Multiple Steady States Research Articles

Dynamic analysis of demographic change and human capital accumulation in an R&D-based growth model

Employing an overlapping-generations model of R&D-based growth with endogenous fertility, mortality, and education choice, we examine how demographic changes and human capital accumulation influence R&D activity. We show that multiple steady states can exist in this economy. One steady state has a high level of human capital and the other has a low level. In the steady state with high (low) level of human capital, there is a high (low) level of R&D activity, a low (high) fertility rate, and a high (low) old-age survival rate. In addition, we show that the government can steer an economy away from a poverty trap trajectory by investing in public health. We also show that an improvement in the government’s public health policy has an inverted U-shaped effect on the growth rate at the steady state.

A computational framework for a Lyapunov-enabled analysis of biochemical reaction networks.

Complex molecular biological processes such as transcription and translation, signal transduction, post-translational modification cascades, and metabolic pathways can be described in principle by biochemical reactions that explicitly take into account the sophisticated network of chemical interactions regulating cell life. The ability to deduce the possible qualitative behaviors of such networks from a set of reactions is a central objective and an ongoing challenge in the field of systems biology. Unfortunately, the construction of complete mathematical models is often hindered by a pervasive problem: despite the wealth of qualitative graphical knowledge about network interactions, the form of the governing nonlinearities and/or the values of kinetic constants are hard to uncover experimentally. The kinetics can also change with environmental variations. This work addresses the following question: given a set of reactions and without assuming a particular form for the kinetics, what can we say about the asymptotic behavior of the network? Specifically, it introduces a class of networks that are “structurally (mono) attractive” meaning that they are incapable of exhibiting multiple steady states, oscillation, or chaos by virtue of their reaction graphs. These networks are characterized by the existence of a universal energy-like function called a Robust Lyapunov function (RLF). To find such functions, a finite set of rank-one linear systems is introduced, which form the extremals of a linear convex cone. The problem is then reduced to that of finding a common Lyapunov function for this set of extremals. Based on this characterization, a computational package, Lyapunov-Enabled Analysis of Reaction Networks (LEARN), is provided that constructs such functions or rules out their existence. An extensive study of biochemical networks demonstrates that LEARN offers a new unified framework. Basic motifs, three-body binding, and genetic networks are studied first. The work then focuses on cellular signalling networks including various post-translational modification cascades, phosphotransfer and phosphorelay networks, T-cell kinetic proofreading, and ERK signalling. The Ribosome Flow Model is also studied.

Multiplicity, travelling waves and spatial patterns in coupled autocatalytic reaction systems

In an autocatalytic reaction, one of the products catalyzes its own formation. In this work, we analyze a system of two species, each participating in two autocatalytic reactions. Each species participates in the autocatalysis of the other. We first analyze how the kinetics of the reactions determines the non-linear characteristics of the lumped system, such as multiple steady-states. Two cases are considered depending on the kinetics of the two reactions. In the first case, both reactions are quadratic in nature, while in the second, reactions are cubic. Singularity theory and bifurcation theory are employed to comprehensively understand the multiplicity features of the steady states and to classify the bifurcation behavior of the lumped system. The analysis here forms the basis to determine operating conditions under which a pure enantiomer can be produced. The occurrence of spontaneous spatio-temporal features like travelling waves and Turing patterns under specific conditions has been established.

Kinetic Modeling of the Blood Oxygenation Level Dependent (BOLD) Signals and Biocatalytic Reactions Observed in the Human Brain Using MRI: An Analysis of Normal and Pathological Conditions.

A kinetic model describing the pulse of increased oxygen concentrations and the subsequent changes in the concentration of N-acetylaspartate in the excited nervous tissue of the human brain in response to an external signal is presented. The model is based on biochemical data, a multistage and nonlinear dynamic process the BOLD signal and N-acetylaspartate. The existence of multiple steady states explains the triggering effect of the system. The inhibitory effect of the substrate is a necessary factor for the autostabilization of N-acetylaspartate. The kinetic model allows the dynamic behavior of previously unmeasurable metabolites, namely, products of the hydrolysis of N-acetylaspartate, such as acetic and aspartic acid, and glutamic acid to be predicted. Kinetic modeling of the BOLD signal and the subsequent hydrolysis of N-acetylaspartate provides information about the biochemical and dynamic characteristics of some pathological conditions (schizophrenia, Canavan disease, and the superexcitation of the neural network).

