Conditions For Existence Of Equilibrium Research Articles

Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations.

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

Comment on “Salesforce Compensation with Inventory Considerations”

This note critically analyzes the conditions for equilibrium existence in a model involving incentive schemes and inventory management. It identifies a technical flaw where equilibrium is asserted to exist under certain parameter values when, in fact, it does not. However, by introducing a reasonable assumption that inventory can only be ordered in sufficiently large increments, the qualitative outcomes and key insights remain intact. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2023.03914 .

Dynamic path for green product development: A game-theoretic analysis of product-process innovation with supply chain collaboration

Two contrasting paths are embraced within the practice of green product development: The first involves market entry with high-quality products in the initial phase, followed by a gradual release of mass-market products at reduced prices. The second entails entering the market with low-quality products, followed by step-by-step improvements in quality. The decision-making on path selection is divided into two stages: in the first stage, firms choose green product technologies, and in the second stage, they determine the path for green product development based on the selected technologies. Our research findings highlight the crucial role of efficiency-cost ratios in product-process innovation for green product development. Equilibriums are only achievable when efficiency-cost ratios are relatively low. Given the singular impact of component cost on profit, the path for process innovation consistently trends downward, whereas the path for product innovation can either descend or ascend. Collaboration positively impacts the conditions for equilibrium existence, making it more attainable. When the manufacturer’s efficiency-cost ratio for product innovation is low, collaboration can transition the path from a descending trend to an ascending one, and vice versa. The selection of appropriate technological parameters is important for achieving a win–win relationship between the environment and profitability. Additionally, we have expanded our models to include competition and marketing scenarios. We find that competition primarily influences green product development by altering the motivation for path selection. Marketing and product-process innovation demonstrate a complementary relationship.

Dynamic Analysis of Progressive Income Taxation and Economic Growth with Endogenous Labor Supply and Public Goods

This research advances the development of a sophisticated economic growth model that intricately integrates gender-differentiated labor, endogenous labor supply, and nonlinear progressive income taxation, with the latter's revenues earmarked for public goods provision. Anchored in a framework that encompasses both a production and a public sector financed through taxation, the model delves into the complex interplay among economic growth, the gender division of labor, and progressive income taxation. By employing rigorous simulation techniques, this study meticulously examines the conditions for equilibrium existence and investigates the system's dynamics in response to various parametric alterations. This exploration not only sheds light on the theoretical underpinnings of economic growth and taxation but also offers insights into the policy implications of gender-differentiated labor contributions within the economy.

Majority choice of taxation and redistribution in a federation

To study redistribution and taxation in a federation, we provide a model with a central government and multiple local governments, the former with power to levy an income tax for redistribution, and the latter choosing a local income tax, property tax, lump-sum tax or subsidy, and a local public good. Policy is set by majority choice at each tier of government by households that differ by income and by ability to move among local jurisdictions. We provide sufficient conditions for existence of equilibrium and examine its properties. Central findings are federal income distribution, little local redistribution, and local preference for property taxation over income taxation to fund local public goods.

Informative advertising in monopolistically competitive markets

Firms spend a half-trillion dollars advertising each year. To model and examine the welfare effects of advertising with heterogeneous goods Grossman and Shapiro (1984) model informative advertising in monopolistic competition find that informative advertising is socially excessive for large numbers of sellers. However, it has been noted that their equilibrium may not exist. We present a tractable model of informative advertising in monopolistic competition replacing the standard assumption of a finite number of firms with a continuum of firms. We derive conditions for equilibrium existence and find that in the monopolistically-competitive model, informative advertising is socially insufficient. We also find that with free entry, the measure of the set of active firms is lower than the socially optimal one. The comparison of our results with the results of Grossman and Shapiro (1984) also shows that a pure-strategy, symmetric equilibrium typically does not exist in the latter model if the number of sellers is large, a fact which accounts for the different conclusions drawn in the two papers.

Existence of Gibbs States and Maximizing Measures on a General One-Dimensional Lattice System with Markovian Structure

Consider a compact metric space $(M, d_M)$ and $X = M^{\mathbb{N}}$. We prove a Ruelle's Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in [Bull. Braz. Math. Soc. 45 (2014), pp. 53-72] which are defined from a continuous function $A : M \times M \to \mathbb{R}$ that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $S^1$. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.

Dynamical analysis of fractional-order Holling type-II food chain model.

This paper proposed a fractional-order Holling type-II food chain model. First, we verified the existence, uniqueness, nonnegativity and boundedness of the solution of the model, and some conditions for equilibrium existence and local stability were studied. Second, a controller was proposed, and the Lyapunov method was used to study the global stability of the positive equilibrium point. Finally, numerical simulations were performed to verify the theoretical results.

