Dynamic Ecological System Analysis

1 DYNAMIC ECOLOGICAL SYSTEM ANALYSIS

2 HUSEYIN COSKUN∗

3 A Holistic Analysis of Compartmental Systems

4 Abstract. In this article, a new mathematical method for the dynamic analysis of nonlinear 5 compartmental systems is developed in the context of ecology. The method is based on the novel 6 dynamic system and subsystem partitioning methodologies through which compartmental systems 7 are decomposed to the utmost level. The dynamic system and subsystem partitioning enable the 8 determination of the distribution of environmental inputs and intercompartmental system flows as 9 well as the organization of the associated storages generated by these inputs and flows individu- 10 ally and separately within the system. Moreover, the transient and the dynamic direct, indirect, 11 acyclic, cycling, and transfer (diact) flows and associated storages transmitted along a given flow 12 path or from one compartment, directly or indirectly, to any other are analytically characterized, 13 systematically classified, and mathematically formulated. Major flow- and stock-related concepts 14 and quantities of the current static network analyses are also extended to nonlinear dynamic set- 15 tings and integrated with the proposed dynamic measures and indices within the proposed unifying 16 mathematical framework. This comprehensive methodology enables a holistic view and analysis of ecological systems.17

18 Key words. dynamic ecological network analysis, nonlinear dynamic compartmental systems, 19 dynamic system and subsystem partitioning, transient ﬂows and storages, diact ﬂows and storages

20 AMS subject classiﬁcations. 34A34, 70G60, 37N25, 92C42, 92D40

21 1. Introduction. Compartmental systems are mathematical abstractions of net- 22 works that approximate behaviors of continuous physical systems composed of discrete 23 living and nonliving homogeneous components. Based on conservation principles, the 24 system compartments are interconnected through the flow of energy, matter, or cur- 25 rency between them and their environment. Therefore, for the quantification of the 26 compartmental system function, accurate and explicit formulation of flows and as- 27 sociated storages are of paramount importance. Various mathematical aspects of 28 compartmental systems are studied in literature [22,2]. While many fields utilize 29 compartmental modeling, this approach proves particularly well-suited for analysis of 30 ecological systems to address environmental phenomena. 31 Environmental issues have taken center stage in human communities due to cur- 32 rent scientific understandings of population and industrial growth, associated resource 33 demands, and technological advances. Despite this increased attention to the envi- 34 ronment, traditional ecology has an applied nature and is still in the empirical stage 35 of development; a first principles-based formal theory has yet to emerge in its main- 36 stream framework. This narrows the field’s scope of applicability and compromises 37 its ability to deal with complex organism-environment relationships. To that extent, 38 ecology and environmental science are limited in their applied reach by a general in- 39 ability to realistically model and analyze the complex systems of man and nature. 40 Mathematical theories and modeling have significant potential to lead the way to a 41 more formalistic and theoretical ecological science devoted to the discovery of scientific 42 laws. Based on this understanding and prediction, more exact, precise, and incisive 43 environmental applications can be expected to materialize. 44 Sound rationales have been offered in literature for ecological network analysis, 45 but these are for specific cases, such as linear and static models. One such environ- 46 mental system theory known as the environ theory, has been developed over recent

∗Department of Mathematics, University of Georgia, Athens, GA 30602 ([email protected]). 1

This manuscript is for review purposes only. 2 HUSEYIN COSKUN

47 decades for static compartmental models. Building on economic input-output analysis 48 [26, 27] introduced into ecology by [17], the concepts of static flow and storage envi- 49 rons are formulated based on conservation principles [29, 28]. Along parallel research 50 lines, ecological networks and complexity in living systems are also analyzed in the 51 context of information theory and thermodynamics [33, 21, 34, 35] and hierarchy the- 52 ory [1], yet only for static systems. Several software developments computerize these 53 static methods [36,7, 13, 23, 30,5]. 54 Although the steady-state analysis is well-established, dynamic analysis of nonlin- 55 ear systems has remained a long-standing, open problem. For example, Finn’s cycling 56 index defined in static network ecology over four decades ago, has still not been made 57 applicable to ecosystem models that change over time [14]. There are earlier dynamic 58 formulations in the literature, but they are essentially designed for the analysis of 59 special cases, such as time dependent linear systems [20] or closed-form abstract for- 60 mulations [16]. There are also computational methods, such as discrete time [31] and 61 individual based techniques [25, 24] that rely on network particle tracking simulations. 62 Today’s major environmental problems–human impact, climate change, biodiver- 63 sity loss, etc.– all involve change, and this makes the need for mathematically dynamic 64 and analytical methods of nonlinear system analysis not only appropriate, but also 65 urgent [6, 19]. In ecosystem ecology, food webs provide a framework to link com- 66 munity structure with flows of energy and material through trophic interactions and, 67 therefore, reconcile biodiversity with ecosystem function. Temporal variation in web 68 architecture and nonlinearity are discussed in the literature, and it is proposed that 69 this dynamic nature is affecting ecosystem attributes [12, 37]. Nonlinearity and dy- 70 namic behavior including extinction in food webs, however, has yet to be addressed 71 methodologically. 72 This is the first manuscript in literature that potentially addresses the mismatch 73 between the current static and computational methods and applied ecological needs. 74 The proposed methodology is a comprehensive approach in the sense that the pro- 75 posed dynamic measures as well as the major flow- and stock-related results of static 76 ecological network analyses are combined and integrated effectively within this novel 77 and unifying mathematical framework. More importantly, the corresponding con- 78 cepts and quantities of ecological network analyses are further extended from static 79 to nonlinear dynamic settings. Therefore, this unifying approach leads to a holistic 80 analysis of ecosystems. The proposed methodology, in effect, brings a novel, formal, 81 deterministic, complex system theory to the service of urgent ecological problems of 82 the day. 83 The proposed method in line with the mathematical theory introduced by [9] is 84 based on the novel dynamic system and subsystem partitioning methodologies. The 85 system partitioning methodology yields the subthroughflow and substorage matrices 86 that represent the flows and storages generated by individual environmental inputs 87 at each compartment separately. Therefore, the system partitioning enables dynami- 88 cally partitioning composite compartmental flows and storages into subcompartmen- 89 tal segments based on their constituent environmental sources. In other words, it 90 enables dynamically tracking the evolution of environmental inputs and associated 91 storages individually and separately within the system. Through the subsystem par- 92 titioning methodology, then, the transient flows and associated storages transmitted 93 along a given subflow path are formulated. Consequently, arbitrary composite inter- 94 compartmental flows and associated storages are dynamically decomposed into the 95 constituent transient subflow and substorage segments along a given set of subflow 96 paths. Therefore, the subsystem partitioning enables dynamically tracking the fate

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 3

97 of arbitrary intercompartmental flows and associated storages within the subsystems. 98 Based on the concept of the transient flow and storage, the dynamic direct, indirect, 99 acyclic, cycling, and transfer (diact) flows and associated storages transmitted from 100 one compartment, directly or indirectly, to any other are also analytically character- 101 ized, systematically classified, and mathematically formulated for the quantification 102 of intercompartmental flow and storage dynamics. 103 In a nutshell, the system and subsystem partitioning methodologies dynamically 104 determine the distribution of environmental inputs and intercompartmental flows as 105 well as the organization of the associated storages generated by these flows indi- 106 vidually and separately within the system. For such system analysis, the dynamic 107 subthroughflows and substorages as well as the transient and diact flows and stor- 108 ages are systematically introduced in the present paper for the first time in literature. 109 Equipped with these measures, the proposed methodology serves as a quantitative 110 platform for testing empirical hypotheses, ecological inferences, and, potentially, the- 111 oretical developments. The method also constructs a base for the development of new 112 dynamic system measures and indices as ecological indicators. Multiple such quanti- 113 tative tools for the dynamic analysis of ecological network models are systematically 114 introduced by [8]. 115 The proposed method is applicable to any conservative compartmental system 116 regardless of its naturogenic or anthropogenic nature. The method can be used, for 117 example, to analyze models designed for material flows in industry [3]. It can also be 118 used to analyze mass or energy transfers between species of different tropic levels in 119 a complex network or along a given food chain of a food web in nonlinear dynamic 120 settings [18,4, 15]. Although the motivating applications are ecological and envi- 121 ronmental for this paper, the applicability of the proposed method extends to other 122 realms, such as economics, pharmacokinetics, chemical reaction kinetics, epidemiol- 123 ogy, biomedical systems, neural networks, and information science–in fact, wherever 124 dynamical compartmental models of conserved quantities can be constructed. 125 The proposed method is applied to two models in Section3 and Appendices to 126 illustrate its wide applicability and practicality. The first case study analyzed in 127 Section3 concerns nutrient transfer within a nutrient-producer-consumer ecosystem 128 comprised of three compartments. The distribution of both environmental inputs 129 and intercompartmental flows of nutrient as well as the organization of the associ- 130 ated storages generated by these inputs and flows within the system are analyzed 131 through the system and subsystem partitioning methodologies. As the case study 132 demonstrates, an important attribute of the system partitioning is that it enables 133 tracking the evolution of environmental nitrogen and energy inputs within the sys- 134 tem. The dynamic subsystem partitioning methodology then enables dynamically 135 tracking the fate of arbitrary intercompartmental nutrient flows and the associated 136 storages generated by these flows in each compartment along given flow paths within 137 the nutrient-producer-consumer ecosystem, as presented in Section3. 138 Analyzing the direct transactions between compartments, even in more complex 139 systems, is relatively straightforward. However, the dynamic analysis of the indirect 140 mass or energy transfer is made possible through the proposed methodology in the 141 present paper as outlined above. The indirect, cycling, and some other transient 142 and dynamic diact flows and associated storages for another linear ecosystem model 143 comprised of two compartments are presented in Section 3.2. The analytic solution 144 to this linear model is also presented in this case study. 145 This paper is organized as follows: the mathematical method is introduced in 146 Section 2.1, the transient and diact flows and storages are formulated in Section 2.4,

This manuscript is for review purposes only. 4 HUSEYIN COSKUN

147 system analysis and measures are discussed in Section 2.6, results and case studies 148 are presented in Section3, and discussion and conclusions follow in Section4 and5.

149 2. Methods. A new mathematical theory for the decomposition of nonlinear 150 dynamic compartmental systems has recently been developed by [9]. In line with 151 this theory, a mathematical method for the dynamic analysis of nonlinear ecological 152 systems is introduced in the present paper. The proposed theory is based on the novel 153 system and subsystem partitioning methodologies. These methodologies together with 154 the corresponding concepts and quantities will be developed and formulated in this 155 section. 156 The terminology and notations used in this paper are adopted from [9]. They are 157 summarized below: 158 n number of compartments t time [t] xi(t) total material (mass) [m] (or energy, currency) stored in com- 159 partment i, i = 1, . . . , n, at time t fij(t, x) non-negative ﬂow from compartment j to i, at time t [m/t] zi(t, x) = fi0(t, x) environmental (j = 0) input into compartment i at time t yi(t, x) = f0i(t, x) environmental output from compartment i at time t 160 The governing equations for compartmental dynamics are

161 (2.1)x ˙ i(t) =τ ˇi(t, x) − τˆi(t, x)

T 162 for i = 1, . . . , n. The state vector x(t) = [x1(t), . . . , xn(t)] is a differentiable function T 163 of compartmental storages with the initial conditions of x(t0) = x0 = [x1,0, . . . , xn,0] . 164 The total inflow,τ ˇi(t, x), and outflow,τ î(t, x), are called the inward and outward 165 throughflows at compartment i, respectively, and formulated as

n n X X 166 (2.2)τ ˇi(t, x) = fij(t, x) andτ ˆi(t, x) = fji(t, x) j=0 j=0

167 for i = 1, . . . , n. The functions fij(t, x) ≥ 0 represent nonnegative ﬂow rates from 168 compartment j to i at time t and fii(t, x) = 0. Index 0 represents the environment. 169 We further assume that fij(t, x) has the following special form:

170 (2.3) fij(t, x) = qij(t, x) xj(t)

