Trivial Equilibrium Research Articles

Equilibrium adjustment of disequilibrium prices

We consider an exchange economy in which price rigidities are present. An always converging price and quantity adjustment process for such an economy is presented that is based on a discrete algorithmic procedure rather than on more traditional adjustment processes, which are based on difference or differential equations. In the short run, all non-numeraire commodities have a flexible price level with respect to the numeraire commodity but their relative prices are mutually fixed. In the long run, prices are assumed to be completely flexible. The adjustment process starts with a trivial equilibrium with a low enough price level and complete demand rationing on all markets. Along the path followed by the adjustment process, initially all relative prices of the non-numeraire commodities are kept fixed and the price level is increased. Rationing schemes are adjusted to keep markets in equilibrium. In doing so, the process reaches a short-run equilibrium with only demand rationing and no rationing on the numeraire and at least one of the other commodities. In the long run, the process allows for a downward price adjustment of unrationed non-numeraire commodities and eventually reaches a Walrasian equilibrium.

An analysis of the transmission of chlamydia in a closed population

In this paper, two models, based on the idea of urn models, for the spread of chlamydia in a closed population are presented. Difference equations are derived and analyzed for the expected number of infections in the population. The stability analysis of the difference equations is accomplished using techniques based on Lyapunov methods. There exist two equilibria, an endemic equilibrium and the trivial equibrium (infection-free). To reduce the level of infection, so that trivial equilibrium is approached, the basic reproductive rateR 0, must be reduced to a level less than unity.

On the Global Structure of the Set of Positive Solutions of Some Semilinear Elliptic Boundary Value Problems

In this work we analyze the structure of the set of positive solutions of a class of semilinear boundary value problems. It is shown that the global continuum of positive solutions emanating from the trivial equilibrium at the principal eigenvalue of the linearization is constituted by a regular curve if the slope of the kinetic at the trivial solution is large enough and Ω is convex. The same result holds if the support region of the species is a bounded simply connected domain of R2 close to a convex domain, in a sense to be precised later. To prove these results we have to find out the exact width of the boundary layer of a singular perturbation problem. The results about the singular perturbation problem are new and of great interest by themselves.

Nonlinear three-dimensional oscillations of elastically constrained fluid conveying viscoelastic tubes with perfect and broken O(2)-symmetry

The loss of the stability of the trivial downhanging equilibrium position of a slender circular tube conveying incompressible fluid flow is studied. The tube is clamped at its upper end and is free at its lower end. Inbetween, the three-dimensional transversal motion is constrained by an elastic support considered to be rotationally symmetric. Tube equations valid for large displacement but small strain based on Kirchhoff's rod theory and the Kelvin-Voigt viscoelastic law are used.

THE NET REPRODUCTIVE VALUE AND STABILITY IN MATRIX POPULATION MODELS

The net reproductive value n is defined for a general discrete linear population model with a non‐negative projection matrix. This number is shown to have the biological interpretation of the expected number of offspring per individual over its life time. The main result relates n to the population's growth rate (i.e. the dominant eigenvalue λ of the projection matrix) and shows that the stability of the extinction state (the trivial equilibrium) can be determined by whether n is less than or greater than 1. Examples are given to show that explicit algebraic formulas for n are often derivable, and hence available for both numerical and parameter studies of stability, when no such formulas for λ are available.

A model for plane elastica with simple shear deformation pattern

Several existing theories of elastic rods that take into account shear effects are classified. Then, a new model for the influence of shearing force, based on simple shear of finite amount, is proposed. The properties of the model are examined on the stability problem for a heavy elastic rod. It is shown that the rod may exhibit sub- and super-critical bifurcation patterns at the trivial equilibrium configuration. A comparison is made between the results for the critical load of the rod predicted by the new and the three different existing models.

A QUALITATIVE ANALYSIS FOR THE LOCAL AND GLOBAL DYNAMIC BUCKLING AND STABILITY OF AUTONOMOUS DISCRETE SYSTEMS

A general analytic approach is presented for the local and global dynamic buckling and stability of discrete dissipative systems described by the most general form of autonomous ordinary differential equations. Attention is focused on nonlinear dynamical gradient systems which under static loading exhibit all types of simple branching points emanating from a trivial fundamental equilibrium path. The relationship between the static stability and the corresponding dynamic stability associated with the Jacobian eigenvalue is thoroughly discussed

Stochastic stability of linear viscoelastic systems

The stochastic almost-sure stability of a single degree-of-freedom linear viscoelastic system subjected to random fluctuation in the stiffness parameter is investigated. For small damping and weak random fluctuation, asymptotic expressions are derived for the Lyapunov exponent and the rotation number using the method of stochastic averaging. From the sign of the Lyapunov exponent, the condition for asymptotic stability with probability 1 of the trivial equilibrium state is obtained.

Bifurcation and stability analysis of a rotating beam

We discuss small oscillations of an elastic beam clamped radially to the interior of a rotating ring. It is well known that if the speed of rotation is sufficiently high, the trivial equilibrium of the beam may lose stability and the beam buckles.