Phase Transitions in Two-Phase Media with the Same Moduli of Elasticity

For a variational problem of the theory of phase transitions in continuum mechanics with the same moduli of elasticity we obtain explicit formulas for the phase transition temperatures and equilibrium energy. The existence of equilibrium states is studied in some particular cases. Bibliography: 15 titles.

Distributive politics with other‐regarding preferences

AbstractThis paper analyzes a nonsmooth model of probabilistic voting with two parties and a broad family of other‐regarding behavior, including fairness and quasi‐maximin preferences, income‐dependent altruism, and inequity aversion. The paper provides conditions for equilibrium existence and uniqueness. It also characterizes the Nash equilibrium in pure strategies when parties hold either symmetric payoffs, or minor forms of asymmetries. The characterization shows that the two parties converge to an equilibrium policy that maximizes a mixture of a “self‐regarding utilitarian” social welfare function and an aggregate of society's other‐regarding preferences. These results are shown to be applicable to other nonsmooth frameworks, such as probabilistic voting with loss averse voters. The characterization also shows that the direction and the size of the inefficiencies emerging from electoral competition depend in a subtle way on the nature of the other‐regarding preferences (and resp., loss aversion).

Molecular dynamics simulations of the colloidal interaction between smectite clay nanoparticles in liquid water

Colloidal interactions between clay nanoparticles have been studied extensively because of their strong influence on the hydrology and mechanics of many soils and sedimentary media. The predominant theory used to describe these interactions is the Derjaguin-Landau-Verwey-Overbeek (DLVO) model, a framework widely applied in colloidal and interfacial science that accurately predicts the interactions between charged surfaces across water films at distances greater than ~ 3 nm (i.e., ten water monolayers). Unfortunately, the DLVO model is inaccurate at the shorter interparticle distances that predominate in most subsurface environments. For example, it inherently cannot predict the existence of equilibrium states wherein clay particles adopt interparticle distances equal to the thickness of one, two, or three water monolayers. Molecular dynamics (MD) simulations have the potential to provide detailed information on the free energy of interaction between clay nanoparticles; however, they have only been used to examine clay swelling and aggregation at interparticle distances below 1 nm. We present the first MD simulation predictions of the free energy of interaction of smectite clay nanoparticles in the entire range of interparticle distances from the large interparticle distances where the DLVO model is accurate (>3 nm) to the short-range swelling states where non-DLVO interactions predominate (<1 nm). Our simulations examine a range of salinities (0.0 to 1.0 M NaCl) and counterion types (Na, K, Ca) and establish a detailed picture of the breakdown of the DLVO model. In particular, they confirm previous theoretical suggestions of the existence of a strong non-DLVO attraction with a range of ~ 3 nm arising from specific ion-clay Coulomb interactions in the electrical double layer.

Behavior-Aware Queueing: The Finite-Buffer Many-Server Setting

Service system design is often informed by queueing theory. Traditional queueing theory assumes that servers work at constant speeds. That is reasonable in computer science and manufacturing contexts. However, servers in service systems are people, and, in contrast to machines, systemic incentives created by design decisions influence their work speeds. We study how server work speed is affected by managerial decisions concerning (i) how many servers to staff and (ii) whether and when to turn away customers, in the context of a finite-buffer many-server queue (an M/M/N/k queue) in which the work speeds emerge as the solution to a noncooperative game. We show that a symmetric equilibrium always exists in a loss system (N=k) and provide conditions for equilibrium existence in a single-server system (N=1). For the general M/M/N/k system, we provide a sufficient condition for the existence of a solution to the relevant first-order condition and bounds on such a solution; however, showing that it is an equilibrium is challenging due to the existence of multiple local maxima in the utility function. Nevertheless, in an asymptotic regime in which demand becomes large, the utility function becomes concave and the first-order condition simplifies to that for an infinite buffer (k=∞) system, allowing us to characterize stable prelimit equilibria.

Signaling in Dynamic Markets with Adverse Selection

We study trade in dynamic decentralized markets with adverse selection. Differently from the literature on the topic so far, we assume that the informed sellers make the offers, so that signaling through prices is possible. We establish basic properties of equilibria, provide necessary and sufficient conditions for equilibrium existence, and discuss the two-type case and separating equilibria in detail. Our main result is that market efficiency---measured by the maximum equilibrium welfare---is invariant to trading frictions. This shows that signaling and screening have markedly different implications for the relationship between market efficiency and trading frictions.

Global synchronization analysis of droop-controlled microgrids—A multivariable cell structure approach

The microgrid concept represents a promising approach to facilitate the large-scale integration of renewable energy sources. Motivated by this, the problem of global synchronization in droop-controlled microgrids with radial topology is considered. To this end, at first a necessary and sufficient condition for existence of equilibria is established in terms of the droop gains and the network parameters. Then, the local stability properties of the equilibria are characterized. Subsequently, sufficient conditions for almost global synchronization are derived by means of the multivariable cell structure approach recently proposed in Efimov and Schiffer (2019). The latter is an extension of the powerful cell structure principle developed by Leonov and Noldus to nonlinear systems that are periodic with respect to several state variables and possess multiple invariant solutions. The analysis is illustrated via numerical examples.