171 where qij(t, x) has the same properties as fij(t, x). We will call qij(t, x) = fij(t, x)/xj(t) 172 the flow intensity directed from compartment j to i per unit storage. Note that 173 qij(t, x) are sometimes called transfer coefficients, technical coefficients in economics, 174 or stoichiometric coefficients in chemistry. The condition, Eq. 2.3, guaranties non- 175 negativity of the compartmental storages, that is xj(t) ≥ 0 for all j. 176 Combining Eqs. 2.1 and 2.2 and separating environmental inputs and outputs, 177 the system takes the following standard form:

 n   n  X X 178 (2.4)x ˙ i(t) = zi(t, x) + fij(t, x) − yi(t, x) + fji(t, x) j=1 j=1

179 with the initial conditions xi(t0) = xi,0, for i = 1, . . . , n. There are n equations; one 180 for each compartment. We assume that zi(t, x) > 0 and xi,0 > 0 for all i. These

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 5

181 assumptions of positive input and initial conditions ensure that the storage values are 182 always strictly positive, xj(t) > 0. 183 The proposed methodology is designed for conservative compartmental systems. 184 A dynamical system is called compartmental if it can be expressed in the form of 185 Eq. 2.4. The compartmental systems is called conservative if all internal ﬂow rates 186 add up to zero when the system is closed, that is, when there is neither environmental 187 input nor output:

n X 188 (2.5) x˙ i(t) = 0 when z(t, x) = y(t, x) = 0 for all t i=1

189 where 0 is the zero column vector with n zero components [9]. 190 For notational convenience, we deﬁne a direct ﬂow matrix function F of size n×n 191 as

192 F (t, x) = (fij(t, x))

193 and the throughﬂow vectors of size n as

T τˇ(t, x) = [ˇτ1(t, x),..., τˇn(t, x)] = z(t, x) + F (t, x) 1 and 194 T T τˆ(t, x) = [ˆτ1(t, x),..., τˆn(t, x)] = y(t, x) + F (t, x) 1

T T 195 where z(t, x) = [z1(t, x), . . . , zn(t, x)] is the input, y(t, x) = [y1(t, x), . . . , yn(t, x)] is 196 the output vector function, and 1 denotes the column vector of size n whose entries 197 are one.

198 2.1. Dynamic System Partitioning Methodology. In this section, we in- 199 troduce the dynamic system partitioning methodology for partitioning the system 200 into mutually exclusive and exhaustive subsystems. This system partitioning deter- 201 mines the distribution of environmental inputs, the organization of the associated 202 storages generated by the inputs, and the dynamics of the initial stocks individually 203 and separately within the system. Therefore, the system partitioning methodology 204 dynamically decomposes composite compartmental throughﬂows and storages into 205 subcompartmental segments based on their constituent environmental inputs. 206 System partitioning involves the dynamic subcompartmentalization and ﬂow par- 207 titioning components, whose mechanisms are explained in this section. The concepts 208 and notations used in this section are summarized below: 209

xik (t) storage generated by environmental input zk(t, x) during [t0, t] and stored in subcompartment k of compartment i, that is, in subcompartment ik, k = 0, . . . , n, at time t fi j (t, x) non-negative ﬂow from subcompartment jk to ik at time t 210 k k yik (t, x) = f0ik (t, x) environmental (j = 0) output from subcompartment ik at time t

zik (t, x) = δik zi(t, x) environmental input into subcompartment ik at time t, where δik is the discrete delta function 211 The system is partitioned explicitly and analytically into mutually exclusive and 212 exhaustive subsystems as follows. Each compartment is partitioned into n+1 subcom- 213 partments; n initially empty subcompartments for n positive inputs and 1 subcom- 214 partment for the initial stock of the compartment. The notation ik is used to represent

This manuscript is for review purposes only. 6 HUSEYIN COSKUN

215 the kth subcompartment of the ith compartment for i = 1, . . . , n and k = 0, . . . , n. 216 The subscript index k = 0 represents the initial subcompartment of compartment i. 217 The storage in subcompartment ik will be called the substorage in ik and de-

218 noted by xik (t). More speciﬁcally, the substorage xik (t) is deﬁned as the storage in 219 compartment i at time t that is derived from the environmental input, zk(t), into 220 compartment k 6= 0 during the time interval [t0, t] (see Fig.1). Consequently, we have n X 221 (2.6) xi(t) = xik (t), i = 1, . . . , n. k=0

222 We deﬁne a new vector variable for the substorages as T 223 x(t) = [x10 (t), . . . , xn0 (t), x11 (t), . . . , xn1 (t), . . . , x1n (t), . . . , xnn (t)] .

y3 x32 z3 x33

x3 f31 1 f32 x30 f f13 23

y1 y2 z2 x13 x12 x22 f12 x23

z1 x10 x11 f21 x21 x20

x1 x2

Fig. 1. Schematic representation of input-oriented dynamic subcompartmentalization in a three- compartment model system. Each subsystem is colored diﬀerently; the second subsystem (k=2) is

blue, for example. Only the subcompartments in the same subsystem (x12 (t), x22 (t), and x32 (t) in the second subsystem, for example) interact with each other. Subsystem k receives environmental input only at subcompartment kk. The initial subsystem has no environmental input. The dynamic flow partitioning is not represented in this figure. Compare this figure with Fig.2, in which the subcompartmentalization and the corresponding flow partitioning are illustrated for x1(t) only. 224 225 We assume that compartment i has a positive environmental input, and it enters th 226 the system at subcompartment ii, for all i. Moreover, no other i subcompartment of 227 any other compartment j, that is, subcompartment ji, receives environmental input. 228 This can be expressed as ( zii (t, x) = zi(t, x), i = k 229 z (t, x) = δ z (t, x) = ik ik i 0. i 6= k

230 The intercompartmental ﬂows are also partitioned in line with the subcompart- 231 mentalization. The intercompartmental ﬂow, fij(t, x), is partitioned based on the

232 assumption that the subcompartmental ﬂow segment, fikjk (t, x), is proportional to

233 the corresponding substorage, xjk (t), with proportionality factors of the ﬂow intensi-

234 ties, qij(t, x), in the ﬂow direction (see Fig.2). The subcompartmental ﬂow fikjk (t, x)

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 7

235 will be called the subﬂow from subcompartment jk to ik at time t. It can be formulated 236 as

fij(t, x) 237 (2.7) fikjk (t, x) = xjk (t) = djk (x) fij(t, x) xj(t)

238 where the coeﬃcients djk (x) = xjk (t)/xj(t) will be called the decomposition factors. 239 Consequently, we have

n X 240 (2.8) fij(t, x) = fikjk (t, x), i, j = 1, . . . , n. k=0

x1 fj1

fj313 x13 x12 fj212

fj111 x10 z1 x11 fj010

Fig. 2. Schematic representation of a dynamic flow partitioning in a three-compartment model system. The figure illustrates subcompartmentalization of compartment i = 1 and the corresponding dynamic flow partitioning from this compartment to others, j. 241 242 Thus, for each k, we explicitly generate a subsystem running within the system 243 through the dynamic system partitioning methodology. This kth subsystem is com- 244 posed of all kth subcompartments and their corresponding substorages and subflows. 245 These mutually exclusive and exhaustive subsystems have the same structure and 246 dynamics as the original system, except for their environmental inputs and initial 247 conditions (see Figs.1 and2). By mutual exclusiveness we mean that transactions 248 are possible only within corresponding subcompartments of the system. By exhaus- 249 tiveness we mean that all the generated subsystems sum to the entire system so 250 partitioned. Except the initial subsystem, each subsystem is driven by a single envi- 251 ronmental input. Consequently, in a system with n compartments, assuming nonzero 252 inputs for each compartment and presence of initial stocks, each compartment has 253 n + 1 subcompartments, and, therefore, the system has n + 1 subsystems, indexed 254 by k = 0, . . . , n. The initial subsystem (k = 0) represents the evolution of the initial 255 stocks, contains no environmental input, and has the same initial conditions as the 256 original system. The initial conditions for all the other subcompartments are zero as 257 explained below. 258 The governing equation for each subcompartment ik then becomes

 n   n  X X 259 (2.9)x ˙ ik (t) = zik (t, x) + fikjk (t, x) − yik (t, x) + fjkik (t, x) j=1 j=1

260 for i = 1, . . . , n, k = 0, . . . , n. There are n × (n + 1) of such governing equations, one 261 for each subcompartment. In order to track the evolution of environmental inputs

This manuscript is for review purposes only. 8 HUSEYIN COSKUN

262 within the system individually and separately, all except the initial subcompartments 263 are assumed to be initially empty. Therefore, the initial conditions become ( xi,0, k = 0 264 (2.10) x (t ) = ik 0 0. k 6= 0

265 The governing system of equations is then solved numerically, and the solution to the 266 subsystems, Eq. 2.9, with the initial conditions, Eq. 2.10, gives the substorages at any

267 time t, that is, xik (t). 268 Total subcompartmental inflows and outflows at each compartment i at time t 269 generated by the environmental input into compartment k during [t0, t] can then be 270 defined, respectively, as n n X X 271 (2.11)τ ˇik (t, x) = zik (t, x) + fikjk (t, x) andτ îk (t, x) = yik (t, x) + fjkik (t, x) j=1 j=1

272 for k = 0, 1, . . . , n. Therefore,τ ˇik (t, x) andτ îk (t, x) will respectively be called inward 273 and outward subthroughflow at subcompartment ik at time t (see Fig.4). Therefore, 274 the dynamic system partitioning methodology determines the distribution of environ- 275 mental inputs and the organization of the associated storages generated by the inputs 276 within the system. Consequently, this methodology enables tracking the evolution of 277 environmental inputs individually and separately within the system. The flows and 278 storages for the kth subsystem in matrix form are formulated in AppendixA. 279 We also define the n × n inward and outward subthroughflow and associated sub- 280 storage matrix functions, Tˇ(t, x), Tˆ(t, x), and X(t), respectively, as follows: ˇ ˆ 281 (2.12) Xik(t) = xik (t), Tik(t, x) =τ ˇik (t, x), and Tik(t, x) =τ îk (t, x)

282 for i, k = 1, . . . , n. The inward and outward subthroughflow and associated substorage 283 vector functions of size n for the initial subsystem,τ ˇ0(t, x),τ ˆ0(t, x), and x0(t), can also T T 284 be defined asτ ˇ0(t, x) = [ˇτ10 (t, x),..., τˇn0 (t, x)] ,τ ˆ0(t, x) = [ˆτ10 (t, x),..., τˆn0 (t, x)] , T 285 and x0(t) = [x10 (t), . . . , xn0 (t)] , respectively. We use the constant notation x0 for 286 the constant initial conditions and the function notation x0(t) for the evolution of 287 these initial conditions for t > t0 with x0(t0) = x0. 288 Let the notation diag(x(t)) represent the diagonal matrix whose diagonal elements 289 are the elements of vector x(t) and diag(X(t)) represent the diagonal matrix whose 290 diagonal elements are the same as the diagonal elements of matrix X(t). The n × n 291 diagonal storage, output, and input matrix functions, X (t), Y(t, x), and Z(t, x) will 292 be defined, respectively, as

293 X (t) = diag(x(t)), Y(t, x) = diag(y(t, x)), and Z(t, x) = diag(z(t, x)).

294 Using Eq. 2.7, the subthroughﬂow matrices can then be formulated as follows: Tˇ(t, x) = Z(t, x) + F (t, x) X −1(t) X(t) 295 (2.13) Tˆ(t, x) = Y(t, x) + diag F T (t, x) 1 X −1(t) X(t) = T (t, x) X −1(t) X(t)

296 where T (t, x) = diag (ˆτ(t, x)) = Y(t, x) + diag F T (t, x) 1. Note that,

x(t) = x0(t) + X(t) 1, 297 τˇ(t, x) =τ ˇ0(t, x) + Tˇ(t, x) 1, andτ ˆ(t, x) =τ ˆ0(t, x) + Tˆ(t, x) 1.

