Sensitivity analysis in A class of dynamic optimization models

A general model of dynamic optimization, deterministic, in discrete time, and with infinite time horizon is considered. We assume that there are parameters in the formulation of the model. Conditions for stability of the optimal solution are studied. Analysis of steady state comparative statics and comparative dynamics are presented. In addition we apply these results to a quadratic case and to an economic example: a one sector growth model. This research was financed by the Universidad Complutense de Madrid, project PR295/95-6073     

Optimal control of locusts in subsistence farming areas

Locust swarms hit subsistence-staple-crop-growing households at random and are not privately controllable. An aerial-spraying optimal control model that supports the said households’ livelihood at least expected cost is therefore developed. The qualitative properties of the model are analysed under economically plausible but mild assumptions. The steady state comparative statics reveal that the locust swarm size and the probability of a household’s crop being destroyed by a swarm decrease with the number of households, yield per household, and the staple crop’s replacement price, and increase with the marginal cost of spraying and the planner’s discount rate. A local comparative dynamics analysis is also conducted, as it provides the necessary economic intuition behind other ostensibly anomalous steady-state comparative statics results.     

Stability analysis of diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type III schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

Stability Analysis of Diffusive Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes

The stability of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes is investigated. A threshold property of the local stability is obtained for a boundary steady state, and sufficient conditions of local stability and un-stability for the positive steady state are also obtained. Furthermore, the global asymptotic stability of these two steady states are discussed. Our results reveal the dynamics of this model system.     

THE QUALITATIVE CONTENT OF RENEWABLE RESOURCE MODELS

This paper analyzes the extent to which standard dynamic renewable resource models possess refutable implications. Both the steady state comparative static and local comparative dynamic properties of the standard model are studied. A unified framework is developed which enables one to analyze the qualitative properties of any standard renewable resource model. This is achieved by explicitly linking the local stability, steady state comparative static, and local comparative dynamic properties of the model.     

NATURAL RESOURCE EXPLOITATION UNDER COMMON PROPERTY RIGHTS

ABSTRACT. Renewable natural resources such as ground‐water, pastures and fisheries are often governed bycommon propertyrights in which members of a group jointlyown the exclusive use of the resource. We develop a formal model of a common propertycontract based on differential game theory and then use the model to examine (i) the incentives of individual users of the common resource; (ii) the resulting harvest and stock time paths; (iii) the local stabilityof the steady state; and (iv) the steadystate comparative statics. Moreover, we compare the qualitative properties of the common propertyregime to those generated under perfectlydefined private rights and open access. We show how common prop‐ertyownership of natural resources can generate rent and be a second‐best solution when private propertyrights are costly to establish.     

MODELING COMMON PROPERTY OWNERSHIP AS A DYNAMIC CONTRACT

Common property ownership is modeled as a joint wealth maximizing egalitarian share contract. Two-stage differential games are developed for various types of common property in order to examine the incentives inherent in common property regimes. Cases in which group members contract over just membership size and cases in which members contract over both group size and resource investment are considered. Envelope methods in optimal control theory are used to generate some comparative statics predictions about the value of the contracts which define property rights to a capital stock.     

Generic stability of Nash equilibria for noncooperative differential games

The stability of Nash equilibria against the perturbation of the right-hand side functions of state equations for noncooperative differential games is investigated. By employing the set-valued analysis theory, we show that the differential games whose equilibria are all stable form a dense residual set, and every differential game can be approximated arbitrarily by a sequence of stable differential games, that is, in the sense of Baire’s category most of the differential games are stable.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

Security investment and information sharing in the market of complementary firms: impact of complementarity degree and industry size

We study a differential game of information security investment and information sharing in a market consisting of n complementary firms. Two game approaches, the non-cooperative game and the totally cooperative game, are employed to investigate the steady state strategy of each firm. Under certain conditions, a unique steady state can be obtained for both games. We find that the steady state security investment and information sharing level are not always less in the non-cooperative game than that in the totally cooperative game. In addition, some theoretical analyses are made on the impacts of the complementarity degree and industry size on firms’ steady state strategies for both games. Finally, some numerical experiments are conducted to give some insights related to the instantaneous profit in the steady state. It can be found that a firm will obtain more instantaneous profit in the steady state of the totally cooperative game than that of the non-cooperative game, which emphasizes the importance of coordinating strategies. The effects of the complementarity degree and industry size on the instantaneous profits in the steady state are also obtained through the numerical experiment results.     

Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations

We consider a nonlinear mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting age-structured model is reduced to a system of three differential equations, with three independent delays, and existence of steady states is investigated. The stability of the trivial steady state, describing cells dying out with a saturation of growth factor concentrations is proven to be asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state allows the determination of a stability area, and shows that instability may occur through a Hopf bifurcation, mainly as a destabilization of the proliferative capacity control, when cell cycle durations are very short. Numerical simulations are carried out and result in a stability diagram that stresses the lead role of the introduction rate compared to the apoptosis rate in the system stability.     

