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