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 9

298 The governing equation, Eq. 2.9, can be expressed as a matrix-vector equation in 299 terms of the vector and matrix functions introduced above as follows: ( X˙ (t) = Tˇ(t, x) − Tˆ(t, x) 300 (2.14) x˙ 0(t) =τ ˇ0(t, x) − τˆ0(t, x)

301 with the initial conditions ( X(t0) = 0 302 (2.15) x0(t0) = x0

303 where 0 is used for both the n × n zero matrix and the zero vector of size n. 304 We also deﬁne an n × n matrix function A(t, x) as

A(t, x) = F (t, x) − Y(t, x) − diag F T (t, x) 1 X −1(t) 305 (2.16) = (F (t, x) − T (t, x)) X −1(t) = Q(t, x) − R−1(t, x)

306 where Q(t, x) := F (t, x) X −1(t) and R−1(t, x) := T (t, x) X −1(t). Note that the first 307 term in the definition of A(t, x), Q(t, x), represents the intercompartmental flow in- 308 tensity defined in Eq. 2.7, and the second term, R−1(t, x), represents the outward 309 throughflow intensity. Consequently, we will call A(t, x) the flow intensity matrix. It 310 is sometimes called the compartmental matrix. As indicated earlier in Eq. 2.3, Q(t, x) 311 is called the coefficient matrix in general, but it will be called the storage distribution 312 matrix in the context of the proposed methodology. The new matrix measure intro- 313 duced in this work, R(t, x), will be called the residence time matrix. The governing 314 equations, Eqs. 2.14 and 2.15, can be expressed using the flow intensity matrix as

X˙ (t) = Z(t, x) + A(t, x) X(t),X(t0) = 0, 315 (2.17) x˙ 0(t) = A(t, x) x0(t), x0(t0) = x0.

316 The dynamic system partitioning methodology that yields a decomposed system 317 of n2 + n governing equations for all subcompartments, Eq. 2.17, from the original 318 system of n governing equations for all compartments, Eq. 2.1, can algebraically be 319 schematized as follows (see Figs.1 and2 for graphical illustrations). x˙(t) = τ(t, x) =     x˙ 1 τ1 x˙ 2  τ2  dynamic system partitioning   =    .   .   .   .  x˙ n τn         x˙ 10 τ10 x˙ 11 ··· x˙ 1n τ11 ··· τ1n

x˙ 20  τ20  x˙ 21 ··· x˙ 2n  τ21 ··· τ2n    =   and   =    .   .   . .. .   . .. .   .   .   . . .   . . . 

x˙ n0 τn0 x˙ n1 ··· x˙ nn τn1 ··· τnn = =

x˙ 0(t) = τ0(t, x) X˙ (t) = T (t, x) 320

This manuscript is for review purposes only. 10 HUSEYIN COSKUN

321 In the diagram above, the net initial throughflow and throughflow vectors, τ0(t, x) and 322 τ(t, x), as well as the net subthroughflow matrix, T (t, x), are defined as the difference 323 between the corresponding inward and outward quantities, that is, τ0(t, x) =τ ˇ0(t, x)− 324 τˆ0(t, x) and τ(t, x) =τ ˇ(t, x) − τˆ(t, x), and T (t, x) = Tˇ(t, x) − Tˆ(t, x). 325 2.2. Analytic Solution to Linear Ecological Systems. In this section, we 326 formulate analytic solutions to linear ecological systems with time dependent inputs 327 based on the proposed dynamic methodology. 328 The system partitioning methodology yields a linear system, if the original system 329 is linear. That is, if Eq. 2.4 is linear, Eq. 2.17 takes the following form:

X˙ (t) = Z(t) + A(t) X(t),X(t0) = 0, 330 (2.18) x˙ 0(t) = A(t) x0(t), x0(t0) = x0.

331 Let V (t) be the fundamental matrix solution to the system Eq. 2.18 as formulated by 332 [9], that is, the unique solution of the system

333 V˙ (t) = A(t) V (t),V (t0) = I.

334 The solutions to Eq. 2.18 for substorage matrix, X(t), and initial substorage vector, 335 x0(t), in terms of V (t) become Z t −1 336 (2.19) X(t) = V (t) V (s) Z(s) ds and x0(t) = V (t) x0. t0

337 Therefore, the solution to the original system, Eq. 2.1, becomes

Z t −1 338 x(t) = V (t) x0 + V (t) V (s) z(s) ds. t0

339 For the particular case of constant diagonalizable ﬂow intensity matrix A, we have

Z t 340 V (t) = exp A ds = e(t−t0) A = Ω e(t−t0)Λ Ω−1 t0

341 where Ω is the matrix whose columns are eigenvectors of A, and Λ is the diagonal 342 matrix whose diagonal elements are the eigenvalues of A. For this particular case, 343 Eq. 2.19 takes the following form: Z t (t−s) A (t−t0) A 344 (2.20) X(t) = e Z(s) ds and x0(t) = e x0. t0

345 Consequently,

Z t (t−t0) A (t−s) A 346 x(t) = e x0 + e z(s) ds. t0

347 A subsystem scaling argument is proposed to analyze system behavior per unit 348 input by [10]. The scaled substorage matrix, S(t) = X(t) Z−1, can be expressed for 349 constant invertible input matrix, Z(t) = Z > 0, as follows

Z t Z t Z t 350 (2.21) S(t) = V (t) V −1(s) ds = e(t−s) A ds = Ω e(t−s)Λ ds Ω−1, t0 t0 t0

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 11

351 using Eq. 2.20. The static version of this measure S(t) is widely used in static eco- 352 logical network analyses as outlined in the next section [10]. 353 An example of the analytic solution to a linear ecosystem model with time de- 354 pendent environmental input is presented in Section 3.2.

355 2.3. Static Ecological Network Analysis. At steady state, the time deriva- 356 tives of the state variables are zero, and all system ﬂows and storages are constant. 357 That is,

358 X˙ (t) = 0 andx ˙ 0(t) = 0.

359 The constant static quantities will be denoted by the same symbols without the time 360 argument, that is X(t) = X, for example. 361 Summing up the equations in Eq. 2.9 side by side over index k yields Eq. 2.4 362 because of the relationship

n X 363 x˙ i(t) = x˙ ik (t), i = 1, . . . , n, k=0

364 deduced from Eq. 2.6 and the deﬁnition of the decomposition factors, dik (x), given in 365 Eq. 2.7. Therefore, if the partitioned system, Eq. 2.9, is at steady state, the original 366 system, Eq. 2.4, is also at steady state. The static version of the proposed dynamic 367 methodology is introduced by [10, 11] as summarized below in this section. 368 Since A is a strictly diagonally dominant constant matrix, it is invertible. It can 369 be expressed as

370 (2.22) A = (F − T ) X −1 = Q − R−1.

371 We then have the following solutions to Eq. 2.17 for the substorage matrix, X(t), and 372 initial substorage vector, x0(t), at steady state:

−1 −1 373 (2.23) X = −A Z = X (T − F ) Z and x0 = 0.

374 From Eq. 2.13 and the fact that τ =τ ˆ =τ ˇ and T = Tˇ = Tˆ at steady state, the 375 throughﬂow matrix can be written in terms of system ﬂows only:

−1 376 (2.24) T = Z + F X −1 X = Z + F T −1 T ⇒ T = I − F T −1 Z.

377 Moreover, the residence time matrix R can be expressed as

378 (2.25) XT −1 = R = XT −1.

379 The scaled versions of X and T can be deﬁned as S = X Z−1 and N = T Z−1, 380 where Z is invertible (Z > 0). Using Eqs. 2.23 and 2.24, they can be expressed as

−1 381 (2.26) S = −A−1 and N = I − F T −1 .

These scaled substorage and subthroughﬂow matrices, S and N, can then be considered as linear transformations that map Z to X and T , respectively, even though there are zero environmental inputs. Note that S(t) formulated in Eq. 2.21 is equivalent to S at steady state, that is lim S(t) = S. t→∞

This manuscript is for review purposes only. 12 HUSEYIN COSKUN

382 Although derivation rationales are diﬀerent, the matrix measures S and N are 383 equivalent to the ones formulated in the current static ecological network analyses as 384 shown by [10]. They are treated separately in the current methodologies, although 385 they are related simply by a factor of the residence time matrix, as implied by Eq. 2.25:

386 T = R−1 X ⇒ S = R N.

387 This relationship enables a holistic view of static ecological systems as introduced by 388 [11]:

389 (2.27) x = S z = R N z = R τ and X = S Z = R N Z = R T.

390 The static systems can be scaled by outputs instead for the output-oriented system 391 analysis by reversing the system ﬂows as

392 F¯ := F T , Y¯ := Z, and Z¯ := Y.

393 The counterparts of the Eq. 2.27 for the output-oriented analysis then become

394 (2.28) x¯ = S¯ y = R N¯ y = R τ¯ and X¯ = S¯ Y = R N¯ Y = R T¯

395 where S¯ = X¯ Y−1 and N¯ = T¯ Y−1. Because x =x ¯ and τ =τ ¯ at steady state, 396 R¯ = R. It is worth noting that the input- and output-oriented scaled subthroughﬂow 397 and substorage matrices are similar:

398 (2.29) S X = X S¯T and NT = T N¯ T .

399 This duality of the input- and output-oriented static analyses is introduced by [11].

400 2.4. Dynamic Subsystem Partitioning Methodology. In this section, we 401 introduce the dynamic subsystem partitioning methodology for further partitioning or 402 segmentation of subsystems along a given set of mutually exclusive and exhaustive 403 subflow paths. This subsystem partitioning determines the distribution of arbitrary 404 intercompartmental flows and the organization of the associated storages generated 405 by these flows within the subsystems. Therefore, the subsystem partitioning method- 406 ology dynamically partitions arbitrary composite intercompartmental flows and the 407 associated storages generated by these flows into the constituent transient subflow 408 and substorage segments along given subflow paths. 409 The dynamic subsystem partitioning methodology will be formulated below using 410 the directed subflow path terminology adopted from the recent paper by [9]. The sub- 411 systems can further be decomposed into subflows and associated substorages along a 412 set of mutually exclusive and exhaustive directed subflow paths. By mutually exclu- 413 sive subflow paths, we mean that no given subflow path in a subsystem is a subpath, 414 that is, completely inside of another path in the same subsystem. The exhaustive- 415 ness, in this context, means that such mutually exclusive subflow paths all together 416 sum to the entire subsystem subflows and associated substorages. We will use the

417 notation Pikjk for a set of mutually exclusive and exhaustive subﬂow paths from sub-

418 compartment jk to ik in subsystem k and wk for the number of subflow paths in Pikjk . 419 The natural subsystem decomposition defined by [9] yields a mutually exclusive and 420 exhaustive decomposition of the entire system. 421 We will first introduce the transient flows and associated storages below. The 422 transient flows and storages will then be used for the formulation of the diact flows 423 and storages in the next section.

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 13

424 No man ever steps in the same river twice. – Heraclitus (535-475 BC)

425 2.4.1. Transient Flows and Storages. As indicated in the famous dictum 426 by Heraclitus that “everything flows”, flows are one of the most important physical 427 phenomena of existence. In this section, we formulate the transient subflows and the 428 associated storages generated by these flows. 429 The transient and cumulative transient subflows along a subflow path between 430 two subcompartments will be defined in what follows. Along a given subflow path 431 pw = i 7→ j → ` n , the transient inflow at subcompartment ` , f w (t), nkik k k k k k `kjkik 432 is the subflow segment transmitted to `k at time t, which is generated by the local 433 input from subcompartment ik (local source) into the first subcompartment of the 434 path, jk, (connection) during [t1, t], t1 ≥ t0. Similarly, the transient outflow at sub- 435 compartment ` , f w (t), is the subflow segment transmitted from ` to the next k nk`kjk k 436 subcompartment, nk, along the path at time t, which is generated by the transient 437 inflow into ` during [t , t]. The associated transient substorage, xw (t), is then k 1 nk`kjk 438 the substorage segment in subcompartment `k at time t, which is derived from the 439 transient inflow and governed by the transient inflow and outflow balance during [t1, t]. 440 The transient outflow at subcompartment `k at time t, from jk to nk along subflow 441 path pw , can be formulated as nkik w xn ` j (t) 442 (2.30) f w (t) = k k k f (t, x), nk`kjk nk`k x`k (t) 443 similar to Eq. 2.7, where the transient substorage xw (t) is determined by the nk`kjk 444 governing equation

