Thermodynamics of Open Systems.

The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle's invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when (4.2) fails to hold and the unique positive steady state $E^*$ becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of $E^*$, albeit with fluctuating population sizes.

Absorption with exothermic reaction in a falling film column

In this article, we propose and analyze an infectious disease model with reinfection and investigate disease dynamics by incorporating saturated treatment and information effect. In the model, we consider the case where an individual’s immunity deteriorates and they become infected again after recovering. According to our findings, multiple steady states and backward bifurcation may occur as a result of treatment saturation. Further, if treatment is available for all, the disease will be eradicated provided {mathcal {R}}_0<1; however, because limited medical resources caused saturation in treatment, the disease may persist even if {mathcal {R}}_0<1. The global stability of the unique endemic steady state is established using a geometric approach. We also establish certain conditions on the transmission rate for the occurrence of periodic oscillations in the model system. Among nonlinear dynamics, we show supercritical Hopf bifurcation, bi-stability, backward Hopf bifurcation, and double Hopf bifurcation. To illustrate and validate our theoretical results, we present numerical examples. We found that when disease information coverage is high, infection cases fall considerably, and the disease persists when the reinfection rate is high. We then extend our model by incorporating two time-dependent controls, namely inhibitory interventions and treatment. Using Pontryagin’s maximum principle, we prove the existence of optimal control paths and find the optimal pair of controls. According to our numerical simulations, the second control is less effective than the first. Furthermore, while implementing a single intervention at a time may be effective, combining both interventions is most effective in reducing disease burden and cost.

Kinetics of the reaction between and to form and in a well‐mixed flow reactor at feed pressures from 0.01 to 0.5 Torr have been studied by monitoring reaction rates with a differentially pumped mass spectrometer to monitor reactant and product partial pressures. A resistively heated tungsten foil was used as the deposition substrate to provide surface temperatures up to 1500°C. This technique provides rapid measurement of gas composition in a well‐mixed flow reactor so that reaction steady states and transients could be examined. Measured film deposition rates agreed well with those determined by monitoring reaction products. Film quality was also monitored by scanning electron microscopy, x‐ray photoelectron spectroscopy, scanning Auger microprobe, and energy dispersive x‐ray analysis. Rates were found to vary from first order to negative order with respect to both and . A simple bimolecular Langmuir‐Hinshelwood rate expression, , was found to describe the steady‐state kinetics remarkably well. This equation suggests reactants and/or intermediates are competitively adsorbed. Temperature dependences of these parameters yielded heats of adsorption of reactants and reaction rate parameters. Under some conditions multiple steady states were observed vs. under nearly isothermal conditions and also vs. surface temperature for fixed reactant pressures. The bifurcation points were found to be analogous to ignition/extinction points observed in combustion systems although these deposition reactions are endothermic. The multiplicity behavior was found to be also consistent with the Langmuir‐Hinshelwood model. However, we occasionally observe as many as three stable steady states at sufficiently high pressures, with only a single steady state at lower reactant pressures.

In this study, we develop a model of overlapping generations where adults make decisions on consumption, fertility, and their personal education. We show that under the assumption of exogenous mortality, there are multiple steady states with club convergence occurring when mortality is sufficiently high. If mortality is sufficiently low, there will be a unique, stable steady state, and the economy will converge to a “good” steady state irrespective of where it starts from. Under the assumption of endogenous mortality with “threshold effects,” we find that club convergence will occur if the threshold is sufficiently high; conversely, a low threshold can help the economy to steer clear of the underdevelopment trap.

Conforming contact is found on many important areas: materials with cracks, inclusions, layered structures, pinned geometries, etc. Coulomb friction is non-associative i.e. does not satisfy Drucker’s postulates for stability in plasticity theory, and this does not permit to use the classical Melan and Koiter theorems for cyclic loading which however remain obvious necessary conditions. On the other hand, the Coulomb friction cone has a peculiarity, its linear self-similarity, and this permits few general results, quite different but also powerful. For example, we show that for conforming contact, the Bree interaction diagram is open and sectors-shaped, and therefore there is no lower and upper bounds as in plasticity, since there is linear dependence on a load parameter \(\lambda\), and not a limit on \(\lambda\). As there is dependence on initial conditions in general, the Bree diagram doesn’t have the same profound meaning as in plasticity, since multiple steady states are possible. However, there is a complete scaling of the possible states (elastic shakedown, cyclic slip, ratchetting) with load factor \(\lambda\), and hence a worst-scenario can be defined. However, in order to develop residual stresses (which generally will take many cycles of steady cyclic loading), the load cannot go to zero, at least for friction coefficient below the critical value for "wedging". More in general, there needs to be a "permanent stick region" in the steady state.Almost conforming contacts, and in particular, in the case of initial interference, loose the independence on \(\lambda\) at low \(\lambda\) only, but "gain" greater possibilities to develop residual stresses. However, their study can be greatly simplified by developing a master curve approach to study the dependence on \(\lambda\). The theorems can also be considered the extension to cyclic loadings of the Dundurs results for conforming contacts under monotonic loading, in which case the contact area doesn’t vary in time and hence the problem is linear (from which the denomination "receding contacts"). However, here the problems remain fully non-linear, and the contact are not in general receding, nor there is a unique steady state, not even in the case of proportional loading. In the latter case it is nevertheless possible to derive some special results.

