Existence Of Equilibrium Research Articles

Hopf bifurcation and patterns formation in a diffusive two prey-one predator system with fear in preys and help

Recognizing the relationship between the spatial patterns in species concentrations and ecological heterogeneity is crucial for understanding demographics and species governance in a given domain, as ecological patterning processes are believed to be imitated in real ecosystems. In this present article, we have considered a two-prey-one-predator system with Holling type-II functional response where both the preys have fear of predator and compete in the absence of predator. Additionally, we have assumed that both the prey help each other when the predator attacks them. We have incorporated self-diffusion in this system under Neumann boundary conditions. The existence and stability conditions of different equilibria are discussed. We have studied the Hopf bifurcation around positive offset and derived the stability and direction of periodic solution. We have done a series of numerical simulations to ensure our theoretical finding. Diffusion-driven instability conditions are obtained. Different instability regions and fascinating patterns, such as spots, mixtures, and stripes, are depicted. Spatiotemporal dynamics also reflects that prey populations are located in isolated areas of low population concentration with increasing level of fear, whereas an increase in handling time forms lower-density stripes. But with increasing help, prey becomes more concentrated in some stripe regions. In the Hopf-Turing region, it is observed that the diffusion coefficient of predators can stabilize the system. Overall, pattern creation in predator-prey systems can help anticipate long-term system behaviour and analyze the influence of ecological factors on system dynamics.

Dynamic analysis and optimal control of HIV/AIDS model with ideological transfer

This paper presents an HIV/AIDS model with ideological transfer in male susceptible individuals, and considers condoms and education as control measures. First, it gives the threshold R0 of the model. Then it proves that the disease-free equilibrium is globally asymptotically stable when R0<1. Under different threshold conditions, the local stability of the boundary equilibria is given. The existence of the endemic equilibrium is also given. We also perform numerical simulations for different parameter values. Furthermore, when considering control measures, the simulation results show that effective and long-term use of condoms significantly reduce the number of infected individuals and education should be vigorously promoted when the epidemic is controllable. This result provides theoretical guidance for effective control the spread of HIV.

A Generalized Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-Convergence Concept for a Class of Equilibrium Problems

A novel generalization of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature.

On the complexity of inverse bivariate multi-unit assignment valuation problems

Inverse and bilevel optimization problems play a central role in both theory and applications. These two classes are known to be closely related due to the pioneering work of Dempe and Lohse (2006), and thus have often been discussed together ever since. In this paper, we consider inverse problems for multi-unit assignment valuations. Multi-unit assignment valuations form a subclass of strong-substitutes valuations that can be represented by edge-weighted complete bipartite graphs. These valuations play a key role in auction theory as the strong substitutes condition implies the existence of a Walrasian equilibrium. A recent line of research concentrated on the problem of deciding whether a bivariate valuation function is an assignment valuation or not. In this paper, we consider an inverse variant of the problem: we are given a bivariate function g, and our goal is to find a bivariate multi-unit assignment valuation function f that is as close to g as possible. The difference between f and g can be measured either in ℓ 1 - or ℓ ∞ -norm. Using tools from discrete convex analysis, we show that the problem is strongly NP -hard. On the other hand, we derive linear programming formulations that solve relaxed versions of the problem.

Robust Stability Analysis of Switched Neural Networks with Application in Psychological Counseling Evaluation System

In this work, the effectiveness and stability of psychological counseling are evaluated using the switched complex-valued neural networks (SCVNN) model, which includes parameter disturbances, impulsive perturbations, variable and continuously distributed delays in the system state, and impulsive delay. How to analyze and judge the stability of the network simply and effectively is the primary prerequisite for its successful application. Therefore, we explore the dynamic behavior of SCVNN with both variable and distributed delays along with impulsive effect. Initially, the proposed conditions for the existence and uniqueness of equilibrium in SCVNN are presented. Subsequently, employing the inequality technique and impulsive average dwell time approach, sufficient conditions for the robust exponential stability of SCVNN under both arbitrary and restricted switching are obtained. Lastly, the psychological counseling evaluation system (PCES) is established, and a simulation example is used to verify the correctness and effectiveness of the presented findings.