τˆ` (t, x) 445 x˙ w (t) = f w (t) − k xw (t), xw (t ) = 0. (2.31) nk`kjk `kjkik nk`kjk nk`kjk 1 x`k (t) 446 The equivalence of the outward throughflow and subthroughflow intensities as well as 447 the flow and subflow intensities in the same direction can be expressed as

fn`(t, x) fnk`k (t, x) −1 τˆ`(t, x) τˆ`k (t, x) 448 qn`(t, x) = = and r` (t, x) = = x`(t) x`k (t) x`(t) x`k (t) 449 for `, n = 1, . . . , n, and k = 0, 1, . . . , n, due to Eqs. 2.7 and 2.13. Therefore, since 450 the fractions in Eqs. 2.30 and 2.31 can be expressed at both compartmental and 451 subcompartmental levels, the subsystem partitioning is actually independent from 452 the system partitioning. Note that the initial condition given in Eq. 2.31 for the 453 initial subsystem (k = 0) is xw (t ) = x , and this initial value of x is not nk`kjk 1 `0 `0 454 considered as a transient substorage. The governing equations, Eqs. 2.30 and 2.31, 455 establishs the foundation of the dynamic subsystem partitioning (see Fig.3). These 456 equations for each subcompartment along a given flow path of interest will then be 457 coupled with the decomposed system, Eq. 2.9, or the original system, Eq. 2.4, and be 458 solved simultaneously. 459 The transient subflows and substorages are defined for linear subflow paths above. 460 The sum of the transient inflows from subcompartment jk to `k and outflows from 461 ` to n at subcompartment ` along a given self-intersecting path pw is called the k k k nkik 462 cumulative transient inflow, f w (t), and outflow, f w (t), respectively, and associ- `kjk nk`k 463 ated total substorage will be called the cumulative transient substorage, xw (t). They `k 464 can be formulated as

mw mw mw w X w,m w X w,m w X w,m 465 (2.32) x (t) = x (t), f (t) = f (t), f (t) = f (t) `k nk`kjk `kjk `kjkik nk`k nk`kjk m=1 m=1 m=1

This manuscript is for review purposes only. 14 HUSEYIN COSKUN

x`k f w nk`kjk xw w nk`kjk f` j i k k k fnk`k

f`kjk τˆ`k

Fig. 3. Schematic representation of the dynamic subsystem partitioning. The transient inflow and outflow, f w (t) and f w (t), and associated substorage, xw (t), at subcompartment `kjkik nk`kjk nk`kjk ` along subflow path pw = i 7→ j → ` n . k nkik k k k k

466 where the superscript m represents the cycle number, and mw is the number of cycles, 467 that is, the number of times the path pw intersects itself. nkik

468 2.4.2. The diact Flows and Storages. In this section, we formulate five im- 469 portant transaction types for ecological systems through the subsystem partitioning 470 methodology: the diact flows and associated storages. The transfer flows (denoted 471 by t) and the associated storages generated by these flows will be formulated in detail 472 below, at both subcompartmental and compartmental levels, and parallel derivations 473 for direct (d), indirect (i), cycling (c), and acyclic (a) flows and associated storages 474 are straightforward. 475 The transfer flow will be defined as the total intercompartmental transient flow 476 from one compartment, directly or indirectly through other compartments, to another. 477 The direct and indirect flow will be defined as the transfer flows from one compartment 478 to another directly and indirectly through other compartments, respectively. The 479 cycling flow will be defined as the transfer flow from a compartment, indirectly through 480 other compartments, back into itself. Lastly, the acyclic flow at a compartment 481 will be defined as the non-cyclic segment of the compartmental throughflow at that 482 compartment. The diact storage will then be defined as the storage generated by the 483 corresponding diact flow. The diact flows and storages at both subcompartmental 484 and compartmental levels are formulated below (see Fig.4). 485 The transfer subflow is defined as the total intercompartmental transient flow 486 from one subcompartment, directly or indirectly through other subcompartments, 487 to another in the same subsystem. Let P t be the set of mutually exclusive and ikjk 488 exhaustive subflow paths pw from subcompartment j directly or indirectly to i in ikjk k k 489 subsystem k. The transfer inflow from subcompartment j to i , τ t (t), is defined k k ikjk 490 as the sum of the cumulative transient inflows generated by local inputs initiated 491 at j during [t , t] and transmitted to i at time t through all paths in P t . The k 0 k ikjk 492 associated transfer substorage, xt (t), at subcompartment i at time t is the sum ikjk k 493 of the cumulative transient substorages derived from transfer inflow τ t (t) during ikjk 494 [t0, t]. The transfer inflow and substorage can then be formulated as

w n w Xk X Xk τ t (t) = f w (t) and xt (t) = xw (t) 495 (2.33) ikjk ik`k ikjk ik w=1 `=1 w=1

496 where w is the number of subflow paths pw ∈ P t . The sum of all the transfer k ikjk ikjk 497 subflows and associated substorages from subcompartment jk to ik for each subsystem t t 498 k is called the transfer flow and storage at time t, τij(t) and xij(t), from compartment

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 15

τ t ikjk τ i ikjk

xik xjk fikjk zi fjiii xji xii τ i jiii τˇji xi xj

t Fig. 4. Schematic representation of the transfer subflow, τ (t), direct subflow, fi j (t, x), in- ikjk k k i direct subflow, τ (t), and inward subthroughflow, τˇi (t, x). Solid arrows represent direct subflows ikjk k and dashed arrows represent indirect subthroughflows through other compartments (not shown).

499 j to i: n n X X τ t (t) = τ t (t) and xt (t) = xt (t). 500 (2.34) ij ikjk ij ikjk k=0 k=0 501 t t 502 For notational convenience, we deﬁne n × n matrix functions Tk (t) and Xk(t), 503 whose (i, j)−elements are τ t (t) and xt (t), respectively, as ikjk ikjk

504 T t(t) = τ t (t) and Xt(t) = xt (t) . (2.35) k ikjk k ikjk

t t th 505 These matrix measures Tk (t) and Xk(t) are called the k transfer subflow and asso- 506 ciated substorage matrix functions. The corresponding transfer flow and associated t t t t 507 storage matrix measures are T (t) = τij(t) and X (t) = xij(t) , respectively. 508 Let P d and P i be defined as the sets of subflow paths pw from subcompart- ikjk ikjk ikjk 509 ment j , directly and indirectly, to i , respectively; P c be the set of subflow paths pw k k ik ik 510 from subcompartment i indirectly back to itself; and P a be the set of linear subflow k ik 511 paths pw from subcompartment k , directly or indirectly, to i in subsystem k. All ik k k 512 these diact subflow sets are assumed to be mutually exclusive and exhaustive. The 513 transfer flows, associated storages, and corresponding matrix functions are formulated 514 in Eqs. 2.33, 2.34, and 2.35 using the subflow set P t . The other flows, associ- ikjk diact 515 ated storages, and matrix functions can then be formulated similarly by substituting 516 the corresponding diact flows and storages for their transfer counterparts in these 517 equations and using the corresponding diact subflow sets instead. Figure5 depicts 518 the complementary nature of the direct, indirect and cycling flows. 519 The direct and indirect subflow are defined as the transfer subflows from one sub- 520 compartment, directly and indirectly through other subcompartments, to another in 521 the same subsystem, respectively. The indirect subflow, τ i (t), from subcompart- ikjk 522 ment jk to ik can be considered as the transfer subflow diminished by the direct 523 subflow from jk to ik at time t (see Fig.4). Therefore, it can also be formulated as

524 τ i (t) = τ t (t) − f (t, x). (2.36) ikjk ikjk ikjk

525 Consequently, we have

526 (2.37) T t(t) = T d(t) + T i(t) and Xt(t) = Xd(t) + Xi(t).

527 There is a functional similarity between T i(t) and T d(t) = F (t, x); the (i, k)−element i i d 528 of T (t), τik(t), is the indirect ﬂow, while that of F (t, x), τik(t) = fik(t, x), is the 529 direct ﬂow from compartment k to i at time t (see Fig.4).

This manuscript is for review purposes only. 16 HUSEYIN COSKUN

τ i ikkk indirect subﬂow zk direct (or cycling) subﬂow xik xkk

fkkik

Fig. 5. Schematic representation of direct and indirect subﬂows in the kth subsystem. The

direct subflow, fkkik (t, x), is represented by solid arrow. This subflow also contributes to the cycling i subflow at subcompartment kk. The indirect subflow, τ (t), through other compartments (not ikkk shown) is represented by dashed arrows.

530 The cycling subflow is defined as the transfer subflow from a subcompartment, 531 indirectly through other subcompartments in the same subsystem, back into itself. 532 Let the cycling subflow and associated substorage matrices be T c(t) and Xc(t). Due 533 to the construction of cycling flow as reflexive transfer or indirect flow, we have

T c(t) = diag (T t(t)) = diag (T i(t)), 534 (2.38) X c(t) = diag (Xt(t)) = diag (Xi(t)).

535 Note that, the cycling ﬂow and subﬂow as well as the cycling storage and substorage 536 matrices are related as

537 T c(t) = diag (T c(t) 1) and X c(t) = diag (Xc(t) 1).

538 Lastly, the acyclic subflow at a subcompartment is defined as the non-cyclic seg- 539 ment of the subthroughflow at that subcompartment. In other words, the acyclic flows 540 and associated storages are generated by environmental inputs directly or indirectly 541 through linear subflow paths. In that sense, they quantify through (non-cyclic) in- 542 fluence of environment on system compartments. The acyclic subflow and associated 543 substorage matrices can be formulated as

544 (2.39) T a(t) = Tˇ(t, x) − T c(t) and Xa(t) = X(t) − Xc(t).

545 Note that, the (i, k)−element of T a(t) and T c(t), τ a (t) and τ c (t), represent the ik ik 546 acyclic flow through linear subflow paths from compartment k to i and cycling flow 547 at subcompartment ik at time t, respectively, generated by environmental input zk(t) 548 during [t0, t]. 549 It is worth mentioning that the indirect subflow from an input-receiving subcom- 550 partment kk to ik can, alternatively, be expressed as

n i X τ (t) = fi j (t, x) =τ ˇi (t, x) − fi k (t, x) − zi (t, x) 551 (2.40) ikkk k k k k k k j=1 j6=k

552 for i, k = 1, . . . , n. Similarly, the transfer subﬂow from kk to ik can be expressed as

n X τ t (t) = f (t, x) =τ ˇ (t, x) − z (t, x). 553 (2.41) ikkk ikjk ik ik j=1

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 17

diact flow and storage distribution matrices flows storages d N d = F T −1 Sd = R N d T d = N d T Xd = Sd T i N i = (N − I) N −1 − F T −1 Si = R N i T i = N i T Xi = Si T a N a = N −1 Sa = RN a T a = N a TXa = Sa T c N c = (N − I) N −1 Sc = RN c T c = N c TXc = Sc T t N t = (N − I) N −1 St = R N t T t = N t T Xt = St T Table 1 The diact flow and subflow and associated storage and substorage matrices.

554 The cycling subflow from an input-receiving subcompartment i to itself, τ c (t), can i ii 555 then, alternatively, be defined in terms of the indirect or transfer subflows as

n X τ c (t) = τ t (t) = τ i (t) = f (t, x) =τ ˇ (t, x) − z (t, x). 556 (2.42) ii iiii iiii iiji ii i j=1

557 Consequently,

558 τ a (t) =τ ˇ (t, x) − τ c (t) = z (t, x). (2.43) ii ii ii i

559 2.5. Static Subsystem Partitioning and diact Transactions. The static 560 version of the dynamic subsystem partitioning methodology formulated in Eqs. 2.30 561 and 2.31 has been recently introduced by [10]. This partitioning methodology is 562 summarized below. 563 Since time derivatives are zero at steady state, we setx ˙ w (t) = 0 in Eq. 2.31. nk`kjk 564 The static transient outflow f w at subcompartment ` , from j to n along subflow nk`kjk k k k 565 path pw , and the transient substate xw generated at ` by the transient inflow nkik nk`kjk k 566 f w are then formulated as `kjkik

x` fn` fn` 567 (2.44) xw = f w and f w = xw = f w . nk`kjk `kjkik nk`kjk nk`kjk `kjkik τ` x` τ`

568 Based on this static subsystem partitioning, the static diact ﬂows and storages are 569 formulated in matrix form by [10], as presented in Table1. All quantities in the table 570 are the static counterparts of their dynamic versions introduced in the present paper 571 and N = diag(N).

