Number Of Steady States Research Articles

The steady-state degree and mixed volume of a chemical reaction network

The steady-state degree of a chemical reaction network is the number of complex steady-states, which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying polyhedral geometry associated with the corresponding polynomial system. We focus on three case studies of infinite families of networks, each generated by joining smaller networks to create larger ones. For each family, we give a formula for the steady-state degree and the mixed volume of the corresponding polynomial system.

Bifurcation analysis of a hybrid continuous stirred tank reactor with imperfect mixing in the cooling jacket

The paper discusses the effects of imperfect mixing on the dynamical behaviour of a partially simulated jacketed reactor. In the partially simulated exothermic reactor, the reactor jacket and vessel are real, but the chemical reactor is simulated by using a mathematical model and electric heaters to generate the reaction heat. The hybrid plant can be used as a benchmark plant for testing control algorithms, which is more realistic case than numerical simulations. It is experimentally shown that the coolant in the jacket is not well-mixed, while the main vessel content is well-mixed by using a recycle pump. Treating flow rate of coolant Fj as the main bifurcation parameter, we study the effect of internal recycle flow rate in the jacket on the number of steady-states and dynamical behaviour of the hybrid plant. It is shown that the imperfect mixing in the jacket is not as significant as the imperfect mixing in the main reactor vessel. The obtained results also show that for a given mixing conditions in the jacket, the reactor jacket model can be simplified by assuming perfect mixing conditions.

COMPARTMENTAL MODELLING OF SOCIAL DYNAMICS WITH GENERALISED PEER INCIDENCE

A generalised compartmental method for investigating the spread of socially determined behaviour is introduced, and cast in the specific context of societal smoking dynamics with multiple peer influence. We consider how new peer influence terms, acting in both the rate at which smokers abandon their habit, and the rate at which former smokers relapse, can affect the spread of smoking in populations of constant size. In particular, we develop a three-population model (comprising classes of potential, current, and former smokers) governed by multiple incidence transfer rates with linear frequency dependence. Both a deterministic system and its stochastic analogue are discussed: in the first we demonstrate that multiple peer influence not only modifies the number of steady-states and nature of their asymptotic stability, but also introduces a new kind of non-linear "tipping-point" dynamic; while in the second we use recently compiled smoking statistics from the Northeast of England to investigate the impact of systemic uncertainty on the potential for societal "tipping". The generality of our assumptions mean that the results presented here are likely to be relevant to other compartmental models, especially those concerned with the transmission of socially determined behaviours.

The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions

This paper proves the uniqueness of the positive linearly stable steady-state for a paradigmatic class of superlinear indefinite parabolic problems arising in population dynamics, under nonhomogeneous Dirichlet conditions on the boundary of the domain. The result is absolutely non-trivial, since examples are known for which the model admits an arbitrarily large number of steady-states. Our proof is based on some local and global continuation techniques. Optimal existence and multiplicity results are also obtained through some additional monotonicity and topological techniques.

Analysis of the effect of substrate material on the steady-state and transient performance of monolith reactors

We study the effect of the substrate material (ceramic versus metallic) on the steady-state and transient performance of monolith reactors using a one-dimensional two-phase model with position dependent transfer coefficients. When the operation of the reactor is on the ignited branch, it is shown that monoliths with metallic substrate clearly lead to a superior steady-state performance compared to those with ceramic substrate. In such cases, the ignited branch extends to lower inlet gas temperatures (typically 60–100°C in after-treatment applications) for the same catalyst loading. For transient operation where the time to light-off is important, it is shown that for the case of back-end ignition (corresponding to low inlet temperatures or low catalyst loading), metallic substrates are again superior. However, for the case of front-end ignition, ceramic converters may lead to lower cumulative emissions at lower inlet gas velocities while the converse is true at higher velocities. It is shown that the transient heating time and hence the cumulative emissions decrease with decrease in the channel hydraulic diameter and thermal capacitance of the substrate but are not monotonic with the inlet gas velocity. We present mathematical analysis and simulations to support these conclusions. Some novel results on the effect of substrate conductivity on the number of steady-states and upstream propagation of temperature fronts are also presented.