The existence of a pure-strategy Nash equilibrium in a discrete ponds dilemma

In a variety of economic situations discrete agents choose one resource among several available resources and, once admitted to the resource of choice, divide it among fellow agents admitted there. The amount of the resource an agent gets is proportional to her relative ability to acquire this particular resource, what we refer to as an agent's weight at the resource. The relevant applications include students self-selecting into colleges, politicians self-selecting into races, and athletes self-selecting into teams. We find that this game has a pure-strategy Nash equilibrium in at least three special cases: 1) when agents have the same weight at each resource, 2) when all resources are the same, 3) when there are only two resources. We also show that this game always has an approximate Nash equilibrium when the number of players is large. Existence in the general case remains an open problem.

Asymmetric auctions: Perturbations, ε- equilibrium, and equilibrium

We consider first-price auctions with independent and private valuations that have asymmetric valuation distributions and supports. We first show the existence of equilibrium in these auctions through a perturbation approach, thereby establishing that the limit of Bayesian Nash equilibria (BNE) of such perturbed auctions is indeed the Bayesian Nash equilibrium (BNE) of the limit auction with asymmetric supports. We then characterize this BNE and show that the ε-equilibrium (ε-BNE) of the auction with asymmetric supports is a BNE of “close” auctions with common supports. We then demonstrate some numerical examples.

A multi-timescale and chance-constrained energy dispatching strategy of integrated heat-power community with shared hybrid energy storage

The community in the future may develop into an integrated heat-power system, which includes a high proportion of renewable energy, power generator units, heat generator units, and shared hybrid energy storage. In the integrated heat-power system with coupling heat-power generators and demands, the key challenges lie in the interaction between heat and power, the inherent uncertainty of renewable energy and consumers’ demands, and the multi-timescale scheduling of heat and power. In this paper, we propose a game theoretic model of the integrated heat-power system. For the welfare-maximizing community operator, its energy dispatch strategy is under chance constraints, where the day-ahead scheduling determines the scheduled energy dispatching strategies, and the real-time dispatch considers the adjustment of generators. For utility-maximizing consumers, their demands are sensitive to the preference parameters. Taking into account the uncertainty in both renewable energy and consumer demand, we prove the existence and uniqueness of the Stackelberg game equilibrium and develop a fixed-point algorithm to find the market equilibrium between the community operator and community consumers. Numerical simulations on integrated heat-power community show that the proposed solution outperforms the constant curtailment strategy with the daily cost reduced by 14.38%.

Edge computing collaborative offloading strategy for space‐air‐ground integrated networks

SummaryDue to geographical factors, it is impossible to build large‐scale communication network infrastructure in remote areas, which leads to poor network communication quality in these areas and a series of delay‐sensitive tasks cannot be timely processed and responded. Aiming at the problem of limited coverage in remote areas, the space‐air‐ground integrated networks (SAGIN) combined with mobile edge computing (MEC) can provide low latency and high reliability transmission for offloading delay‐sensitive tasks for users in remote areas. Considering the strong limitation of satellite resources in the space‐ground integrated network and insufficient energy of local user equipment, firstly, a satellite‐UAV cluster‐ground three‐layer edge computing network architecture is proposed in this paper. Under the condition that the delay requirements of various ground tasks are met, the task offloading problem is transformed into a Stackelberg game between ground user equipment and edge servers. In addition, it is proved that the existence of Nash equilibrium in non‐cooperative game between ground user equipment by using potential game. Finally, a Nash equilibrium iterative offloading algorithm based on Stackelberg game (NEIO‐SG) is proposed to find the optimal offloading strategy for tasks to minimize the system offloading cost and the optimal forwarding percentage strategy for offloading tasks to maximize the utility function of the edge server. Simulation results show that compared to other baseline algorithms, NEIO‐SG reduces the total system latency during task offloading by about 13 and the energy consumption of the edge server by about 35.