572 2.6. System Measures and Indices. The dynamic system partitioning method- 573 ology yields the subthroughflow and substorage matrices that measure the environ- 574 mental influence on system compartments in terms of the flow and storage generation. 575 For the quantification of intercompartmental flow and storage dynamics, the dynamic 576 subsystem partitioning methodology then formulates the transient and dynamic diact 577 flows and associated storages. 578 Many other dynamic and static system analysis tools, such as measures and in- 579 dices for the diact effect, utility, exposure, and residence time, as well as the system 580 efficiency, stress, and resilience can be formulated based on the proposed methodology. 581 These dynamic measures and indices of matrix, vector, and scalar types have recently 582 been introduced as quantitative ecosystem indicators by [8]. The static versions of 583 these system analysis tools have also been formulated in separate works [10, 11].

584 3. Results. The proposed dynamic methodology is applied to two dynamic 585 ecosystem models: a nonlinear and a linear model. Analysis of both models are

This manuscript is for review purposes only. 18 HUSEYIN COSKUN

586 presented in this section. The numerical results for the quantitative system analy- 587 sis tools developed in the present paper, such as the substorage and subthroughflow 588 matrix measures as well as the transient and dynamic diact flows and associated 589 storages, are presented. 590 The results indicate that the proposed methodology precisely quantifies dynamic 591 system functions, properties, and behaviors, tracks the evolution of environmental 592 inputs and intercompartmental flows as well as associated storages within the system, 593 is sensitive to perturbations due to even a brief unit impulse, and, thus, can be used 594 for rigorous analysis of nonlinear ecological systems. It is worth noting, however, that 595 this present work proposes a mathematical method—a systematic technique designed 596 for solving and analyzing any nonlinear compartmental model—and it is not a model. 597 Therefore, we focus more on demonstrating the efficiency and wide applicability of the 598 method. It is expected that once the method is accessible to a broader community of 599 environmental ecologists, it can be used for ecological inferences and detailed analyses 600 of specific models of interest.

601 3.1. Case Study. In this section, a nonlinear resource-producer-consumer ecosys- 602 tem model proposed by [16] is analyzed through the proposed methodology. A com- 603 parison of the results is not possible, as the authors did not provide any computational 604 or explicit results in the article. Moreover, the system is further examined for a time 605 dependent, symmetric Gaussian impulse to illustrate the eﬃciency of the proposed 606 method in capturing the system response to disturbances. Such analysis can be used 607 to quantify the system resistance and resilience in the face of disturbances and per- 608 turbations. 609 The resource-producer-consumer model by [16] consists of the dynamics for three 610 components: resource, x1(t) = r(t), which is the nutrient storage (such as phosphorus 611 or nitrogen) present at time t; producer, x2(t) = s(t), which denotes the nutrient stor- 612 age in the producer (such as phytoplankton) population; and consumer, x3(t) = c(t), 613 which denotes the nutrient storage in the consumer (such as zooplankton) population 614 (see Fig.6). The conservation of nutrient is the basic model assumption. The system 615 ﬂows are described as follows:

 0 d s(t) d c(t)     1 2 z1(t) r(t)  α1 s(t) r(t) 0 0  616 F (t, x) = α2+r(t) , z(t) = z2(t) , y(t) = s(t)  β s(t) c(t)  0 1 0 z3(t) c(t) β2+s(t)

617 where the constant input is z(t) = [1, 1, 1]T and the parameters are given as

618 d1 = 2.7, d2 = 2.025, α2 = 0.098, β1 = 2, β2 = 20, and α1 = 1.

619 The value for α1 was not provided in [16] and was chosen arbitrarily for this example. 620 The governing equations take the following form:

α1 s(t) r(t) r˙(t) = −r(t) + d1 s(t) + d2 c(t) − + z1(t) α2 + r(t)

α1 s(t) r(t) β1 c(t) s(t) 621 (3.1) s˙(t) = −(1 + d1) s(t) + − + z2(t) α2 + r(t) β2 + s(t)

β1 c(t) s(t) c˙(t) = −(1 + d2) c(t) + + z3(t) β2 + s(t)

622 with the initial conditions of [r0, s0, c0] = [1, 1, 1].

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 19

x3(t)

p1 0101

x1(t) x2(t)

Fig. 6. Schematic representation of the model network. Subﬂow path p1 along which the 0101 transient subﬂows and substorages are computed is red (subsystems are not shown) (Case Study 3.1).

623 Let the subcompartmentalization become

624 x1k (t) = rk(t), x2k (t) = sk(t), and x3k (t) = ck(t), k = 0,..., 3.

625 The ﬂow partitioning then yields

 0 d d s d d c     2k 1 3k 2 δ1k z1 d1k r α1 s r d1 0 0 626 Fk(t, x) =  k α2+r  , zˇk(t, x) = δ2k z2 , yˆk(t, x) = d2k s , β1 s c 0 d2 0 δ3k z3 d3 c k β2+s k

th 627 where Fk,z ˇk, andy ˆk describe the k direct ﬂow matrix, input, and output vectors th 628 for the k subsystem, and the decomposition factors dik (x) are deﬁned by Eq. 2.7. 629 Therefore, the dynamic system partitioning methodology yields

α1 s(t) rk(t) r˙k(t) = δ1k z1(t) + d1 sk(t) + d2 ck(t) − rk(t) − α2 + r(t)

α1 s(t) rk(t) β1 c(t) sk(t) 630 (3.2) s˙k(t) = δ2k z2(t) + − sk(t) − d1 sk(t) − α2 + r(t) β2 + s(t)

β1 c(t) sk(t) c˙k(t) = δ3k z3(t) + − ck(t) − d2 ck(t) β2 + s(t)

631 with the initial conditions ( 1, k = 0 632 x (t ) = ik 0 0, k 6= 0

633 for i = 1,..., 3. There are n × (n + 1) = 3 × 4 = 12 equations in this system. 634 The system is solved numerically and the graphs for selected elements of the 635 substorage and the subthroughﬂow matrices are depicted in Fig.7. As seen from the 636 graphs, the system converges to a steady-state quickly at about t ≈ 6 units. The 637 results show, for example, that the nutrient storage in the resource compartment

638 (i = 1) derived from nutrient input into the consumers compartment (3), x13 (t), 639 increases from 0 to 0.62 units until the system reaches the steady state, while the

640 initial nutrient storage in the resource compartment, x10 , ﬁrst increases from 1 to 1.36 641 units and then vanishes. The throughﬂow into the resource compartment generated

642 by nutrient input into the producers compartment (2),τ ˇ12 (t, x), increases until about

This manuscript is for review purposes only. 20 HUSEYIN COSKUN

1.4 4 1.2

1 3 0.8

0.6 2

0.4 1 0.2 substorage matrix elements

0 subthroughflow matrix elements 0 0 2 4 6 8 10 0 2 4 6 8 10 time time

Fig. 7. The numerical results for selected elements of the substorage, X(t), and the subthrough- ﬂow matrix functions, Tˇ(t, x) and Tˆ(t, x), for the system with constant input z(t) = [1, 1, 1]T (Case Study 3.1).

643 t ≈ 2. The outward throughﬂow at the same subcompartment,τ ˆ12 (t, x), is slightly

644 smaller than inward throughﬂow,τ ˇ12 (t, x), but has a similar behavior. As seen from 645 these results, the distribution of environmental nutrient inputs and the organization of 646 the associated nutrient storages generated by the inputs can be analyzed individually 647 and separately within the system. 648 In general terms, the state variable xi(t) of the original system for the resource- 649 producer-consumer dynamics, Eq. 3.1, gives the nutrient storage in compartment i at 650 time t based on its initial stock, xi(t0). It cannot be used to distinguish the nutrient 651 storage derived from individual environmental nutrient inputs. On the other hand, the

652 state variable xik (t) of the decomposed system, Eq. 3.2, represents the nutrient storage 653 in compartment i that is derived from the speciﬁc environmental nutrient input into

654 compartment k, zk(t). Similarly, the state variable xi0 (t) of the decomposed system 655 represents the dynamics of the initial nutrient stocks in compartment i. Parallel 656 interpretations are possible for the net throughﬂow function of the original system,

657 τi(t, x), and the net subthroughflow function of the decomposed system, τik (t, x), as 658 well. 659 The proposed dynamic system partitioning methodology, consequently, enables 660 partitioning the compartmental composite nutrient flows and storages into subcom- 661 partmental segments based on their constituent environmental nutrient sources. In 662 other words, the system partitioning enables tracking the evolution of environmental 663 nutrient inputs and the associated storages generated by the inputs individually and 664 separately within the system. Therefore, this partitioning also allows for compiling a 665 history of compartments visited by individual nutrient inputs separately. −(t−15)2 666 The system is also perturbed with a Gaussian input z2(t) = e 2 + 0.1, which 667 represents a brief, unit local impulse about t = 15 to demonstrate the capability 668 of the proposed method to analyze the influence of time dependent inputs on the 669 system. The other two environmental nutrient inputs are kept constant as before for 670 a comparison, that is, z1(t) = z3(t) = 1. The graphical representations for the selected 671 elements of the substorage and subthroughflow matrices are given in Fig.8. It is clear 672 from the graphs that the dynamic substorage and subthroughflow matrix measures 673 reflect the impact of the unit impulse about t = 15. Note that, the system completely 674 recovers after the disturbance in about 10 time units. This time interval can be taken 675 as a quantitative measure for the restoration time or system resilience. Therefore, the

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 21

1.4 4 1.2

1 3 0.8

0.6 2

0.4 1 0.2 substorage matrix elements

0 subthroughflow matrix elements 0 0 5 10 15 20 25 0 5 10 15 20 25 time time

Fig. 8. The numerical results for the selected elements of the substorage, X(t), and subthrough- ﬂow matrix functions, Tˇ(t, x) and Tˆ(t, x), for the system with time dependent environmental input −(t−15)2 of a Gaussian impulse function z2(t) = e 2 + 0.1, z1(t) = 1, and z3(t) = 1 (Case Study 3.1).

676 proposed measures can also be used as quantitative ecological indicators for various 677 ecosystem characteristics and behaviors. 678 The subsystem partitioning methodology is also applied to this model to track 679 the fate of arbitrary intercompartmental flows and associated storages generated with 680 this flow within the subsystems. Along the subflow path p1 = 0 7→ 1 → 2 3 3111 1 1 1 1 681 from subcompartment 11 to 31 in subsystem 1, the transient subflows and associated 682 substorages are obtained as formulated in Eqs. 2.30 and 2.31. The numerical results for 683 the transient subflows, f 1 (t) and f 1 (t), and associated substorage functions, 211101 312111 684 x1 (t) and x1 (t), are presented in Fig.9. 211101 312111 685 The path is also extended to p1 , where 0101

686 p1 = 0 7→ 1 → 2 → 3 → 1 → 2 → 1 → 0 0101 1 1 1 1 1 1 1 1

687 to compute the local output f 1 (t) (a segment of the environmental output y (t)) 011121 1 688 derived from local (environmental) input z1(t) along that particular path (see Fig.6). 689 That is, the fate of z (t) within the system along path p1 is determined. The corre- 1 0101 690 sponding transient subflow and associated substorage functions at each step (subcom- 691 partment) along the path are also presented in Fig.9. Since f 1 (t) ≤ 6.28 × 10−5 011121 692 for all time t, at most about %0.006 of z1(t) = 1 exits the system through the given 693 path p1 at any time. 0101 694 These results indicate that the proposed dynamic subsystem partitioning method- 695 ology enables dynamically tracking the fate of an arbitrary amount of nutrient flow 696 and the associated nutrient storages generated by this flow in each compartment along 697 a given flow path. Therefore, the effect of one compartment on any other in terms 698 of the nutrient transfer, through not only direct but also indirect interactions, can be 699 determined. 700 As shown in this case study, the detailed information and inferences enabled by 701 the proposed methodology cannot be obtained through the analysis of the original 702 system by the state of the art techniques.

703 3.2. Case Study. An additional linear ecosystem model introduced by [20] is 704 analyzed in this section to demonstrate the analytic solutions to linear systems with 705 time dependent inputs through the proposed methodology. The graphical represen- 706 tations of the solutions are presented.

