro up in the Depar tment o f Mathematics at UGA, for useful discussio ns and v aluable feedback on a prior draft of this pap er during his v isit in 2016 . In par- ticular, the author is indebted to Caner Ka zanci for intro ducing the static environ theory to the author, iden tifying some o p en problems in the ar ea, a nd reviewing a prior dra ft of the pap er, and other gro up mem b ers, Bernar d C. Patten for his detailed review of a prio r draft of the pap er and Malcolm Adams for us e ful discuss ions and comments. The a uthor a ls o thanks Hasan Coskun, Sang won Suh, Stuart Borett, and Brian F ath for their r e v iew of a prio r draft o f the pa p er and helpful co mment s. REFERENCES [1] T. Allen and M. G iampietro , Holons, cr e aons, genons, envir ons, in hier ar c hy t he ory: Wher e we have g one , Ecological M odell i ng, 293 (2014) , pp. 31–41, h ttps://doi.org/10.1016/j. ecolmodel.2014.06.017 . 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