Unique Equilibrium Solution Research Articles

Stationary patterns in a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis

Stationary patterns of a two-cell coupled isothermal chemical system with arbitrary powers of autocatalysis are considered. Firstly, we obtain the stability of the unique positive constant equilibrium solution for the system. Then, based on a priori estimates, non-existence and existence of nontrivial steady state solutions are shown by using implicit function theorem and topological degree theory, respectively. The effects of autocatalysis order and diffusion coefficients to the pattern formation are discussed.

A time since onset of injection model for hepatitis C spread amongst injecting drug users

Studies of hepatitis C virus (HCV) infection amongst injecting drug users (IDUs) have suggested that this population can be separated into two risk groups (naive and experienced) with different injecting risk behaviours. Understanding the differences between these two groups and how they interact could lead to a better allocation of prevention measures designed to reduce the burden of HCV in this population. In this paper we develop a deterministic, compartmental mathematical model for the spread of HCV in an IDU population that has been separated into two groups (naive and experienced) by time since onset of injection. We will first describe the model. After deriving the system of governing equations, we will examine the basic reproductive number R₀, the existence and uniqueness of equilibrium solutions and the global stability of the disease free equilibrium (DFE) solution. The model behaviour is determined by the basic reproductive number, with R₀ = 1 a critical threshold for endemic HCV prevalence. We will show that when R₀ ≤ 1, and HCV is initially present in the population, the system will tend towards the globally asymptotically stable DFE where HCV has been eliminated from the population. We also show that when R₀ > 1 there exists a unique non-zero equilibrium solution. Then we estimate the value of R₀ from epidemiological data for Glasgow and verify our theoretical results using simulations with realistic parameter values. The numerical results suggest that if R₀ > 1 and the disease is initially present then the system will tend to the unique endemic equilibrium.

Solution of the Hotelling’s game in secure strategies

We show that the price-setting subgame in the classic Hotelling’s model (1929) with the linear transport costs has the unique equilibrium solution for all location pairs under the assumption that duopolists secure themselves against being driven out of the market by undercutting. In contrast to the modified zero conjectural variation approach of Eaton and Lipsey (1978) we assume that the players consider the threat of ‘pressing out’ of the market as a real possibility. We employ the concept of equilibrium in secure strategies (EinSS) as the generalization of the Nash equilibrium. The EinSS coincides with the Nash Equilibrium when Nash Equilibrium exists and postulates the incentive of players to maximize their profit under the condition of security against the actions of other players.

Cross-diffusion induced Turing instability for a three species food chain model

In this paper, we study a strongly coupled reaction–diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. We first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ODE system. The positive equilibrium solution remains linearly stable for the reaction–diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. We further proved that the positive equilibrium solution is globally asymptotically stable for the reaction–diffusion system without cross-diffusion by constructing a Lyapunov function. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction–diffusion system, hence the instability is driven solely from the effect of cross-diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions.

The Newsvendor Problem and Price‐Only Contract When Bankruptcy Costs Exist

We study a supply chain of a supplier selling via a wholesale price contract to a financially constrained retailer who faces stochastic demand. The retailer might need to borrow money from a bank to execute his order. The bank offers a fairly priced loan for relevant risks. Failure of loan repayment leads to a costly bankruptcy (fixed administrative costs, costs proportional to sales, and a depressed collateral value). We identify the retailer's optimal order quantity as a function of the wholesale price and his total wealth (working capital and collateral). The analysis of the supplier's optimal wholesale price problem as a Stackelberg game, with the supplier the leader and the retailer the follower, leads to unique equilibrium solutions in wholesale price and order quantity, with the equilibrium order quantity smaller than the traditional newsvendor one. Furthermore, in the presence of the retailer's bankruptcy risks, increases in the retailer's wealth lead to increased supplier's wholesale prices, but without the retailer's bankruptcy risks the supplier's wholesale prices stay the same or decrease in retailer's wealth.