This manuscript is for review purposes only. 22 HUSEYIN COSKUN

1 0.16

0.14 0.8 0.12

0.6 0.1 0.08

0.4 0.06

transient subflows 0.04 transient substorages 0.2 0.02

0 0 0 5 10 15 20 25 0 5 10 15 20 25 time time -4 10 10-3 7 1

6 0.8 5

4 0.6

3 0.4

2 transient subflows transient substorages 0.2 1

0 0 0 5 10 15 20 25 0 5 10 15 20 25 time time

Fig. 9. The numerical results for the transient subﬂows and substorages at each step along the paths p1 and p1 . In the ﬁrst row, the functions f 1 (t) and x1 (t) are scaled up by a 3111 0101 312111 312111 factor of 10 for clarity of the presentation (Case Study 3.1).

707 The linear dynamic ﬂow model proposed by [20] has two compartments, x1(t) 708 and x2(t) (see Fig. 10). The system ﬂows are described as

2 1 0 3 x2(t) z1(t) 3 x1(t) 709 F (t, x) = 4 , z(t, x) = , y(t, x) = 5 . 3 x1(t) 0 z2(t) 3 x2(t)

710 The governing equations take the following form:

2 4 1 x˙ (t) = z (t) + x (t) − + x (t) 1 1 3 2 3 3 1 711 4 2 5 x˙ (t) = z (t) + x (t) − + x (t) 2 2 3 1 3 3 2

T T 712 with the initial conditions [x1,0, x2,0] = [3, 3] . 713 Let the subcompartmentalization become

714 x1k (t) and x2k (t), k = 0, 1, 2.

715 The ﬂow partitioning then yields

2 1 0 3 d2k x2 δ1k z1 3 d1k x1 716 Fk(t, x) = 4 , zˇk(t, x) = , yˆk(t, x) = 5 3 d1k x1 0 δ2k z2 3 d2k x2

717 where the decomposition factors dik (x) are deﬁned by Eq. 2.7. Componentwise, the

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 23

p1 11

x1(t) x2(t)

Fig. 10. Schematic representation of the model network. Subﬂow path p1 , along which the 11 cycling ﬂow and associated substorage are computed, is red (subsystems are not shown) (Case Study 3.2).

718 dynamic system partitioning methodology then yields

2 4 1 x˙ (t) = z (t) + x (t) − + x (t) 1k 1k 3 2k 3 3 1k 719 4 2 5 x˙ (t) = z (t) + x (t) − + x (t) 2k 2k 3 1k 3 3 2k

720 with the initial conditions ( 3, k = 0 721 x (t ) = ik 0 0, k 6= 0

722 for i = 1, 2. There are n × (n + 1) = 2 × 3 = 6 equations in the system. 723 The system can be written in matrix form as given in Eq. 2.18:

X˙ (t) = Z(t) + AX(t),X(t0) = 0 724 (3.3) x˙ 0(t) = A x0(t), x0(t0) = x0

725 where the constant ﬂow intensity matrix A, as formulated in Eq. 2.22, is −(4/3 + 1/3) 2/3 −5/3 2/3 726 A = = . 4/3 −(2/3 + 5/3) 4/3 −7/3

727 The governing partitioned system, Eq. 3.3, is linear. We can, therefore, solve 728 it analytically, as formulated in Section 2.2. Since the ﬂow intensity matrix A is 729 constant, we have the following fundamental matrix solution as given in Eq. 2.20:

" −t −3 t −t −3 t # 2 e + e e − e 730 3 3 3 3 V (t) = 2 e−t 2 e−3 t e−t 2 e−3 t . 3 − 3 3 + 3

731 For z = [1, 1]T , the solution for the matrix equation, Eq. 3.3, as formulated in Eq. 2.20 732 then becomes

" 7 e−3 t 2 e−t 2 e−3 t e−t # −t 9 − 9 − 3 9 + 9 − 3 3 e 733 (3.4) X(t) = −3 t −t −3 t −t , x (t) = . 4 2 e 2 e 5 2 e e 0 3 e−t 9 + 9 − 3 9 − 9 − 3 734 Therefore, the solution to the original system, Eq. 3.3, in vector form, is

−t x1(t) 2 e + 1 735 x(t) = x0(t) + X(t) 1 = = −t . x2(t) 2 e + 1

736 These solutions are equivalent to the results provided by [20]. The throughﬂow ma- 737 trices, Tˇ(t) and Tˆ(t), can be expressed as formulated in Eq. 2.13 using the solution 738 for the substorage matrix X(t).

This manuscript is for review purposes only. 24 HUSEYIN COSKUN

739 The steady state solutions as formulated in Eq. 2.23 then are

7 2 −1 −1 9 9 740 (3.5) X = −A = X (diag(ˇτ) − F ) = 4 5 and x0 = 0. 9 9

741 It can easily be seen that, this steady-state solution is the same as the limit of the 742 dynamic solution, Eq. 3.4, as t tends to inﬁnity. 743 We also analyzed the system with a time dependent input z(t) = [3 + sin(t), 3 + 744 sin(2t)]T as suggested by [20]. Similar computations with the same fundamental

745 matrix, V (t), lead us to the following initial subsystem storage vector, xi0 (t), and

746 substorage matrix components, xik (t):

−t x10 (t) = x20 (t) = 3 e , 7 11 cos (t) 13 sin (t) 5 e−t 3 e−3 t x (t) = − + − − , 11 3 30 30 3 10 2 16 cos (2 t) 2 sin (2 t) 13 e−t 11 e−3 t x (t) = − − − + , 747 (3.6) 12 3 195 195 15 39 4 4 cos (t) 2 sin (t) 5 e−t 3 e−3 t x (t) = − + − + , 21 3 15 15 3 5 5 46 cos (2 t) 43 sin (2 t) 13 e−t 22 e−3 t x (t) = − + − − . 22 3 195 195 15 39

748 From these, we have the following solutions for the original system, Eq. 3.3:

2 2 X X 749 x1(t) = x1k (t) and x2(t) = x2k (t). k=0 k=0

750 The proposed dynamic method solves linear systems analytically. These analytic 751 solutions can be evaluated at ay time t to dynamically determine the corresponding 752 quantities.

753 The elements of the inward throughﬂow vector for the initial system,τ ˇi0 (t), and

754 the subthroughﬂow matrix,τ ˇik (t), can be computed using Eq. 2.13 as follows.

−t −t τˇ10 (t) = 2 e , τˇ20 (t) = 4 e , 35 8 cos (t) 49 sin (t) 10 e−t 2 e−3 t τˇ (t) = − + − + , 11 9 45 45 9 5 742 184 cos2 (t) 86 sin (2 t) 26 e−t 44 e−3 t τˇ (t) = − + − − , 755 (3.7) 12 585 585 585 45 117 28 22 cos (t) 26 sin (t) 20 e−t 2 e−3 t τˇ (t) = − + − − , 21 9 45 45 9 5 2339 128 cos2 (t) 577 sin (2 t) 52 e−t 44 e−3 t τˇ (t) = − + − + . 22 585 585 585 45 117

756 Graphical representations of the substorage, inward subthroughflow, and storage dis- 757 tribution matrices are depicted in Fig. 11. Using the substorage and inward sub- 758 throughflow functions given in Eqs. 3.6 and 3.7, the dynamic distribution of environ- 759 mental inputs as inward throughflows and the organization of the associated storages 760 generated by the inputs within the system can be analyzed individually and separately.

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 25

3 1

2.5 0.8 4

2 0.6 3 1.5 0.4 2 1 substorage matrix

0.2 subthroughflow matrix 1 0.5 storage distribution matrix

0 0 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 time time time

Fig. 11. The graphical representation of the substorage, X(t), storage distribution, S(t), and subthroughﬂow matrix, Tˇ(t), for time dependent input z(t) = [3 + sin(t), 3 + sin(2 t)]T (Case Study 3.2).

761 The limit of the storage distribution matrix S(t) = X(t) Z−1 for invertible con- 762 stant input matrix Z can be calculated at inﬁnity as follows:

 7 11 cos(t) 13 sin(t) 2 16 cos(2 t) 2 sin(2 t)  3 − 30 + 30 3 − 195 − 195 −1 sin(t)+3 sin(2 t)+3 763 lim S(t) = lim X(t) Z =  4 4 cos(t) 2 sin(t) 5 46 cos(2 t) 43 sin(2 t)  . t→∞ t→∞ 3 − 15 + 15 3 − 195 + 195 sin(t)+3 sin(2 t)+3

764 Graphs of the elements of S(t) are depicted in Fig. 11. 765 The subsystem partitioning methodology allows for the further analysis of the 766 system and brings out additional insights that are not available through the linear 767 methodology introduced by [20]. The transfer ﬂow and associated storage from com- t t 768 partment 2 to 1, τ12(t) and x12(t), are computed below as an application of the 769 proposed subsystem partitioning methodology. They can be expressed as

2 2 X X 770 (3.8) τ t (t) = τ t (t) and xt (t) = xt (t), 12 1k2k 12 1k2k k=0 k=0

771 as formulated in Eq. 2.34. The sets of mutually exclusive and exhaustive subﬂow

772 paths from 2k to 1k, P1k2k , for k = 0, 1, 2, can be formulated as follows: P1020 = 773 {p1 , p2 }, P = {p1 }, P = {p1 } where p1 = 1 7→ 1 2 → 1 , 1020 1020 1121 1121 1222 1222 1020 0 0 0 0 774 p2 = 2 7→ 2 → 1 2 , p1 = 0 7→ 1 2 → 1 , and p1 = 0 7→ 2 → 1020 0 0 0 0 1121 1 1 1 1 1222 2 2 775 12 22. 776 There are two subﬂow paths in the initial subsystem, p1 and p2 , and there- 1020 1020 777 fore, w0 = 2. Since fii(t) = 0, the corresponding transfer subﬂow and associated 778 substorage functions as formulated in Eq. 2.33 become

2 2 X X τ t (t) = f w (t) = f 1 (t) + f 2 (t), 1020 10`0 1020 1020 w=1 `=1 779 (3.9) 2 X xt (t) = xw (t) = x1 (t) + x2 (t). 1020 10 10 10 w=1

780 Similarly, we have (3.10) 1 2 1 X X X 781 τ t (t) = f w (t) = f 1 (t) and xt (t) = xw (t) = x1 (t), 1k2k 1k`k 1k2k 1k2k 1k 1k w=1 `=1 w=1

This manuscript is for review purposes only. 26 HUSEYIN COSKUN

4 2.5 2

3.5 2 3 1.5

2.5 1.5

2 1 1 1.5

1 transfer subflows 0.5 transfer substorages 0.5

0.5 cycling flows and storages

0 0 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 time time time

t Fig. 12. The graphical representation of the transfer flow and associated storage, τ12(t) and xt (t), together with the contributing transfer subflows and substorages, τ t (t) and xt (t), and 12 1k2k 1k2k c c cycling flows and associated storages, τi (t) and xi (t) (Case Study 3.2).

782 since there is only one subflow path in these subsystems (wk = 1 for k = 1, 2). The 783 links on this path p1 that directly contribute to the inflow are numbered with the 1121 784 red cycle numbers, m, in the extended subflow path diagram below:

1 2 3 785 p1 = 0 7→ 1 2 −→ 1 2 −→ 1 2 −→ 1 ··· 1121 1 1 1 1 1 1 1 1 786 The cumulative transient inflow f 1 (t) and the substorage x1 (t) at subcompartment 1121 11 787 1 along p1 will be approximated by two terms (m = 2) using Eq. 2.32: 1 1121 1 2 X x1 (t) ≈ x1,m (t) = x1,1 (t) + x1,2 (t), 11 211121 211121 211121 m=1 788 (3.11) 2 X f 1 (t) ≈ f 1,m (t) = f 1,1 (t) + f 1,2 (t). 1121 112111 112111 112111 m=1 789 The governing equations, Eqs. 2.30 and 2.31, for the transient subflows and associated 1,m 1,m 790 substorages, f (t) and x (t), and other transient subflow and substorages 112111 211121 791 involved in Eqs. 3.9 and 3.10, are solved simultaneously, together with the decomposed 792 system, Eq. 2.9. Numerical results for the transfer subflows and associated substorages 793 are presented in Fig. 12.