Second-order game problem: Decision dynamics in gaming phenomena

The paradigm of decision dynamics (Ref. 1) is used to describe the decision dynamics involving more than one decision-maker. The framework supplied in this paper is different from traditional game theory or differential games. Traditional simplicity assumptions are replaced by a more complicated, but more realistic, setting. Although many mathematically beautiful results in traditional game theory or differential games have disappeared in second-order games, the more realistic setting of the latter does make it easier for the decision-makers to find agood decision. Concepts of time optimality and time stability, and their necessary and/or sufficient conditions are described. Unconventional concepts of strategies and uncertainty involved in gaming phenomena are discussed. A highlight of the paper is a systematic discussion on reframing tactics of gaming situations, which do not exist in the context of traditional game theory or differential games. Various research topics are discussed at the end of the paper.     

Characterization of equilibrium strategies in a class of difference games

We formulate a class of N player difference games and derive open—loop and Markov equilibria. It turns out that both types of equilibria can be characterized by a set of difference equations that describe the equilibrium dynamics. We analyze the stability properties of the difference equations that correspond to an equilibrium and find that in both the open—loop and the Markov game there is convergence towards a steady state equilibrium     

Network capacity management under competition

We consider capacity management games between airlines who transport passengers over a joint airline network. Passengers are likely to purchase alternative tickets of the same class from competing airlines if they do not get tickets from their preferred airlines. We propose a Nash and a generalized Nash game model to address the competitive network revenue management problem. These two models are based on well-known deterministic linear programming and probabilistic nonlinear programming approximations for the non-competitive network capacity management problem. We prove the existence of a Nash equilibrium for both games and investigate the uniqueness of a Nash equilibrium for the Nash game. We provide some further uniqueness and comparative statics analysis when the network is reduced to a single-leg flight structure with two products. The comparative statics analysis reveals some useful insights on how Nash equilibrium booking limits change monotonically in the prices of products. Our numerical results indicate that airlines can generate higher and more stable revenues from a booking scheme that is based on the combination of the partitioned booking-limit policy and the generalized Nash game model. The results also show that this booking scheme is robust irrespective of which booking scheme the competitor takes.     

Stability and Hopf bifurcation of a delayed reaction–diffusion neural network

In this paper, a delayed reaction–diffusion neural network with Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed. The existence of Hopf bifurcation at the trivial steady state is established. Using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2011 John Wiley & Sons, Ltd.     

A dynamical characterization of evolutionarily stable states

Evolutionary stability, the central solution concept in evolutionary game theory, is closely related to local asymptotic stability in a certain nonlinear dynamical system operating on the state space, the so-called "replicator dynamics". However, a purely dynamical characterization of evolutionary stability is not available in an elementary manner. This characterization can be achieved by investigating so-called "derived games" which consist of mixed strategies corresponding to successful states in the original game. Using well-known facts, several characterization results are obtained within this context. These also may shed light on the extremality properties of evolutionary stability.     

Non-constant positive steady state of a diffusive Leslie-Gower type food web system

A three species food web comprising of two preys and one predator in an isolated homogeneous habitat is considered. The preys are assumed to grow logistically. The predator follows modified Leslie-Gower dynamics and feeds upon the prey species according to Holling Type II functional response. The local stability of the constant positive steady state of the corresponding temporal system and the spatio-temporal system are discussed. The existence and non-existence of non-constant positive steady states are investigated.     

Bifurcation analysis in a diffusive Logistic population model with two delayed density-dependent feedback terms

The present paper is concerned with a diffusive population model of Logistic type with an instantaneous density-dependent term and two delayed density-dependent terms and subject to the zero-Dirichlet boundary condition. By regarding the delay as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation for the sufficiently small positive steady state solution are shown. It is found that under the suitable condition, the positive steady state solution of the model will become ultimately unstable after a single stability switch (or change) at a certain critical value of delay through a Hopf bifurcation. However, under the other condition, the positive steady state solution of the model will become ultimately unstable after multiple stability switches at some certain critical values of delay through Hopf bifurcations. In addition, the direction of the above Hopf bifurcations and the stability of the bifurcating periodic solutions are analyzed by means of the center manifold theory and normal form method for partial functional differential equations. Finally, in order to illustrate the correction of the obtained theoretical results, some numerical simulations are also carried out.     

Stability and Hopf bifurcation of a delayed Cohen–Grossberg neural network with diffusion

In this paper, a delayed Cohen–Grossberg neural network with diffusion under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equation, the local stability of the trivial uniform steady state and the existence of Hopf bifurcation at the trivial steady state are established, respectively. By using the normal form theory and the center manifold reduction of partial function differential equations, formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamics analysis of a reaction–diffusion system with Beddington–DeAngelis functional response and strong Allee effect

In this paper, the dynamics of a diffusive predator–prey model with modified Leslie–Gower term and strong Allee effect on prey under homogeneous Neumann boundary condition is considered. Firstly, we obtain the qualitative properties of the system including the existence of the global positive solution and the local and global asymptotical stability of the constant equilibria. In addition, we investigate a priori estimate and the nonexistence of nonconstant positive steady state solutions. Finally, we establish the existence and local structure of steady state patterns and time-periodic patterns for the system.