Applying Graph Theory to Some Problems of Economic Dynamics

This paper studies dynamic models of production and exchange on graph with consideration of transportation costs. Using graph and set of matrices, we introduce superlinear multivalued mappings which describe the exchange ratio in considered system. Effective trajectories of these models are studied. It is shown that trajectories can be constructed using the simplest equilibrium type mechanisms. Characteristics of effective trajectories in Neumann type models are given. Conditions for the existence of equilibrium state of the considered model are found.

Experimental and Theoretical Study on Dynamic Hydraulic Fracture

Hydraulic fracturing is vital in the stimulation of oil and gas reservoirs, whereas the dynamic process during hydraulic fracturing is still unclear due to the difficulty in capturing the behavior of both fluid and fracture in the transient process. For the first time, the direct observations and theoretical analyses of the relationship between the crack tip and the fluid front in a dynamic hydraulic fracture are presented. A laboratory-scale hydraulic fracturing device is built. The momentum-balance equation of the fracturing fluid is established and numerically solved. The theoretical predictions conform well to the directly observed relationship between the crack tip and the fluid front. The kinetic energy of the fluid occupies over half of the total input energy. Using dimensionless analyses, the existence of equilibrium state of the driving fluid in this dynamic system is theoretically established and experimentally verified. The dimensionless separation criterion of the crack tip and the fluid front in the dynamic situation is established and conforms well to the experimental data. The dynamic analyses show that the separation of crack tip and fluid front is dominated by the crack profile and the equilibrium fluid velocity. This study provides a better understanding of the dynamic hydraulic fracture.

Pesin’s entropy formula for $$C^1$$ C 1 non-uniformly expanding maps

We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric spaces and continuous potentials. Moreover, we formulate a C$^1$ generalization of Pesin's Entropy Formula: all ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula for $C^1$ non-uniformly expanding maps. We show that for weak-expanding maps such that $\Leb$-a.e $x$ has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin's Entropy Formula and are equilibrium states for the potential $\psi=-\log|\det Df|$. In particular, this holds for any $C^1$-expanding map and in this case the set of invariant probability measures that satisfy Pesin's Entropy Formula is the weak$^*$-closed convex hull of the ergodic weak-SRB-like measures.

Non Stationary Markov Perfect Equilibria in Mean-Field Games

In this paper, we consider both finite and infinite horizon discounted dynamic mean-field games where there is a large population of homogeneous players sequentially making strategic decisions and each player is affected by other players through an aggregate population state. Each player has a private type that only she observes. Such games have been studied in the literature under simplifying assumption that population state dynamics are stationary. In this paper, we consider non-stationary population state dynamics and present a novel backward recursive algorithm to compute Markov perfect equilibrium (MPE) that depend on both, a player's private type, and current (dynamic) population state. We present sufficient conditions for existence of equilibria. Using this algorithm, we study a security problem in cyber-physical system where infected nodes put negative externality on the system, and each node makes a decision to get vaccinated. We numerically compute MPE of the game.

Analysis of HIV Transmission of Commercial Sex Wokers and Their Clients with Condom Use Treatment

A model of the HIV/AIDS epidemic among sex workers and their clients is discussed to study the effects of condom use in the prevention of HIV transmission. The model is addressed to determine the existence of equilibrium states, and then analyze the global stability of disease free and endemic equilibrium states. The global stability of equilibria depends on the vales of the basic reproduction ratio derived from the next generation matrix of the model. The endemic equilibrium state is globally stable when the ratio exceeds unity. The simulation results are presented to discuss the effect of condom use treatment in preventing the spread of HIV/AIDS among sex workers and their clients. The results show that the effectiveness level in using condoms in sexual intercourse corresponds to the decreasing level of the spread of HIV/AIDS. We use Maple and Matlab software to simulate the impact of condom use.

Thermodynamic formalism for topological Markov chains on standard Borel spaces

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space $X\equiv E^{\mathbb{N}}$, where $E$ is a general Borel standard space. In particular, we introduce meaningful concepts of entropy and pressure for shifts acting on $X$ and obtain the existence of equilibrium states as additive probability measures for any bounded continuous potential. Furthermore, we establish convexity and other structural properties of the set of equilibrium states, prove a version of the Perron-Frobenius-Ruelle theorem under additional assumptions on the regularity of the potential and show that the Yosida-Hewitt decomposition of these equilibrium states do not have a purely additive part. We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite dimensional diffusion.