Analysis of open-loop and [formula omitted] controlled closed-loop behavior of the Cholette’s bioreactor under different operating conditions

The scope of this paper is the analysis of open-loop and L2/D controlled closed-loop behavior of the Cholette’s bioreactor under different operating conditions. The open-loop behavior is studied by means of the steady-state multiplicity areas formed by substrate feed concentration and substrate concentration in the output using the nonlinear model of the bioreactor. The stability of each steady-state inside areas is determined by a polynomial representation. The nonlinear dynamic and steady-state multiplicity areas of the Cholette’s bioreactor are shown to depend on the operating conditions and concentration required. The closed-loop behavior is studied by the tuning of a set of Pareto-optimal L2/D linear controllers at a stable operating point. Numerical simulations and phase portraits are used to evaluate the control system. This paper could be a guide for theoretical and practical applications in the fields of biotechnology and automatic control.

A Petlyuk distillation column dynamic analysis: Hysteresis and bifurcations

Thermally coupled distillation columns (TCDC) have proven to be a superior and feasible alternative compared to conventional distillation. However, their inherent system complexity presents optimization and operational challenges. In this work, we analyzed these problems from a different perspective: understanding the steady-state behavior by the means of a steady state and bifurcation analysis. This approach used a dynamic model programmed in Matlab 2019b and validated with Aspen Plus v8.8. The analysis demonstrated that the steady-state multiplicity is triggered by the magnitude of the internal flows and that the interlinking trays may develop isolated bifurcation branches. Moreover, it is shown that the “operational holes” or intermittent discontinuities in the TCDC steady states are formed by saddle-node bifurcations. This suggests that a non-smooth optimization algorithm is more suitable for TCDC design.

Critical fan-induced pressure rise for preventing smoke flow multiplicity in longitudinally ventilated sloping tunnel fires: Potential function analysis and numerical simulations

Tunnel fires remain a major concern around the world, especially in sloping tunnels. We investigated the critical conditions for preventing smoke flow multiplicity in longitudinally ventilated sloping tunnel fires and identified multiple flow patterns for smoke movement (i.e., different steady states could be achieved under identical boundary conditions). This results in significant challenges for the design and efficient operation of longitudinal ventilation systems in tunnels. We conducted a potential function analysis and numerical simulations to investigate the critical fan-induced pressure rise for preventing smoke flow multiplicity in a downhill sloping tunnel. Calculation formulas regarding the critical fan-induced pressure rise for preventing smoke flow multiplicity, Pj,max′, were obtained. We demonstrated that a fan-dominated pattern with back-layering flow is physically unstable, and that the fire source location and source strength have an evident impact on smoke flow multiplicity, and on critical values (i.e., Pj,max′andPj,min′) of fan-induced pressure rise. The magnitude of pressure rise derived from critical velocity Pj,cr′ is not sufficient to prevent smoke from spreading upstream in sloping tunnels. Multiple steady states can be achieved if the magnitude of fan-induced pressure rise is between Pj,cr′and Pj,max′. Moreover, Pj,max′ increases more rapidly than Pj,cr′ as the strength of the source increases. The Pj,cr′ is approximately 10–30% less than Pj,max′ when the strength of the source surpasses 20 MW. Therefore, in applications, Pj,max′, rather than Pj,cr′ is essential to ensure the effectiveness of longitudinal ventilation design for smoke control.

Nonlinear dynamic analysis of solution multiplicity of buoyancy ventilation in a typical underground structure

Buoyancy ventilation is widely used in underground buildings, such as underground hydropower stations. Multiple solutions of buoyancy ventilation may exist in those underground structures. In this study, we developed a transient model comprising an ordinary differential equation system to describe buoyancy ventilation patterns in typical two-zone underground structures. Additionally, the accuracy of the model was validated. Nonlinear dynamical analysis was conducted to study multiple steady-state airflow. According to mathematical derivation, the configuration of one local heat source at the bottom corner introduces two stable solutions. The criterion to determine the stability and existence of solutions for more general scenarios was developed. Using this criterion, we obtained the multiple steady states of any two-zone underground buildings for different stack height ratios and the strength ratios of the heat sources. This criterion can be adopted for the design of buoyancy ventilation or natural smoke ventilation systems. Designers can change the height ratio of the stack or the heat ratio of two zones to induce the desired ventilation patterns. Finally, a case study was conducted with field measurements to demonstrate the use of the nonlinear dynamical analysis method to investigate the multiple steady states of buoyancy ventilation. Through the case study, we validated that the proposed criterion could produce the same result as the nonlinear dynamical analysis.

Dynamics of opinion formation under majority rules on complex social networks

We study opinion dynamics on complex social networks where each individual holding a binary opinion on a certain subject may change her/his mind to match the opinion of the majority. Two rules of interactions between individuals, termed as classic majority and influence majority rules, respectively, are imposed on the social networks. The former rule allows each individual to adopt an opinion following a simple majority of her/his immediate neighbors, while the latter one lets each individual calculate the influence of each opinion and choose to follow the more influential one. In this calculation, the influences of different opinions are counted as the sum of the influences of their respective opinion holders in neighborhood area, where the influence of each individual is conveniently estimated as the number of social connections s/he has. Our study reveals that in densely-connected social networks, all individuals tend to converge to having a single global consensus. In sparsely-connected networks, however, the systems may exhibit rich properties where coexistence of different opinions, and more interestingly, multiple steady states of coexistence can be observed. Further studies reveal that low-degree and high-degree nodes may play different roles in formulating the final steady state, including multi-steady states, of the systems under different opinion evolution rules. Such observations would help understand the complex dynamics of opinion evolution and coexistence in social systems.

