A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

AnN-person noncooperative discounted vector-valued dynamic game with a metric space

AnN-person noncooperative discounted vector-valued game with a metric space is investigated. We define aD-convex equilibrium point and prove that aD-convex equilibrium point exists in this game system. A modified version of the original game system is formulated so that an equilibrium point of the modified version is aD-convex equilibrium point of the original game system. Under appropriate conditions, the converse holds in the sense that aD-convex equilibrium point is also an equilibrium point of the modified game system.     

TIME INCONSISTENCY AND REPUTATION IN MONETARY POLICY: A STRATEGIC MODELLING IN CONTINUOUS TIME

This article develops a model to examine the equilibrium behavior of the time inconsistency problem in a continuous time economy with stochastic and endogenized dis-tortion. First, the authors introduce the notion of sequentially rational equilibrium, and show that the time inconsistency problem may be solved with trigger reputation strategies for stochastic setting. The conditions for the existence of sequentially rational equilibrium are provided. Then, the concept of sequentially rational stochastically stable equilibrium is introduced. The authors compare the relative stability between the cooperative behavior and uncooperative behavior, and show that the cooperative equilibrium in this monetary policy game is a sequentially rational stochastically stable equilibrium and the uncooper-ative equilibrium is sequentially rational stochastically unstable equilibrium. In the long run, the zero inflation monetary policies are inherently more stable than the discretion rules, and once established, they tend to persist for longer periods of the time.     

Global dynamics of a delayed predator–prey model with stage structure for the predator and the prey

A delayed predator–prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each feasible equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory for infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using suitable Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator–extinction equilibrium and the coexistence equilibrium do not exist, and that the predator–extinction equilibrium is globally stable when the coexistence equilibrium does not exist. Further, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

具强制条件的向量均衡问题

研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性．     

On the notion of perfect Bayesian equilibrium

Often, perfect Bayesian equilibrium is loosely defined by stating that players should be sequentially rational given some beliefs in which Bayes rule is applied “whenever possible.” We argue that there are situations in which it is not clear what “whenever possible” means. Then we provide an elementary definition of perfect Bayesian equilibrium for general extensive games that refines both weak perfect Bayesian equilibrium and subgame perfect equilibrium.     

Global dynamics of a predator–prey model with time delay and stage structure for the prey

A stage-structured predator–prey system with Holling type-II functional response and time delay due to the gestation of predator is investigated. By analyzing the characteristic equations, the local stability of each of feasible equilibria of the system is discussed and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory on infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functionals and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator-extinction equilibrium and the coexistence equilibrium are not feasible, and that the predator-extinction equilibrium is globally asymptotically stable if the coexistence equilibrium does not exist, and sufficient conditions are derived for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Implementation by mediated equilibrium

Implementation theory tackles the following problem given a social choice correspondence (SCC), find a decentralized mechanism such that for every constellation of the individuals’ preferences, the set of outcomes in equilibrium is exactly the set of socially optimal alternatives (as specified by the correspondence). In this paper we are concerned with implementation by mediated equilibrium; under such an equilibrium, the players’ strategies can be coordinated in a way that discourages deviation. Our main result is a complete characterization of SCCs that are implementable by mediated strong equilibrium. This characterization, in addition to being strikingly concise, implies that some important SCCs that are not implementable by strong equilibrium are in fact implementable by mediated strong equilibrium.     

Persistent equilibria in strategic games

A perfect equilibrium [Selten] can be viewed as a Nash equilibrium with certain properties of local stability. Simple examples show that a stronger notion of local stability is needed to eliminate unreasonable Nash equilibria. The persistent equilibrium is such a notion. Properties of this solution are studied. In particular, it is shown that in each strategic game there exists a pesistent equilibrium which is perfect and proper.     

Vector equilibrium flows with nonconvex ordering relations

In this note we introduce the concept of vector network equilibrium flows when the ordering cone is the union of finitely many closed and convex cones. We show that the set of vector network equilibrium flows is equal to the intersection of finitely many sets, where each set is a collection of vector equilibrium flows with respect to a closed and convex cone. Sufficient and necessary conditions for a vector equilibrium flow are presented in terms of scalar equilibrium flows.     

微生物连续培养过程平衡点的稳定分析

§1Introduction1,3-propanediol(1,3-PD)isakindofimportantindustrialchemicals.Themethodsof1,3-PDfromfermentationofglycerolbymicrobialhascausedgreatinterestintheworld foritslowcost,highproductionandnopollution,etc.Therearemainstudiesof fermentationofglycerolto1,3-PDsuchasthekineticsofproductformation,growthof cells,substrateconsumptionandinhibition[1—3],andtheanalysisofmultiplicity,hysteresis,bifurcation,etc[4—6].Intheprocessofcontinuousculture,thesystemsapproachthe equilibriumstatebyautocata…     

Nondominated equilibrium solutions of a multiobjective two-person nonzero-sum game in extensive form and corresponding mathematical programming problem

In most of studies on multiobjective noncooperative games, games are represented in normal form and a solution concept of Pareto equilibrium solutions which is an extension of Nash equilibrium solutions has been focused on. However, for analyzing economic situations and modeling real world applications, we often see cases where the extensive form representation of games is more appropriate than the normal form representation. In this paper, in a multiobjective two-person nonzero-sum game in extensive form, we employ the sequence form of strategy representation to define a nondominated equilibrium solution which is an extension of a Pareto equilibrium solution, and provide a necessary and sufficient condition that a pair of realization plans, which are strategies of players in sequence form, is a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. Finally, giving a numerical example, we demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem.     