Simple mnemonic models named as ‘Thermodynamic Circle-1’ and ‘Thermodynamic Circle-2’ have been reported earlier by the author to represent the Maxwell relations of closed and open thermodynamic systems graphically so that these relations for simple compressible systems can be easily recalled. As these models do not involve any complicated rules for determining the algebraic signs in the relationships, recalling the network of Maxwell relations of closed and open thermodynamic systems is made easier. These two models can also be of much help to the students in recalling the network of Gibbs and other important thermodynamic relations of closed and open systems. This paper illustrates the systematic ways by which Gibbs and other thermodynamic relations of closed and open systems can be deduced directly from the models devised.

The dynamics of risk-sensitive allocations

Trejos–Wright with a 2-unit bound: Existence and stability of monetary steady states

This paper investigated high cell density cultivation of Haematococcus pluvialis for astaxanthin production in 3.7-L bioreactors. A biomass concentration of 2.74 g L-1and an astaxanthin yield of 64.4 mg L-1 were obtained. Based on the experimental results, a new and simple dynamic model is proposed, differing from Monod kinetics, to describe cell growth, product formation and substrate consumption. Good agreement was found between the model predictions and experimental data. The model revealed that there was cell growth inhibition on product formation and product feedback compensation for substrate consumption, but no substrate inhibition or product inhibition of cell growth. Stability analysis demonstrated that no multiplicity of steady states was observed; the unique positive steady state was locally asymptotically stable; and the effect of dilution rate on steady states was greater than that of the initial substrate concentration.

A system comprising two conjugated feedback loops was studied with the classical continuous formalism using differential equations and with a discrete formalism using logical equations. Although the logical equations describing the system are very simple, the graph of the sequences of states allowed by this logical system is quite complex. Depending on the initial state and values of the time delays, the system will proceed to a stable state or it will follow a stable cycle, which can display various profiles depending on the time delays. The main interest of this work resides probably in the complementarity of the continuous and logical approaches. As regards the continuous approach, the relative complexity of the system lead us to limit the analytical treatment to cases which permitted simplifications, but turned out to be rather uninteresting. In view of this situation, we started from the qualitative results of the logical analysis, and checked to what extent they could be extrapolated to the continuous system. This approach was extremely rewarding. In particular, it seems clear that when the logical analysis concludes : "depending on the initial state and values of the time delays, the system has a choice between a stable state and a stable cycle" it means that in the corresponding continuous system "there exist values of the parameters for which the system has two attractors, of which one is a stable steady state and the other is either a second stable steady state or a limit cycle depending on the values of parameters".

The spatial, temporal, and spatiotemporal dynamics of a reaction–diffusion predator–prey system with ratio-dependent Holling type III functional response, under homogeneous Neumann boundary conditions, are studied in this paper. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equation is presented. For the reaction–diffusion model, firstly the parameter regions for the stability or instability of the unique constant steady state are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next, it is proved that the model exhibits Hopf bifurcation, which produces temporal inhomogeneous patterns. Finally, the existence and nonexistence of nonconstant steady- state solutions are established by bifurcation method and energy method, respectively. Numerical simulations are presented to verify and illustrate the theoretical results.

Networks of phase oscillators are studied in various contexts, in particular, in the modeling of the electric power grid. A functional grid corresponds to a stable steady state such that any bifurcation can have catastrophic consequences up to a blackout. Also, the existence of multiple steady states is undesirable as it can lead to transitions or circulatory flows. Despite the high practical importance there is still no general theory of the existence and uniqueness of steady states in such systems. Analytic results are mostly limited to grids without Ohmic losses. In this article, we introduce a method to systematically construct the solutions of the real power load-flow equations in the presence of Ohmic losses and explicitly compute them for tree and ring networks. We investigate different mechanisms leading to multistability and discuss the impact of Ohmic losses on the existence of solutions.

We present a deep neural network which predicts the stability of isotropic steady states of the asymptotically flat, spherically symmetric Einstein–Vlasov system in Schwarzschild coordinates. The network takes as input the energy profile and the redshift of the steady state. Its architecture consists of a U-Net with a dense bridge. The network was trained on more than ten thousand steady states using an active learning scheme and has high accuracy on test data. As first applications, we analyze the validity of physical hypotheses regarding the stability of the steady states.

Phase resetting and bifurcation in the ventricular myocardium