Thermal stability analysis of cold start processes in PEM fuel cells

Cold-start issue of proton exchange membrane (PEM) fuel cells is one of the major factors hindering its commercialization. The key element involved is the competition between ice formation and melting, behind which is the thermal behavior accompanied by nonlinear effects. In this work, the steady-state multiplicity feature during cold-start has been discovered based on the thermal stability analysis approach. The nonlinear heat generation term Qg and linear removal term Qr are separated from conservation equations and constructed against ice fraction. The plot thus obtained could identify the existence of multiplicity which is the intersection of the above two functions. Results indicate that one or two steady states arise for different operating conditions. The startup with only one steady state, namely the “extinguished” state is the worst condition and bound to fail. The startup with two steady states also incorporates an unstable one which denotes not only the critical transition from ice formation to melting but also the limit of the operational domain. Based on the theoretical analysis, simple and explicit criteria are quantitatively developed for the prediction of startup feasibility. Moreover, in order to ensure a reasonable and efficient operation, the impacts of key parameters are further summarized and discussed on the cold-start operability. The proposed approach that develops analytical expressions for the steady-state points provides a systematic yet simple way to reveal the essential physics of cold start.

Pathway Complexity in Fuel-Driven DNA Nanostructures with Autonomous Reconfiguration of Multiple Dynamic Steady States

We introduce pathway complexity on a multicomponent systems level in chemically fueled transient DNA polymerization systems, achieving autonomous evolution over multiple structural dynamic steady states from monomers to dimers, oligomers of dimers, and randomized polymer structures before being ultimately degraded back to monomers once the fuel is consumed. The enabling key principle is to design monomer species having kinetically selected molecular recognition in the structure-forming step and which are reconfigured in an enzymatic reaction network. This nonequilibrium systems chemistry approach to pathway complexity provides new conceptual insights into fuel-driven automatons and autonomous materials design.

Nonlinear Model Predictive Control for an Exothermic Reaction in an Adiabatic CSTR

We model and simulate an exothermic reaction conducted in an adiabatic continuous stirred tank reactor (CSTR). The system has multiple steady states in part of its operating window. We demonstrate that the three-state model representing the mass and energy balances of the system can be well approximated with a two-state as well as a one-state model. The reduced-order models are relevant for efficient implementation of nonlinear model predictive control (NMPC). We extend the drift term in the system of deterministic ordinary differential equations (ODEs) with a diffusion term such that the process is modeled by a system of stochastic differential equations (SDEs). The SDE representation is well-suited for development of the NMPC that is based on a continuous-discrete extended Kalman filter (CD-EKF) and an optimal control problem. The optimal control problem (OCP) is solved using a multiple-shooting algorithm as the NMPC must be able to stabilize the system at stable as well as unstable steady states. Simulations demonstrate that the NMPC based on a one-state reduced SDE can track any reactor temperature setpoint at stable as well as unstable steady states.

Modeling the effects of density dependent emigration, weak Allee effects, and matrix hostility on patch-level population persistence.

The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect. In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density. The habitat patch is one-dimensional and is surrounded by a tuneable hostile matrix. We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies. Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them. Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch. Our results are obtained mathematically through the method of subsuper solutions, time map analysis, and numerical computations using Wolfram Mathematica.

Convergence, Divergence, or Multiple Steady States? New Evidence on the Institutional Development Within the European Union