Coalition-proof equilibria in a voluntary participation game

We examine the coalition-proof equilibria of a participation game in the provision of a (pure) public good. We study which Nash equilibria are achieved through cooperation, and we investigate coalition-proof equilibria under strict and weak domination. We show that under some incentive condition, (i) a profile of strategies is a coalition-proof equilibrium under strict domination if and only if it is a Nash equilibrium that is not strictly Pareto-dominated by any other Nash equilibrium and (ii) every strict Nash equilibrium for non-participants is a coalition-proof equilibrium under weak domination.     

On Exact Controllability of Networks of Nonlinear Elastic Strings in 3-Dimensional Space

This paper concerns a system of nonlinear wave equations describing the vibrations of a 3-dimensional network of elastic strings.The authors derive the equations and appropriate nodal conditions,determine equilibrium solutions,and,by using the methods of quasilinear hyperbolic systems,prove that for tree networks the natural initial,boundary value problem has classical solutions existing in neighborhoods of the "stretched" equilibrium solutions.Then the local controllability of such networks near such equilibrium configurations in a certain specified time interval is proved.Finally,it is proved that,given two different equilibrium states satisfying certain conditions,it is possible to control the network from states in a small enough neighborhood of one equilibrium to any state in a suitable neighborhood of the second equilibrium over a suffciently large time interval.     

Deterministic Time-inconsistent Optimal Control Problems - an Essentially Cooperative Approach

A general deterministic time-inconsistent optimal control problem is formulated for ordinary differential equations. To find a time-consistent equilibrium value function and the corresponding time-consistent equilibrium control, a non-cooperative N-person differential game (but essentially cooperative in some sense) is introduced. Under certain conditions, it is proved that the open-loop Nash equilibrium value function of the N -person differential game converges to a time-consistent equilibrium value function of the original problem, which is the value function of a time-consistent optimal control problem. Moreover, it is proved that any optimal control of the time-consistent limit problem is a time-consistent equilibrium control of the original problem.     

A note on the dynamics of incentive contracts

Laffont and Tirole (Econometrica 56:1153–1175) show that when uncertainty about an agent’s ability is small, the equilibrium must involve a large amount of pooling, but, whether the continuation equilibrium induced by an optimal first-period menu of contracts is partitional or not, remains unclear. They construct a non-partitional continuation equilibrium for a given first-period menu of contracts and conjecture that this continuation equilibrium need not be suboptimal for the whole game under small uncertainty. We show that, irrespective of the amount of uncertainty, this non-partitional continuation equilibrium generates a strictly smaller payoff for the principal than a different menu of contracts with a partitional continuation equilibrium. In this sense, Laffont and Tirole’s menu of contracts, giving rise to a non-partitional continuation equilibrium, is not optimal.     

带风险指数的金融均衡的存在性定理及其结构特征(下)

本文主要论证了在不完全市场条件下带风险指数的金融均衡的存在性，并揭示其均衡结构的特征．本文中建立的模型是一、二期货币投入产出金融经济且具有可微的资产结构，这一模型包括了许多具有特殊资产结构的均衡模型，如实资产结构、虚资产结构、恒秩资产结构的均衡模型．因此本文的这一模型具有广泛的应用前景和实用价值．接着给出了本文的金融均衡的存在性定理，再借助微分拓扑给出它的证明过程，这一证明过程较之以前证明均衡存在性的经典方法（如Duffie，D＆W．Shfer（1985）的方法）要简便得多．同时也应注意到本文的这一结论既适用于资产市场下会随机风险因素的情形，也适用于商品空间为无限维的情形，除此之外，还给出了怎样判别资产结构是否属于T类的判别法，为检验均衡存在性提供了更为便利的途径．最后，本文论证了在金融市场里，尽管由于稀缺性的存在，从而导致均衡分配的多样化，然而均衡分配集却形成了一光滑子流，但该流形的维数与稀缺性有关．换句话说，尽管市场是不完全的，但均市分配不确定性的反却是可比的．如此使得人们对均衡资产结构的认识更进一步．     

Dynamics of a rate equation describing cluster-size evolution

In this paper, we show dynamics of Smoluchowski's rate equation which has been widely applied to studies of aggregation processes (i.e., the evolution of cluster-size distribution) in physics. We introduce dissociation in the rate equation while dissociation is neglected in previous works. We prove the positiveness of solutions of the equation, which is a basic guarantee for the effectiveness of the model since the possibility that some solution may be negative is excluded. For the case of cluster coalesce without dissociation, we show both the equilibrium uniqueness and the equilibrium stability under the condition that the monomer deposition stops. For the case that clusters evolve with dissociation and there is no monomer deposition, we show the equilibrium uniqueness and prove the equilibrium stability if the maximum cluster size is not larger than three while we show the equilibrium stability by numerical simulations if the maximum size is larger than three.     

On Nash–Cournot oligopolistic market equilibrium models with concave cost functions

We consider Nash–Cournot oligopolistic market equilibrium models with concave cost functions. Concavity implies, in general, that a local equilibrium point is not necessarily a global one. We give conditions for existence of global equilibrium points. We then propose an algorithm for finding a global equilibrium point or for detecting that the problem is unsolvable. Numerical experiments on some randomly generated data show efficiency of the proposed algorithm.     

一类带有潜伏期和部分免疫的传染病模型的全局分析

通过假设被接种者具有部分免疫,建立了一类具有潜伏期和接种的SEIR传染病模型,借助再生矩阵得到了确定此接种模型动力学行为的基本再生数.当基本再生数小于1时,模型只有无病平衡点;当基本再生数大于1时,除无病平衡点外,模型还有唯一的地方病平衡点.借助Liapunov函数,证明了无病平衡点和地方病平衡点的全局稳定性.