Heat and mass transfer correlations and bifurcation analysis of catalytic monoliths with developing flows

We review and compare the literature correlations for estimating the heat and mass transfer coefficients as well as pressure drop in catalytic monoliths with simultaneously developing velocity, concentration and temperature profiles. We present accurate correlations for estimating the local Nusselt and Sherwood numbers for developing flows with constant flux (slow reaction) and constant wall concentration or temperature (fast reaction) cases for a channel of arbitrary shape. These new correlations need only a single parameter, namely, the asymptotic value, which depends on the channel geometric shape. We establish the accuracy of the proposed correlations by comparing the predicted values with the exact numerical values available for a few cases. We use the new correlations to analyze the effect of flow conditions near the inlet of the channel on the ignition and extinction behavior of catalytic monoliths used in combustion and after-treatment applications as well as laboratory experiments. It is shown that the bifurcation behavior, such as the number and location of the ignition/extinction points, the number of stable steady-states and the hysteresis locus is sensitive to the flow conditions in the entry region, and hence the heat and mass transfer correlations used, especially for large values of the transverse Peclet number (high space velocities or very short monoliths) or adiabatic temperature rise or when the axial catalyst loading is not uniform.

Nonlinear analysis of the effect of maintenance in continuous cell cultures

The effect of maintenance on steady-state behavior in continuous cell cultures is studied theoretically with particular emphasis on stability and multiplicity characteristics. Bifurcation theory and stability analysis is applied to determine how the number and stability of steady-states depend on dilution rate and feed composition for Klebsiella pneumoniae feeding on mixed glucose–xylose substrates. The biological system is analyzed using a cybernetic modeling approach. We propose a numerical strategy which enables us to determine all possible steady-states for a given set of operating conditions. The algorithm exploits the polynomial nature of the equations. Results for a model without maintenance are compared to the predictions of a model with maintenance. It is shown that the model without maintenance effects and constitutive enzyme synthesis rate admits spurious steady-state solutions, which disappear when these effects are taken into account. In both cases multiple steady-states are observed over a range of dilution rates. In addition, the model with maintenance also predicts dynamic instability in the form of self-sustained oscillations. We have also investigated the effect of various parameters on the region of multiplicity to determine operating conditions which would enable experimental verification of our predictions.

The uniqueness of the stable positive solution for a class of superlinear indefinite reaction diffusion equations

In this paper we characterize the existence and prove the uniqueness of the stable positive steady-state for a general class of superlinear indefinite reaction diffusion equations in the absence of $L_\infty$ a priori bounds. More precisely, it will be shown that the model possesses a linearly stable positive steady-state if, and only if, the trivial solution is linearly unstable and the model possesses some positive steady-state. Moreover, it is unique if it exists. Actually, the minimal positive steady-state provides us with the unique linearly stable positive steady-state of the model. This is an extremely striking result since these problems can have an arbitrarily large number of positive steady-states as a result of having spatial inhomogeneities or varying the geometry of the support domain where the reaction takes place.

Dynamic modelling and steady-state multiplicity in high pressure multizone LDPE autoclaves

A comprehensive mathematical model is developed to simulate the ethylene polymerization in high-pressure autoclave reactors. Two generalised macromixing models, namely the external recycle and the backflow model are established to describe the complex mixing patterns occurring in multizone, multifeed low density polyethylene (LDPE) autoclaves. According to these models, each zone in the reactor is divided into a sequence of perfectly mixed vessels, called segments. To represent the kinetics of ethylene polymerization a general reaction mechanism is considered and the method of moments is employed to calculate the molecular weight developments. A new technique the so-called ‘numerical fractionation’ is applied to predict the formation of gel and the number chain length distribution in LDPE autoclaves. Finally, a detailed stability analysis is carried out for an LDPE autoclave to identify the number of multiple steady-states.

Reverse engineering: a model for T-cell vaccination.

A class of minimal models is constructed that can exhibit several salient phenomena associated with T-cell inoculations that prevent and cure autoimmune disease. The models consist of differential equations for the magnitude of two populations, the effectors E (which cause the disease), and an interacting regulator population R. In these models, normality, vaccination and disease are identified with stable steady-states of the differential equations. Thereby accommodated by the models are a variety of findings such as the induction of vaccination or disease, depending on the size of the effector inoculant. Features such as spontaneous acquisition of disease and spontaneous cure require that the models be expanded to permit slow variation of their coefficients and hence slow shifts in the number of steady-states. Other extensions of the basic models permit them to be relevant to vaccination by killed cells or by antigen, or to the interaction of a larger number of cell types. The discussion includes an indication of how the highly simplified approach taken here can serve as a first step in a modeling program that takes increasing cognizance of relevant aspects of known immunological physiology. Even at its present stage, the theory leads to several suggestions for experiments.