Non-linear vibration of a traveling tensioned beam

Free non-linear vibration of an axially moving, elastic, tensioned beam is analyzed over the sub- and supercritical transport speed ranges. The pattern of equilibria is analogous to that of Euler column buckling and consists of the straight configuration and of non-trivial solutions that bifurcate with speed. The governing equations for finite local motion about the trivial equilibrium and for motion about each bifurcated solution are cast in the standard form of continuous gyroscopic systems. A perturbation theory for the near-modal free vibration of a general gyroscopic system with weakly non-linear stiffness and/or dissipation is derived through the asymptotic method of Krylov, Bogoliubov, and Mitropolsky. The method is subsequently specialized to non-linear vibration of a traveling beam, and of a traveling string in the limit of vanishing flexural rigidity. The contribution of non-linear stiffness to the response increases with subcritical speed, grows most rapidly near the critical speed, and can be several times greater for a translating beam than for one that is not translating. In the supercritical speed range, asymmetry of the non-linear stiffness distribution biases finite-amplitude vibration toward the straight configuration and lowers the effective modal stiffness. The linear vibration theory underestimates stability in the subcritical range, overestimates it for supercritical speeds, and is most limited in the near-critical regime.

Reversible systems. Stability at 1:1 resonance

Some general properties of reversible systems are studied: the nature of the stability of the trivial equilibrium position, the conditions for the existence of certain periodic solutions and symmetry of the phase portrait. It is shown that a discrete automorphism (symmetry) group generates integral manifolds. A detailed investigation of the stability of the trivial solution at 1:1 resonance is presented. The necessary and sufficient conditions for the stability of a model system are obtained and it is shown that instability of the system implies instability of the complete system.

Dynamic Stability Analysis of High Speed Axially Moving Bands With End Curvatures

Measurements of vibration on continuous bands driven by rotating wheels show that the end curvature of the spans can greatly affect the system dynamic characteristics [1,2]. Significant error in the predicted response of a span can occur if the end curvature is neglected. To understand the global dynamic behavior of such systems under high speed operating conditions, a nonlinear model of axially moving bands with finite end curvatures is developed and analyzed in this paper. It is shown that a pitch-fork bifurcation from the trivial equilibrium state will occur at the critical speed of a perfectly straight, axially moving band and the original trivial equilibrium configuration will become unstable. However, for bands with finite end curvatures, the classical critical speed theory does not apply. The equilibrium configuration will continuously change with increasing band speed, but will remain stable. Multiple equilibrium states will occur at sufficiently high band speed, and the minimum speed for this phenomenon to happen increases with the end curvature of the band. Large oscillation coupling the multiple equilibrium states will be induced by large enough initial disturbance. The disturbance required for this to occur varies with the band end curvature and transport speed. The study provides us with new insight and guidelines toward optimizing this class of high speed mechanical systems.

A note on reactive equilibria in the discounted prisoner's dilemma and associated games

Necessary and sufficient conditions are derived under which the Tit-for-Tat strategy pair is a subgame perfect equilibrium in the standard discounted prisoner's dilemma. The sensi- tivity of the equilibrium to changes in the discount parameters is shown to be a common feature of all such nontrivial "reactive" equilibria. This appears to rule them out as reasonable solutions to the discounted prisoner's dilemma. In remarks, we indicate how these results can be extended to a fairly general class of games. The purpose of this note is to investigate the subgame perfection (see Selten (7)) properties of certain pure strategy equilibria in infinitely repeated versions of the classic prisoner's dilemma and related classes of games. Concentrating on the prisoner's dilemma for the moment, the equilibria we consider consist of strategy pairs in which at least one of the players bases current period action only on the prior actions of his opponent, ignoring the history of his own actions in the supergame. A strategy of this type is called reactive for obvious reasons. The corresponding equilibria will also be called reactive. In the example below, we demonstrate the existence of a perfect reactive equilibrium in which the collusive outcome obtains along the equilibrium path. This equilibrium is seen to be quite special, however; being extremely sensitive to variations in the discount rates of the players. We show that this is a feature com- mon to all such nontrivial reactive equilibria. In particular, the class of perfect reactive equilibria which are robust to changes in the discount rates consists of the single trivial equilibrium in which both players choose their noncooperative action in every

Global stability of single-species diffusion Volterra models with continuous time delays.

In this paper we consider the stability property of single-species patches connected by diffusion with a within-patch dynamics of Volterra type and with continuous time delays. We prove that this system can only have two kinds of equilibria: the positive and the trivial one. By the assumption that the delay kernels are convex combinations of suitable non-negative and normalized functions, the linear chain trick gives an expanded system of O.D.E. with the same stability properties as the original integro-differential system. Homotopy function techniques provide sufficient conditions for the existence of the positive equilibrium and for its global stability. We also prove the local stability of any positive equilibrium and the local instability both of positive and trivial equilibria. The biological meanings of the results obtained are compared with known results from the literature.