Managing a Noncooperative Supply Chain with Limited Capacity

We consider a two-stage serial supply chain with capacity limits, where each installation is operated by managers attempting to minimize their own costs. A multiple-period model is necessitated by the multiple stages, capacity limits, stochastic demand, and the explicit consideration of inventories. With appropriate salvage value functions, a Markov equilibrium policy is found. Intuitive profit dominance allows for existence of a unique equilibrium solution, which is shown to be a modified echelon base-stock policy. This equilibrium policy structure is sustained in the infinite horizon. A numerical study compares the behavior of the decentralized system with the first-best integrated capacitated system. The performance of this decentralized system relative to the integrated system across other parameters can be very good over a broad range of values. This implies that an acceptable system performance may be attained without the imposition of a contract or other coordinating mechanism, which themselves may encounter difficulties in implementation in the form of negotiation, execution, or enforcement of these agreements. We find instances where tighter capacities may actually enhance channel efficiency. We also examine the effect of capacity utilization on the system suboptimality.

ANALYSIS OF MANUFACTURING COST MISREPORTING BEHAVIOR OF MANUFACTURER UNDER CONSIGNMENT CONTRACT WITH REVENUE SHARING

The two-echelon supply chain under stochastic manufacturing demand has a single component supplier selling to a single manufacturer.The model with perpetual manufacturing demand and many replenishment opportunities is considered by analytic method. The problem can be modeled as a supplier-led Stackelberg dynamic game.Supply chain performances under consignment and non-consignment are considered respectively.The supplier, acting as a designer of supply chain,offers the manufacturer a backorder penalty cost share allocation,and then the manufacturer decides the optimal base stock level.In this paper the following conclusions are obtained.Under consignment,there is a unique equilibrium solution in the game,and the supply chain system is optimal.Under non-consignment,there is a unique equilibrium solution in the game,and the supply chain system is optimal.Supply chain performance under consignment is superior to that under non-consignment.

ON THE SUPERCRITICALITY OF THE FIRST HOPF BIFURCATION IN A DELAY BOUNDARY PROBLEM

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

Dynamics of solution for a class of delayed diffusive neural networks with mixed boundary conditions

In this paper, we investigate a class of cellular neural networks model with delays and diffusive terms. By using the method of upper and lower solutions, we obtain that if the neuronal output signal functions in system possess mixed quasimonotone property and the corresponding elliptic system has upper and lower solutions the model has a unique nonconstant equilibrium solution. Under some additional conditions we further obtain that the solution of the neural networks converges to this nonconstant equilibrium solution.

Spatiotemporal pattern formation and multiple bifurcations in a diffusive bimolecular model

In this paper, we have investigated a homogeneous reaction–diffusion bimolecular model with autocatalysis and saturation law subject to Neumann boundary conditions. We mainly consider Hopf bifurcations and steady state bifurcations which bifurcate from the unique constant positive equilibrium solution of the system. Our results suggest the existence of spatially non-homogeneous periodic orbits and non-constant positive steady state solutions, which implies the possibility of rich spatiotemporal patterns in this diffusive biomolecular system. Numerical examples are presented to support our theoretical analysis.

Stochastic Differential Game of Transboundary Pollution and International Environmental Policy

The purpose of this paper is to investigate a stochastic differential game of transboundary pollution and environmental policy. We considered two asymmetric countries. The production process results in the emission of pollutants that are added to the existing stock common to both countries. The pollution accumulation dynamics is affected by an additive shock, the mean and variance are known. In both countries, the government controls emission levels. Uncertainty is incorporated into the model through a shock. The stock of pollution evolves stochastically according to geometric Brownian motion. In order to maximize the expected net present value of social welfare, the two governments should set the emission levels. We formulate stochastic differential games that need to be solved by the two governments and derive unique Markov-perfect Nash and cooperative equilibrium solutions. Subsequently, we propose a mechanism to derive the payoff distribution procedures of subgame consistent solutions in cooperative stochastic differential games for the two governments, and are able to derive a subgame consistent solution based on the Nash bargaining axioms. In conclusion, we show how the governments agree to maximize the sum of their expected payoffs and divide the total cooperative payoff so that the Nash bargaining outcome is maintained at every instant of time. Moreover, we show conditions guaranteeing individual rationality.JFL Classification: F18, L13, Q58

Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand

Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution.

Successive duopoly under moral hazard: Will incentive contracts persist?