794 The subflow paths in P1k2k for each subsystem k are mutually exclusive and t 795 exhaustive. Therefore, x1(t) and x12(t) + x211101 (t) must be the same, as well as t t 796 f12(t) and τ12(t). The term added to x12(t) for comparison, x211101 (t), is the storage 797 generated by environmental input at 11 (01 7→ 11) and, therefore, is not included t 798 in transfer storage x12(t). They, however, are approximately equal as presented in t t 799 Fig. 12: x12(t) + x211101 (t) ≈ x1(t) and τ12(t) ≈ f12(t). The difference is caused by 800 the truncation errors in the computation of cumulative transient subflows as indicated 801 in Eq. 3.11, and larger mw values improve these approximations. Closed paths are 802 approximated by an infinite series of functions due to the method construction. These 803 close approximations by just two terms of the series justify the validity and indicate 804 the accuracy of the proposed subsystem partitioning methodology. 805 The cycling flows and the associated storages generated by these flows are also 806 calculated below along the following subflow paths. The sets of mutually exclusive 807 and exhaustive subflow paths from subcompartment 1 to itself, P c , are given as k 1k 808 P c = {p1 , p2 }, P c = {p1 }, P c = {p1 } where p1 = 1 7→ 1 2 → 1 , 10 10 10 11 11 12 12 10 0 0 0 0 809 p2 = 2 7→ 2 1 2 → 1 , p1 = 0 7→ 1 2 → 1 , and p1 = 0 7→ 10 0 0 0 0 0 11 1 1 1 1 12 2 810 2 1 2 → 1 (see Fig. 10). The sets of subflow paths for P c , k = 0, 1, 2, can 2 2 2 2 2k 811 similarly be defined.

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 27

812 The cycling subﬂow at subcompartment 12 along the only subﬂow path (w2 = 1) 813 p1 ∈ P c and associated substorage are 12 12

1 2 1 X X X 814 τ c (t) = f w (t) = f 1 (t) and xc (t) = xw (t) = x1 (t), 12 12`2 1222 12 12 12 w=1 `=1 w=1

815 as formulated in Eq. 2.33. The links contributing to the cycling ﬂow along the path 816 are numbered with red cycle numbers in the extended subﬂow diagram below:

1 2 817 p1 = 0 7→ 2 1 2 −→ 1 2 −→ 1 ··· 12 2 2 2 2 2 2 2

818 The cumulative transient inﬂow f 1 (t) and substorage x1 (t) can be approximated 1222 12 819 by two terms (m1 = 2) as formulated in Eq. 2.32:

2 X x1 (t) ≈ x1,m (t) = x1,1 (t) + x1,2 (t), 12 221222 221222 221222 m=1 820 2 X f 1 (t) ≈ f 1,m (t) = f 1,1 (t) + f 1,2 (t). 1222 122212 122212 122212 m=1

w,m w,m 821 The transient subflows and associated substorages f (t) and x (t) and the 122212 221222 822 other transient subflows and substorages involved in Eq. 3.12, as formulated in Eqs. 2.30 823 and 2.31, are solved simultaneously together with the decomposed system, Eq. 2.9. 824 The numerical results for the cycling flow and associated storage functions

2 2 X X 825 (3.12) τ c(t) = τ c (t) and xc(t) = xc (t) i ik i ik k=0 k=0

826 for i = 1, 2, are presented in Fig. 12.

827 4. Discussion. Environment is not an easy concept to define and analyze math- 828 ematically. Although sound rationales are offered in the literature for considering nat- 829 ural system dynamics under specific cases, such as linear models and static systems, 830 realistically nature is always on the move and its systems are always changing to meet 831 ever-renewing circumstances. In recent decades, there has been several attempts to 832 analyze dynamic ecological networks, but they all have disadvantages. The need for 833 dynamic and nonlinear methodologies has always been present. We propose a novel 834 mathematical methodology for the analysis of nonlinear dynamical compartmental 835 systems to comprehensively address these shortcomings in the present manuscript. 836 Considering a hypothetical ecosystem with several interacting species for which 837 the effect of a specific poison needs to be investigated, one of the most critical in- 838 quiries would be about the evolution of the poison within the system. Assuming that 839 the intercompartmental interactions are formulated deterministically, current mathe- 840 matical methods can analyze the composite throughflow and storage of the toxin at 841 each species within the system, given the initial poison amount in each species. The 842 evolution of a single environmental poison input within the system, however, cannot 843 be determined through the current methodologies. If multiple species are exposed to 844 the same poison from the environment, the proposed system partitioning methodology 845 enables dynamically partitioning the composite poison flow and storage in any species 846 into subcompartmental segments based on their constituent environmental inputs. In

This manuscript is for review purposes only. 28 HUSEYIN COSKUN

847 other words, the system partitioning enables dynamically tracking the evolution of en- 848 vironmental poison inputs and associated storages individually and separately within 849 the system. 850 Unlike the state of the art techniques, the proposed subsystem partitioning method- 851 ology can then dynamically track the fate of arbitrary intercompartmental poison flows 852 and the associated storages generated by these transient flows in each species along 853 a given chain of interspecific interactions. Therefore, the spread of the arbitrary 854 amount of toxin from one specific species to the entire system or along a given chain 855 of interspecific interactions can be determined and monitored dynamically. Through 856 the subsystem partitioning, the direct, indirect, acyclic, cycling, and transfer (diact) 857 flows and associated storages of the poison from one compartment, directly or indi- 858 rectly, to any other—including itself—can also be determined dynamically. 859 More technically, the proposed system partitioning methodology explicitly gener- 860 ates mutually exclusive and exhaustive subsystems, each driven by a single environ- 861 mental input, that are running within the original system and have the same struc- 862 tures and dynamics as the system itself. Therefore, the system partitioning yields 863 the subthroughflow and substorage matrix functions, which respectively represent 864 the throughflows and storages generated by environmental inputs in each compart- 865 ment. Equipped with these matrix measures, the system partitioning ascertain the 866 dynamic distribution of environmental inputs and the organization of the associated 867 storages generated by the inputs individually and separately within the system. Con- 868 sequently, the composite compartmental storages and throughflows, xi(t) and τi(t), 869 are dynamically partitioned into the subcompartmental substorage and subthrough-

870 flow segments, xik (t) and τik (t), based on their constituent environmental sources, 871 zk(t). In other words, the system partitioning enables dynamically tracking the evo- 872 lution of environmental inputs individually and separately within the system. The 873 system partitioning methodology, therefore, refines system analysis from the current 874 static, linear, compartmental to the dynamic, nonlinear, subcompartmental level to 875 explore the full complexity of the ecological systems. 876 The subsystems are then further decomposed into subflows and associated sub- 877 storages along a set of mutually exclusive and exhaustive directed subflow paths. 878 The subsystem partitioning methodology yields the transient and the dynamic direct, 879 indirect, acyclic, cycling, and transfer (diact) flows and the associated storages gen- 880 erated by these flows. The transient subflows and associated substorages determine 881 the dynamic distribution of arbitrary intercompartmental flows and the organization 882 of the associated storages generated by these flows along given subflow paths within 883 the subsystems. Consequently, arbitrary composite intercompartmental flows and as- 884 sociated storages are dynamically partitioned into the constituent transient subflow 885 and substorage segments along the given set of subflow paths. In other words, the 886 subsystem partitioning enables dynamically tracking the fate of arbitrary intercom- 887 partmental flows and associated storages within the subsystems. Moreover, a history 888 of compartments visited by arbitrary system flows and storages can be also compiled. 889 Based on the concept of transient flows and storages, the dynamic direct, indirect, 890 acyclic, cycling, and transfer (diact) flows and associated storages from one com- 891 partment, directly or indirectly, to any other—including itself—within the system are 892 also formulated systematically for the quantification of intercompartmental flow and 893 storage dynamics. Therefore, the proposed mathematical method, as a whole, yields 894 the decomposition to the utmost level or “atomization” of the system flows and stor- 895 ages. The illustrative case studies in Section3 demonstrate that these measures are 896 rigorous and efficient ecological system analysis tools.

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 29

897 For a comparison of the proposed methodology with the state of the art tech- 898 niques, we first note that, at steady state, the proposed dynamic methodology agrees 899 with the current techniques for static ecological network analysis, as shown by [10]. In 900 recent decades, there have been several attempts to analyze dynamic ecological net- 901 works. All of these attempts have disadvantages. The first actual dynamic analysis, 902 which was limited to linear systems with time dependent input, was introduced by 903 [20]. The proposed method is applied to the linear ecosystem model introduced by 904 [20] as an illustrative case study. It is shown that the analytic solutions obtained by 905 the proposed methodology agree with those by Hippe’s approach. Further results that 906 are not available through Hippe’s methodology, such as the diact flows and storages 907 are also presented in Section 3.2. 908 The dynamic approach is extended from linear to nonlinear systems by [16]. The 909 authors provided, however, only closed-form, abstract formulations that are hard 910 to apply to real cases. The proposed methodology is also applied to the nonlinear 911 ecosystem model analyzed by [16], and the results, together with their ecological 912 interpretations, are presented in Section3. A comparison of our results with the 913 ones provided by the authors was not possible because, unlike the complete dynamic 914 analysis enabled by the proposed methodology for nonlinear systems, the authors 915 could only provide asymptotic solutions to the model at steady state through their 916 methodology. 917 Individual-based algorithms that rely on network particle tracking simulations are 918 also proposed for nonlinear ecological models in the literature. A truncated infinite 919 series formulation was proposed by [31]. The authors’ approach was approximate, 920 computationally resource-intensive, and offered no guarantees of series convergence. 921 A guarantee of convergence is added by [24], but computation remained very resource- 922 intensive due to the individual based simulation methodology [25, 32]. 923 This is the first manuscript in literature that comprehensively addresses all the 924 previously identified problems and shortcomings. The main limitation of the method 925 is that it is designed for analysis of conservative models as defined by Eq. 2.5. A large 926 class of real world problems are formulated based on conservation principles in many 927 fields as a consequence of their being the fundamental laws of nature. On the other 928 hand, there are still various non-conservative systems that cannot be analyzed by the 929 proposed methodology in its current form.

930 5. Conclusions. In the present paper, we developed a comprehensive mathe- 931 matical method for the analysis of nonlinear dynamic ecological systems. The pro- 932 posed method is based on the novel analytical and explicit, mutually exclusive and ex- 933 haustive system and subsystem partitioning methodologies. As the proposed dynamic 934 system partitioning yields the subthroughflow and substorage matrices to determine 935 the distribution of environmental inputs and the organization of associated storages 936 individually and separately within the system, the subsystem partitioning yields the 937 transient and diact flow and storage matrices to determine the distribution of ar- 938 bitrary intercompartmental flows and the organization of associated storages within 939 the subsystems. Consequently, the evolution of environmental inputs and arbitrary 940 intercompartmental system flows as well as the associated storages generated by these 941 inputs and flows can be tracked individually and separately within the system. 942 Traditional ecology is still largely a descriptive empirical science. This narrows 943 the field’s scope of applicability and compromises its ability to deal with complex 944 ecological networks. The proposed dynamic method extends the strength and appli- 945 cability of the state of the art techniques and provides significant advancements in

This manuscript is for review purposes only. 30 HUSEYIN COSKUN

946 theory, methodology, and practicality. It serves as a quantitative platform for testing 947 empirical hypotheses, ecological inferences, and, potentially, theoretical developments. 948 Therefore, this method has the potential to lead the way to a more formalistic eco- 949 logical science. We consider that the proposed methodology brings a novel complex 950 system theory to the service of urgent and diﬃcult environmental problems of the 951 day. Several case studies from ecosystem ecology are presented to demonstrate the 952 accuracy and eﬃciency of the method. 953 The proposed methodology also constructs a base for the development of new 954 dynamic system analysis tools as ecological indicators. The time dependent nature of 955 these quantities enables also their time derivatives and integrals to be formulated as 956 novel quantitative analysis tools. Multiple such dynamic diact measures and indices 957 of matrix, vector, and scalar type are systematically introduced in a separate paper 958 by [8].

959 Appendices. The dynamic system partitioning methodology decomposes an eco- 960 logical system into mutually exclusive and exhaustive subsystems through a set of gov- 961 erning equations derived from subcompartmentalization and ﬂow partitioning. This 962 methodology enables the analysis of each subsystem generated by an environmental 963 input and the initial conditions individually and separately. The subsystem ﬂows and 964 storages in matrix form are formulated in this section.