A novel optimal control strategy for nutrient–phytoplankton–zooplankton model with viral infection in plankton

This paper presents a mathematical model that explores the intricate dynamics between nutrients, phytoplankton, and zooplankton populations, incorporating viral infection phenomena in the frame of the Caputo fractional derivative. The conceptual properties such as the existence, uniqueness, and stability of multiple equilibria are analyzed under specific conditions. We have defined some threshold parameters whose biological meaning helps in strengthening the stability conditions. Furthermore, an optimal control strategy is proposed for the suggested model and the impacts of the optimal control strategy on both commensurate and non-commensurate systems are investigated. A dynamic framework is developed to investigate the optimal use of resources, stock sustainability, and resource benefits. The control variables considered are concerned with the infection of the susceptible phytoplankton and the nutrient consumption of infected phytoplankton. Pontryagin’s maximum principle is employed to determine the best control strategy. Using the Adam-Bashforth-Moulton predictor–corrector method we have performed the numerical simulation that provides the justification of the theoretical findings.

Evaluation of cutout factors with small and narrow fields using various dosimetry detectors in electron beam keloid radiotherapy.

The inherent problems in the existence of electron equilibrium and steep dose fall-off pose difficulties for small- and narrow-field dosimetry. To investigate the cutout factors for keloid electron radiotherapy using various dosimetry detectors for small and narrow fields. The measurements were performed in a solid water phantom with nine different cutout shapes. Five dosimetry detectors were used in the study: pinpoint 3D ionization chamber, Farmer chamber, semiflex chamber, Classic Markus parallel plate chamber, and EBT3 film. The results demonstrated good agreement between the semiflex and pinpoint chambers. Furthermore, there was no difference between the Farmer and pinpoint chambers for large cutouts. For the EBT3 film, half of the cases had differences greater than 1%, and the maximum discrepancy compared with the reference chamber was greater than 2% for the narrow field. The parallel plate, semiflex chamber and EBT3 film are suitable dosimeters that are comparable with pinpoint 3D chambers in small and narrow electron fields. Notably, a semiflex chamber could be an alternative option to a pinpoint 3D chamber for cutout widths≥3 cm. It is very important to perform patient-specific cutout factor calibration with an appropriate dosimeter for keloid radiotherapy.

Optimal control and bifurcation analysis of a delayed fractional-order epidemic model for the COVID-19 pandemic

In this paper, a delayed fractional-order epidemic model with general incidence rate and incubation period is proposed for the Corona Virus Disease 2019 (COVID-19) pandemic. The corresponding sufficient conditions are established to analyze the existence and stability of disease-free equilibrium and endemic equilibrium of the proposed model. The conditions for the existence of Hopf bifurcation are obtained by selecting the time delay as the bifurcation parameter. The control strategies for the COVID-19 pandemic are designed, and the corresponding delay fractional order optimal control problem (DFOCP) is analyzed based on the generalized Euler–Lagrange equation. The parameters of the model are identified based on the data of multiple types of the COVID-19 pandemic. Further, the effectiveness of the model in describing the trend of the COVID-19 pandemic is verified. Based on the results of parameter identification, the influence of incubation period on the COVID-19 pandemic is discussed. The forward–backward sweep method (FBSM) is adopted to numerically solve DFOCP, and the control effects under different control measures are analyzed.

Joint optimization of deployment, user association, channel, and resource allocation for fairness‐aware multi‐UAV network

AbstractThis paper studies the problem of joint deployment, user association, channel, and resource allocation in unmanned aerial vehicle‐enabled access network. Since different user equipments performing different tasks and have different data rate requirements, the priority‐based traffic fairness problem is investigated. This problem, however, is a mixed integer nonlinear programming problem with NP‐hard complexity, making it challenging to be solved. To address this issue, a self‐organized and distributed framework “sense‐as‐you‐fly” based on the decomposition process, which divides the original problem into several subproblems, is proposed. Assuming without central controller, we derive the closed‐form resource allocation scheme and propose distributed many‐to‐one matching to optimize user association subproblem. Considering the coupled characteristics, the multi‐unmanned aerial vehicle deployment and channel allocation subproblems are modelled as a local altruistic game. The existence of Nash equilibrium is proved with the aid of exact potential game and efficient best response learning‐based algorithm is proposed. The original problem is finally addressed by solving the sub‐problems alternately and iteratively. Simulation results verify its effectiveness. By jointly optimizing multidimensional variables, the proposed algorithm unlocks network performance gains, especially in resource‐limited regimes.