The positive link between good institutions and economic development is well documented in the literature on the deep determinants of economic growth. Consequently, convergence in institutional quality should be a necessary precondition for income convergence; however, only recently, researchers started to investigate empirically the nature of institutional development. This topic seems to be of particular relevance for the European Union (EU) since convergence in institutions and in per capita income across the EU Member States are key goals of the European integrations process. Especially in the course of the various EU enlargement waves starting in 2004, it was intensively discussed whether institutional and structural homogeneity are necessary preconditions for real convergence and the smooth functioning of the EU or whether a (further) catching up in the institutional and economic development will endogenously occur after the EU accession. Our paper is dedicated to the analysis of these institutional dynamics within the EU. In particular, we analyze the formation of institutional convergence clusters using Phillips and Sul’s (2007, 2009) log t test over the period 2002 to 2018. It is the first study that takes into account the possibility of multiple equilibria when analyzing the institutional catching-up process within the EU, whereas previous contributions focused on the empirical concepts of beta- and sigma-convergence. Our results indicate the existence of multiple institutional clubs (for the dimensions government effectiveness, regulatory quality, rule of law, and control of corruption of the Worldwide Governance Indicators) with various countries being stuck in a poor institutional trap. Moreover, we find that institutional convergence clubs are formed mainly on the basis of geographic region. Northern and Western European countries are typically in the higher institutional clubs whereas the lower institutional clubs comprise primarily Eastern and Southern European countries. When analyzing per capita income clubs, a rather similar picture emerges, suggesting that the underlying institutional clusters determine the formation of income clubs. Among the countries that joined the EU after 2004, Estonia, Lithuania, and Latvia perform best with respect to their institutional catching-up, whereas most of the remaining new member countries are found to be stuck in poor institutional traps. The situation is particularly severe for Slovakia, Slovenia, Bulgaria, Croatia, and Hungary. We also study the factors that determine club membership by using an ordered probit model. Most importantly, we find that the initial levels of human capital and institutional quality are decisive for determining whether a country is on a high or low institutional growth path.

Spatial pattern formation in reaction-diffusion models: a computational approach.

Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton's method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses. In this paper, we present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states. We applied the method to two classic reaction-diffusion models and compared our results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities.

Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone

<p style='text-indent:20px;'>In this paper we study the protection zone problem to a predator-prey model subject to Beddington-DeAngelis functional responses and small prey growth rate. This is a successive work to a previous paper of the authors [X. He, S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol. 75 (2017) 239-257], where the model with large prey growth rate was considered. At first we establish the existence and multiplicity of positive steady state solutions, and then give the dynamic behavior of the evolution problem. It is proved that there may be no positive steady state, or may have at leat one, two, or even three positive steady states, depending on the parameters involved such as the growth rate, the predation rate, and the food handling time of the predators, the growth rate and the refuge ability of the preys, and the sizes of the habitat with protection zone. In addition, it is shown that the dynamics of the solutions rely on the initial state as well, e.g., though there could be multiple positive steady states, the prey will go to extinction as time tends to infinity if its initial value is small.

Liquidity Fluctuations in Over-the-Counter Markets

This paper presents a model of opaque secondary markets. Investors meet over-the-counter to trade heterogeneous assets under asymmetric information. An endogenous composition effect emerges whereby high liquidity alters the quality of the pool of sellers and decreases future liquidity. With impatient investors, cyclical equilibria arise: Price and volume oscillate without any fundamental shock. With patient investors, the equilibrium is driven instead by a resale effect: Liquidity depends primarily on investors' expectations about future liquidity, and multiple steady states coexist. In each case, equilibrium liquidity is driven by the opaqueness about asset quality and trade history typical of over-the-counter markets.

Programming Multistationarity in Chemical Replication Networks

AbstractSystems Chemistry generates platforms for studying the emergence of function in networks operating far from equilibrium. In this area, we have previously used experiments and simulations towards characterization and manipulation of small peptide‐based networks, driven by reversible self‐replication processes, that exhibit bistability. We now show how coupling two such networks, each exhibiting bistability, yields new dynamic systems that reach multiple (up to four) steady states. Furthermore, we demonstrate how such multistationarity, rarely analyzed before, can be systematically programmed and tuned. Our results suggest that the key to mimic biological complexification lies not only in the applied network size, the number of molecules involved, or even in the emergence of elaborate structures, but also in the network nature and topology.

Buying First or Selling First in Housing Markets

Abstract Housing transactions by moving homeowners take two steps—buying a new house and selling the old one. This paper argues that the transaction sequence decisions of moving homeowners have important effects on the housing market. Moving homeowners prefer to buy first whenever there are more buyers than sellers in the market. However, this congests the buyer side of the market and increases the buyer–seller ratio, further strengthening the incentives of other moving owners to buy first. This endogenous strategic complementarity leads to multiple steady state equilibria and large fluctuations, which are broadly consistent with stylized facts about the housing cycle.

Direct numerical simulation of a non-isothermal non-adiabatic packed bed reactor

A fundamental continuum-based numerical model was developed to simulate a non-isothermal non-adiabatic reactor which does not employ any empirical closures. The model was able to capture unique features of an exothermic catalytic reactor such as parametric sensitivity, hot-spot formations and multiplicity of steady states. Furthermore, the model inherently accounts for the various aspects of classical phenomenological models such as axial and radial dispersion of heat and mass and the intrinsic coupling of heat and mass transport between the fluid phase and the solid phase. The numerical procedure was validated with existing literature data before moving on to the simulation of a bed consisting of 340 spherical particles packed using the Discrete Element Method. Five simulations were performed by varying the rate of reaction and keeping all other parameters constant to capture the ignition/extinction phenomena exhibited by exothermic packed bed reactors.