965 Appendix A. Subsystem flows and storages. th th 966 We define the k direct subflow matrix function for the k subsystem as Fk(t, x) =

967 (fikjk (t, x)), for k = 0, . . . , n. Using the relationships formulated in Eq. 2.7, it can be 968 expressed as

−1 969 (A.1) Fk(t, x) = F (t, x) X (t) Xk(t)

970 where Xk(t) = diag ([x1k (t), . . . , xnk (t)]) is the diagonal matrix of the substorage th th 971 values in the k subsystem. The matrix Xk(t) will accordingly be called the k 972 substorage matrix function. The kth output and input matrix functions become

−1 973 Yk(t, x) = Y(t, x) X (t) Xk(t) and Zk(t, x) = diag (zk ek)

th 974 where ek is the elementary unit vector whose components are all zero except the k th 975 element, which is 1. We set e0 = 0. The k direct subflow matrix, Fk(t, x), and the th 976 k input and output vector functions,y ˆk(t, x) = Yk(t, x) 1 andz ˇk(t, x) = Zk(t, x) 1, 977 are the counterparts for subsystem k of the direct flow matrix, F (t, x), and input and 978 output vectors, y(t, x) and z(t, x), for the original system. Altogether, they represent 979 the subflow regime of the kth subsystem. 980 Using the notations and definitions of Eqs. 2.13 and A.1, the kth inward and out- ˇ ˆ 981 ward subthroughflow matrices, Tk(t, x) = diag ([ˇτ1k (t, x),..., τˇnk (t, x)]) and Tk(t, x) = th 982 diag ([ˆτ1k (t, x),..., τˆnk (t, x)]), for the k subsystem can be expressed as

−1 Tˇk(t, x) = Zk(t, x) + diag F (t, x) X (t) Xk(t) 1 , T −1 983 (A.2) Tˆk(t, x) = Y(t, x) + diag F (t, x) 1 X (t) Xk(t) −1 = T (t, x) X (t) Xk(t).

th 984 We deﬁne the decomposition and k decomposition matrices, D(x) = (dik (x)) and

985 Dk(x) = diag ([d1k (x), . . . , dnk (x)]), as

−1 −1 −1 −1 986 D(x) = X (t) X(t) = T (t) Tˆ(t) and Dk(x) = X (t) Xk(t) = T (t, x) Tˆk(t, x).

This manuscript is for review purposes only. DYNAMIC ECOLOGICAL SYSTEM ANALYSIS 31

987 The second equalities in the deﬁnitions of D(x) and Dk(x) are due to Eq. 2.13 and A.2, 988 respectively. th 989 The k direct subﬂow and substorage matrices, Fk(t, x) and Xk(t), can then be 990 written in various forms as follows: −1 −1 Fk(t, x) = F (t, x) Dk(x) = F (t, x) X (t) Xk(t) = F (t, x) T (t, x) Tˆk(t, x), 991 (A.3) Xk(t) = R(t, x) Tˆk(t, x).

992 Note that D(x) and Dk(x) partition the compartmental throughflow matrix, T (t, x), 993 into the subthroughflow and kth subthroughflow matrices as indicated in Eqs. 2.13 994 and A.2, similar to the decomposition of Fk(t, x) as formulated in Eq. A.3. That is,

995 (A.4) Tˆ(t, x) = T (t, x) D(x) and Tˆk(t, x) = T (t, x) Dk(x).

996 Acknowledgments. The author would like to thank the members of the ecosys- 997 tem ecology group in the Department of Mathematics at UGA, for useful discussions 998 and valuable feedback during his visit in 2016 from which the work has beneﬁted. In 999 particular, the author is indebted to Caner Kazanci for introducing the static environ 1000 theory to the author, identifying some open problems in the area, and reviewing the 1001 paper, and other group members, Bernard C. Patten for his detailed review of the 1002 paper and Malcolm Adams for useful discussions and comments. The author also 1003 thanks Hasan Coskun, Stuart Borett, Sangwon Suh, and Brian Fath for their review 1004 of the paper and helpful comments that improved the manuscript.

REFERENCES1005

1006 [1] T. Allen and M. Giampietro, Holons, creaons, genons, environs, in hierarchy theory: Where 1007 we have gone, Ecological Modelling, 293 (2014), pp. 31–41, https://doi.org/10.1016/j. ecolmodel.2014.06.017.1008 1009 [2] D. Anderson, Compartmental Modeling and Tracer Kinetics, Springer Berlin Heidelberg, 2013. 1010 [3] R. Bailey, J. K. Allen, and B. Bras, Applying ecological input-output flow analysis to 1011 material flows in industrial systems: Part i: Tracing flows, Journal of Industrial Ecology, 1012 8 (2004), pp. 45–68, https://doi.org/10.1162/1088198041269346. 1013 [4] A. Belgrano, U. M. Scharler, J. Dunne, and R. E. Ulanowicz, Aquatic Food Webs: An 1014 Ecosystem Approach, Oxford University Press, Oxford, U.K, 2005. 1015 [5] S. R. Borrett and M. K. Lau, enaR : An R package for ecosystem network analysis, Methods 1016 in Ecology and Evolution, 5 (2014), pp. 1206–1213, https://doi.org/10.1111/2041-210x. 12282.1017 1018 [6] X. Chen and J. E. Cohen, Transient dynamics and food–web complexity in the lotka–volterra 1019 cascade model, Proceedings of the Royal Society of London. Series B: Biological Sciences, 268 (2001), pp. 869–877.1020 1021 [7] V. Christensen and D. Pauly, ECOPATH II-a software for balancing steady-state ecosystem 1022 models and calculating network characteristics, Ecological modelling, 61 (1992), pp. 169– 185.1023 1024 [8] H. Coskun, Dynamic ecological system measures. Preprint, 2018, https://doi.org/10.31219/ osf.io/j2pd3.1025 1026 [9] H. Coskun, Nonlinear decomposition principle and fundamental matrix solutions for dynamic 1027 compartmental systems. Preprint, 2018, https://doi.org/10.31219/osf.io/cyrzf. 1028 [10] H. Coskun, Static ecological system analysis. Preprint, 2018, https://doi.org/10.31219/osf.io/ zqxc5.1029 1030 [11] H. Coskun, Static ecological system measures. Preprint, 2018, https://doi.org/10.31219/osf. io/g4xzt.1031 1032 [12] A. I. Dell, G. D. Kokkoris, C. Bana˜sek-Richter, L. F. Bersier, J. A. Dunne, M. Kondoh, 1033 T. N. Romanuk, and N. D. Martinez, How do Complex Food Webs Persist in Nature?, 1034 Elsevier Inc., 2006, pp. 425–436, https://doi.org/10.1016/B978-012088458-2/50040-0. 1035 [13] B. D. Fath and S. R. Borrett, A Matlab R function for Network Environ Analysis, Environ- 1036 mental Modelling and Software, 21 (2006), pp. 375–405, https://doi.org/10.1016/j.envsoft. 2004.11.007.1037

This manuscript is for review purposes only. 32 HUSEYIN COSKUN

1038 [14] J. Finn, Measures of structure and functioning derived from analysis of flows, Journal of Theoretical Biology, 56 (1976), pp. 363–380.1039 1040 [15] P. J. S. Franks, Npz models of plankton dynamics: Their construction, coupling to physics, 1041 and application, Journal of Oceanography, 58 (2002), pp. 379–387, https://doi.org/10. 1042 1023/a:1015874028196, hGotoISIi://WOS:000175729100012. n/a. 1043 [16] T. G. Hallam and M. N. Antonios, An environ analysis for nonlinear compartment mod- 1044 els, Bulletin of Mathematical Biology, 47 (1985), pp. 739–748, https://doi.org/10.1007/ BF02469300.1045 1046 [17] B. Hannon, The structure of ecosystems, Journal of theoretical biology, 41 (1973), pp. 535–546. 1047 [18] A. Hastings, What Equilibrium Behavior of Lotka-Volterra Models Does Not Tell Us About 1048 Food Webs, Springer US, Boston, MA, 1996, pp. 211–217, https://doi.org/10.1007/ 978-1-4615-7007-3{ }21.1049 1050 [19] A. Hastings, Transients: the key to long-term ecological understanding?, Trends in Ecology 1051 & Evolution, 19 (2004), pp. 39–45, https://doi.org/10.1016/j.tree.2003.09.007. 1052 [20] P. W. Hippe, Environ analysis of linear compartmental systems: The dynamic, time-invariant 1053 case, Ecological Modelling, 19 (1983), pp. 1–26, https://doi.org/10.1016/0304-3800(83) 90067-4.1054 1055 [21] H. Hirata and R. E. Ulanowicz, Information theoretical analysis of the aggregation and 1056 hierarchical structure of ecologial networks, Journal of Theoretical Biology, 116 (1985), pp. 321–341.1057 1058 [22] J. Jacquez and C. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), pp. 43–79, https://doi.org/10.1137/1035003.1059 1060 [23] C. Kazanci, Network calculations II: a user’s manual for EcoNet, in Handbook of Ecological 1061 Modelling and Informatics, WITPRESS LTD., 2009, pp. 325–350, https://doi.org/10.2495/ 978-1-84564-207-5/18.1062 1063 [24] C. Kazanci and Q. Ma, Extending ecological network analysis measures to dynamic ecosys- 1064 tem models, Ecological Modelling, 242 (2012), pp. 180–188, https://doi.org/10.1016/j. ecolmodel.2012.05.021.1065 1066 [25] C. Kazanci, L. Matamba, and E. W. Tollner, Cycling in ecosystems: An individual based 1067 approach, Ecological Modelling, 220 (2009), pp. 2908–2914, https://doi.org/10.1016/j. ecolmodel.2008.09.013.1068 1069 [26] W. W. Leontief, Quantitative input and output relations in the economic systems of the 1070 united states, The review of economic statistics, 18 (1936), pp. 105–125. 1071 [27] W. W. Leontief, Input-output economics, Oxford University Press on Demand, New York, 1986.1072 1073 [28] J. Matis and B. Patten, Environ analysis of linear compartmental systems: the static, time 1074 invariant case, Statistical Ecology, 48 (1981), pp. 527–565. 1075 [29] B. C. Patten, Systems approach to the concept of environment, Ohio Journal of Science, 78 (1978), pp. 206–222.1076 1077 [30] J. R. Schramski, C. Kazanci, and E. W. Tollner, Network environ theory, simulation, 1078 and EcoNet R , 2.0, Environmental Modelling and Software, 26 (2011), pp. 419–428, https: //doi.org/10.1016/j.envsoft.2010.10.003.1079 1080 [31] J. Shevtsov, C. Kazanci, and B. C. Patten, Dynamic environ analysis of compartmental 1081 systems: A computational approach, Ecological Modelling, 220 (2009), pp. 3219–3224, https://doi.org/10.1016/j.ecolmodel.2009.07.022.1082 1083 [32] E. W. Tollner, J. R. Schramski, C. Kazanci, and B. C. Patten, Implications of net- 1084 work particle tracking (NPT) for ecological model interpretation, Ecological Modelling, 1085 220 (2009), pp. 1904–1912, https://doi.org/10.1016/j.ecolmodel.2009.04.018. 1086 [33] R. E. Ulanowicz, Mass and energy flow in closed ecosystems, Journal of Theoretical Biology, 1087 34 (1972), pp. 239–253, https://doi.org/10.1016/0022-5193(72)90158-0. 1088 [34] R. E. Ulanowicz, Quantitative methods for ecological network analysis, Computational Biol- 1089 ogy and Chemistry, 28 (2004), pp. 321–339, https://doi.org/10.1016/j.compbiolchem.2004. 09.001.1090 1091 [35] R. E. Ulanowicz, R. D. Holt, and M. Barfield, Limits on ecosystem trophic complexity: 1092 insights from ecological network analysis, Ecology Letters, 17 (2013), pp. 127–136, https: //doi.org/10.1111/ele.12216.1093 1094 [36] R. E. Ulanowicz and J. J. Kay, A package for the analysis of ecosystem flow networks, 1991, https://doi.org/10.1016/0266-9838(91)90024-K.1095 1096 [37] M. J. V. Zanden, J. D. Olden, C. Gratton, and T. D. Tunney, Food Web Theory and 1097 Ecological Restoration, Island Press/Center for Resource Economics, Washington, DC, USA 978-1-61091-698-1, 2016, pp. 301–329.1098

This manuscript is for review purposes only.