A reaction-diffusion system of a competitor-competitor-mutualist model

We investigate the homogeneous Dirichlet problem and Neumann problem to a reaction-diffusion system of a competitor-competitor-mutualist model. The existence, uniqueness, and boundedness of the solutions are established by means of the comparison principle and the monotonicity method. For the Dirichlet problem, we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities. For the Neumann problem, we analyze the contant equilibrium solutions and their stabilities. The main method used in studying of the stabilities is the spectral analysis to the linearized operators. The O.D.E. problem to the same model was proposed and studied by B. Rai, H. I. Freedman, and J. F. Addicott ( Math. Biosci. 65 (1983), 13–50).

A reaction-diffusion system of a predator-prey-mutualist model

Mutualism is part of many significant processes in nature. Mutualistic benefits arising from modification of predator-prey interactions involve interactions of at least three species. In this paper we investigate the Homogeneous Neumann problem and Dirichlet problem for a reaction-diffusion system of three species—a predator, a mutualist-prey, and a mutualist. The existence, uniqueness, and boundedness of the solution are established by means of the comparison principle and the monotonicity method. For the Neumann problem, we analyze the constant equilibrium solutions and their stability. For the Dirichlet problem, we prove the global asymptotic stability of the trivial equilibrium solution. Specifically, we study the existence and the asymptotic behavior of two nonconstant equilibrium solutions. The main method used in studying of the stability is the spectral analysis to the linearized operators. The O.D.E. problem for the same model was proposed and studied in [13]. Through our results, we can see the influences of the diffusion mechanism and the different boundary value conditions upon the asymptotic behavior of the populations.

Numerical study of thermal equilibrium and a catastrophe model for major disruptions in tokamaks

The thermal equilibrium of tokamak plasmas is studied by solving the electron and ion energy balance equations numerically. Three basic types of thermal equilibria, depending on different boundary temperatures, are found. The presence of a fold or cusp catastrophe structure of the equilibrium curves for temperature versus density in the case of low-boundary temperature indicates the existence of a density limit beyond which no equilibrium or only a trivial equilibrium exists. Major disruptions are considered to occur because of the loss of thermal equilibrium of the tokamak plasma. The study shows that major disruptions can be parametrically controlled. Thermal equilibria with higher boundary temperature display a monotonic structure of the temperature versus density curve, indicating good confinement characteristics and hence the absence of disruptions.

Anomalous vibrational states of a monatomic surface film

Vibrational states of a one-dimentional chain of atoms which models peculiarities of a monatomic surface film are considered. A set of nontrivial equilibrium states of the system is determined in whose vicinity almost linear band vibrations are possible. Peculiarities of anomalous branches of vibrational states are discussed. It is shown that the low-frequency gap of vibrational states disappears under certain conditions. This determines the possibility of appearance of low-frequency sound vibrations which are not possible for a trivial equilibrium distribution of atoms in the film.

Asymptotic stability results for system of quasilinear parabolic equations

A system of quasilinear parabolic equations arising in the theory of nuclear reactors is studied. After characterizing the appearance of a nontrivial equilibrium solution, its asymptotic stability is proved. It is also shown that, when the nontrivial solution is absent, the trivial equilibrium solution is asymptotically stable.

Zufallspaarung mit teilweiser Selbstung unter verschiedenen Modellen von häufigkeitsabhängiger Selektion /Mixed Random Mating and Selfing with Different Models of Frequency-Dependent Selection

Abstract Given a large population with mixed random mating and selfing (one locus-two alleles) different models of frequency-dependent selection were discussed - including a simple biometrical model for considering and analysing the competitive effects between neighbouring individuals in plant populations. For each model there were studied: changes in gene frequencies, population genetic equilibria, times until reaching these equilibria etc. - in dependence of the different parameters used: composition of the initial population, probability of selfing, selection-coefficients, competition-parameters. Apart from only few differing results it follows from the studies performed in these investigations, that the different composition of the initial population is of no particular importance as well for the gene frequencies p̂ at equilibrium as for the time t̂ until reaching these equilibria. This result is especially right for p̂. Different probabilities of selfing and different degrees of dominance in the selection coefficients are indeed of some influence on the existence and location of the population genetic equilibria, but here too we find an disproportionately stronger dependence with the time t̂ until reaching the equilibrium than with the gene frequency p̂ at equilibrium . The special importance of overdominance for the maintenance of genetic polymorphisms, which is well known in the case of non­ frequency-dependent selection (see: model 1 of the present studies) turn out to be of some other meaning in the models of frequency-dependent selection, which were analysed in the present paper: Depart from only few special situations (model 2 ; complete self-fertilization in models 5 and 6 ; extremely high probabilities of selfing in model 7) nontrivial equilibria are reached for all degrees of dominance. Therefore, the special importance of overdominance mentioned above, not proves right in the case of frequency-dependent selection. The investigations of the present paper have shown, that existence and location of the non­ trivial population genetic equilibria are determined not so much by degree of dominance and probability of selfing, but the equilibria are mainly determined by the model of the investigation used in the concerning studies. In the case of frequency-dependent selection, therefore, the explicit form of the fitness values as functions of the frequencies plays the decisive role in maintaining genetic polymorphisms.