Sustainable system dynamic models

Because systems and processes are not always operating at steady state, dynamic approaches should be also accounted for when discussing sustainable systems. Chapter Sustainable System Dynamics A Complex Network Analysis describes the implementation of complex network analysis to examine some integrated dynamic models of sus-tainabihty. In addition, it shows by using optimal techno-socio-economic policies how the system can be brought toward stable sustainability. 

Finally, Section 2.4 analyses a simplified model of a bursting pancreatic /3-cell [12]. The purpose of this section is to underline the importance of complex nonlinear dynamic phenomena in biomedical systems. Living systems operate under far-from-equilibrium conditions. This implies that, contrary to the conventional assumption of homeostasis, many regulatory mechanisms are actually unstable and produce self-sustained oscillatory dynamics. The electrophysiological processes of the pancreatic /3-cell display (at least) two interacting oscillatory processes A fast process associated with the K+ dynamics and a much slower process associated with the Ca2+ dynamics. Together these two processes can explain the characteristic bursting dynamics in the membrane potential. 

We started this chapter by delineating the two fundamental types of equations, either nonlinear or linear. We then introduced the few techniques suitable for nonlinear equations, noting the possibility of so-called singular solutions when they arose. We also pointed out that nonlinear equations describing model systems usually lead to the appearance of implicit arbitrary constants of integration, which means they appear within the mathematical arguments, rather than as simple multipliers as in linear equations. The effect of this implicit constant often shows up in startup of dynamic systems. Thus, if the final steady state depends on the way a system is started up, one must be suspicious that the system sustains nonlinear dynamics. No such problem arises in linear models, as we showed in several extensive examples. We emphasized that no general technique exists for nonlinear systems of equations. 

Flalog and Manik [12] developed AISMF LCSA through the combination of the E-LCA, LCC, and S-LCA frameworks, incorporated with multistakeholders analysis. The authors used multicriteria decision analysis to obtain the key indicators for LCSA, which were then used as critical variables for agent-based and/or system dynamics (e.g., use of causal loop relationships) modeling to ascertain the final results of sustainability decisions. 

Jin, W. Xu, L Yang, Z, (2009) Modeling a policy making framework for urban sustainability Incorporating system dynamics into the Ecological Footprint In Ecological Economics, 68(12), 2938-2949. 

Together, all the inferences from both computational modeling and simulation (which can reveal novel aspects of the receptor mechanisms, based on the dynamic properties of the proteins) serve as mechanistic working hypotheses for new and more focused experiments. This mode of closely considered interactions and synergy between computational developments and experimental probing of the receptor systems has become a sustained characteristic of current studies of structure-function... 

Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter 0 TP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Further extensions of the model are required to address the dynamical consequences of these additional regulatory loops and of the indirect nature of the negative feedback on gene expression. Such extended models have been proposed for Drosophila [112, 113] and mammals [113]. The model for the circadian clock mechanism in mammals is schematized in Fig. 3C. The presence of additional mRNA and protein species, as well as of multiple complexes formed between the various clock proteins, complicates the model, which is now governed by a system of 16 or 19 kinetic equations. Sustained or damped oscillations can occur in this model for parameter values corresponding to continuous darkness. As observed in the experiments on the mammalian clock. Email mRNA oscillates in opposite phase with respect to Per and Cry mRNAs [97]. The model displays the property of entrainment by the ED cycle... 

In a series of experiments we have tested the type and range of entrainment of glycolytic oscillations by a periodic source of substrate realizing domains of entrainment by the fundamental frequency, one-half harmonic and one-third harmonic of a sinusoidal source of substrate. Furthermore, random variation of the substrate input was found to yield sustained oscillations of stable period. The demonstration of the subharmonic entrainment adds to the proof of the nonlinear nature of the glycolytic oscillator, since this behavior is not observed in linear systems. A comparison between the experimental results and computer simulations furthermore showed that the oscillatory dynamics of the glycolytic system can be described by the phosphofructokinase model. 

In this fermentation process, sustained oscillations have been reported frequently in experimental fermentors and several mathematical models have been proposed. Our approach in this section shows the rich static and dynamic bifurcation behavior of fermentation systems by solving and analyzing the corresponding nonlinear mathematical models. The results of this section show that these oscillations can be complex leading to chaotic behavior and that the periodic and chaotic attractors of the system can be exploited for increasing the yield and productivity of ethanol. The readers are advised to investigate the system further. 

In general, the condition of systems H and N can be described by vectors xH t) = xlH,..., Xff and xN(t) = x, ..., x , respectively. The combined trajectory of these systems in n + m-dimensional space is described by the function rj t) = F(xh,xn) which is determined by solutions of the global model equations. The form of F is determined by knowledge of the laws of co-evolution, and therefore there is a possibility of investigations in different spheres of science. The available estimates of F (Krapivin, 1996) reveal a correlation between the notions survivability and sustainability. According to Ashby (1956), the dynamic system is alive within the time interval (ta, tb), if its determining phase coordinates are within admissible limits xlH>min N< x/N>max. And since systems H and N have a biological basis and limited resources, one of the indicated boundary conditions turns out to be unnecessary (i.e., for the components of vector... 

In the early 1990s, Brenner and coworkers [163] developed interaction potentials for model explosives that include realistic chemical reaction steps (i.e., endothermic bond rupture and exothermic product formation) and many-body effects. This potential, called the Reactive Empirical Bond Order (REBO) potential, has been used in molecular dynamics simulations by numerous groups to explore atomic-level details of self-sustained reaction waves propagating through a crystal [163-171], The potential is based on ideas first proposed by Abell [172] and implemented for covalent solids by Tersoff [173]. It introduces many-body effects through modification of the pair-additive attractive term by an empirical bond-order function whose value is dependent on the local atomic environment. The form that has been used in the detonation simulations assumes that the total energy of a system of N atoms is ... 

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