The central purpose of this paper is to examine the incentive contract as an equilibrium phenomenon. We analyse a model of vertical differentiation in which we deal with the strategic role of the competitor’s decisions in a successive duopoly. Is it better for a processor to offer an incentive contract to an upstream producer or the spot market? We determine the equilibrium of a game in which the processors simultaneously decide whether to offer an incentive contract or to continue at the spot market to acquire their input. Our results show that under successive duopoly, offering an incentive contract constitutes the unique equilibrium solution, which highlights the incentive contract persistence.

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u ≡ 1 . We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain.

Successive duopoly under moral hazard: Will incentive contracts persist?

The central purpose of this paper is to examine the incentive contract as an equilibrium phenomenon. We analyse a model of vertical differentiation in which we deal with the strategic role of the competitor’s decisions in a successive duopoly. Is it better for a processor to offer an incentive contract to an upstream producer or the spot market? We determine the equilibrium of a game in which the processors simultaneously decide whether to offer an incentive contract or to continue at the spot market to acquire their input. Our results show that under successive duopoly, offering an incentive contract constitutes the unique equilibrium solution, which highlights the incentive contract persistence.

An approach to solution uniqueness in game problems

A new concept of strong conflict equilibrium is proposed that supplements the well-known fundamental system of conflict equilibria and considerably increases the possibility of finding a unique strongest equilibrium (solution) in any game problem. The efficiency of this new equilibrium is illustrated by static and dynamic game problems.

Global asymptotic stability analysis of nonlinear differential equations in hybrid bidirectional associative memory neural networks with distributed time-varying delays

In this paper, the problem on an unique equilibrium solution of the bidirectional associative memory neural networks with distributed time-varying delays is investigated via Lyapunov stability theory. By using topological degree theory and M-matrix theory, several sufficient conditions ensuring the existence, uniqueness and global asymptotic stability of equilibrium solution are derived. An illustrative example is given to demonstrate the effectiveness of the obtained results.

A note on channel performance under consignment contract with revenue sharing

Wang et al. [Y. Wang, L. Jiang, Z.J. Shen, Channel performance under consignment contract with revenue sharing. Management Science 50 (2004), 34–47] indicate that a decentralized supply chain cannot be perfectly coordinated. This note provides a cooperative game model that implements profit sharing between the manufacturer and the retailer to achieve their cooperation. When the manufacturer and the retailer are assumed to be risk-neutral, under a very mild restriction on the demand distribution function, the cooperative game model can achieve its unique equilibrium solution in iso-price-elastic or linear demand case. Under the revenue sharing agreement attached with the equilibrium payment scheme, the decentralized supply chain can be perfectly coordinated.

A REGIONAL ECONOMY, LAND USE, AND TRANSPORTATION MODEL (RELU‐TRAN©): FORMULATION, ALGORITHM DESIGN, AND TESTING*

RELU is a dynamic general equilibrium model of a metropolitan economy and its land use, derived by unifying in a theoretically valid way, models developed by one of the authors [Anas (1982), Anas–Arnott (1991, 1997), Anas–Kim (1996), Anas–Xu (1999)]. RELU equilibrates floor space, land and labor markets, and the market for the products of industries, treating development (construction and demolition), spatial interindustry linkages, commuting, and discretionary travel. Mode choices and equilibrium congestion on the highway network are treated by unifying RELU with the TRAN algorithm of stochastic user equilibrium [Anas–Kim (1990)]. The RELU-TRAN algorithm's performance for a stationary state is demonstrated for a prototype consisting of 4-building, 4-industry, 4-labor-type, 15-land-use-zone, 68-link-highway-network version of the Chicago MSA. The algorithm solves 656 equations in a special block-recursive convergent procedure by iterations nested within loops and loops within cycles. Runs show excellent and smooth convergence from different starting points, so that the number of loops within successive cycles continually decreases. The tests also imply a numerically ascertained unique stationary equilibrium solution of the unified model for the calibrated parameters.

Criterion for stable reentry in a ring of cardiac tissue

We model electrical wave propagation in a ring of cardiac tissue using an mth-order difference equation, where m denotes the number of cells in the ring. Under physiologically reasonable assumptions, the difference equation has a unique equilibrium solution. Applying Jury's stability test, we prove a theorem concerning the local asymptotic stability of this equilibrium solution. Our results yield conditions for sustained reentrant tachycardia, a type of cardiac arrhythmia.