Multiple-Population Discrete-Time Mean Field Games with Discounted and Total Payoffs: The Existence of Equilibria

In the paper we present a model of discrete-time mean-field game with several populations of players. Mean-field games with multiple populations of the players have only been studied in the literature in the continuous-time setting. The main results of this article are the first stationary and Markov mean-field equilibrium existence theorems for discrete-time mean-field games of this type. We consider two payoff criteria: β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-discounted payoff and total payoff. The results are provided under some rather general assumptions on one-step reward functions and individual transition kernels of the players. In addition, the results for total payoff case, when applied to a single population, extend the theory of mean-field games also by relaxing some strong assumptions used in the existing literature.

Bayesian Nash equilibrium in all-pay auctions with interdependent types

We prove the existence of a behavioral-strategy Bayesian Nash equilibrium in all-pay auctions with statistically interdependent types (signals) under quite general assumptions on the values, costs and tie-breaking rules. Moreover, the set of equilibria is shown to be the same for any tie-breaking rule used in the auction.

Sustainable management of predatory fish affected by an Allee effect through marine protected areas and taxation

Ecological balance and stable economic development are crucial for the fishery. This study proposes a predator–prey system for marine communities, where the growth of predators follows the Allee effect and takes into account the rapid fluctuations in resource prices caused by supply and demand. The system predicts the existence of catastrophic equilibrium, which may lead to the extinction of prey, consequently leading to the extinction of predators, but fishing efforts remain high. Marine protected areas are established near fishing areas to avoid such situations. Fish migrate rapidly between these two areas and are only harvested in the nonprotected areas. A three-dimensional simplified model is derived by applying variable aggregation to describe the variation of global variables on a slow time scale. To seek conditions to avoid species extinction and maintain sustainable fishing activities, the existence of positive equilibrium points and their local stability are explored based on the simplified model. Moreover, the long-term impact of establishing marine protected areas and levying taxes based on unit catch on fishery dynamics is studied, and the optimal tax policy is obtained by applying Pontryagin’s maximum principle. The theoretical analysis and numerical examples of this study demonstrate the comprehensive effectiveness of increasing the proportion of marine protected areas and controlling taxes on the sustainable development of fishery.

Robust contracts in common agency

AbstractBusiness activities often involve a common agent managing a variety of projects on behalf of investors with potentially conflicting interests. The extent of the agent's actions is also often unknown to investors, who have to design contracts that provide incentives to the manager despite this lack of crucial knowledge. We consider a game between several principals and a common agent, where principals know only a subset of the actions available to the agent. Principals demand robustness and evaluate contracts on a worst‐case basis. This robust approach allows for a crisp characterization of the equilibrium contracts and payoffs and provides a novel proof of equilibrium existence in common agency by constructing a pseudo‐potential for the game. Robust contracts make explicit how the efficiency of the equilibrium outcome relative to collusion among principals depends on the principals' ability to extract payments from the agent.

Control dengue with Wolbachia: Insights from a two-strain vector-host model with larval competition

Due to deficient treatments and vaccines for dengue viruses, dengue control mainly depends on controlling mosquitoes. Suppressing wild mosquito population size by releasing male-only Wolbachia-carrying mosquitoes into the field has become an eco-friendly alternative for traditional mosquito control methods. By assuming an ideal continuous proportional release strategy, we formulate a two-strain vector-host model incorporating larval competition to describe the affected cross-transmission dynamics of two dengue viruses serotypes. Moreover, antibody-dependent enhancement (ADE) and temporal cross-immunity are considered in this model. Theoretical analysis including the well-posedness, the existence and local stability of dengue-free equilibria in terms of the basic reproduction numbers is conducted. Sensitivity analysis indicates that the basic reproduction numbers are most sensitive to the adult female mortality rate and mosquito bite rate. As expected, intensifying larval competition can prevent the spread of dengue viruses. Unexpectedly, increasing the release ratio can delay the peak time while increase the peak level of both primary and secondary infections. To efficiently eliminate dengue outbreaks, other control methods such as spraying larvicides/adult insecticides and avoiding mosquito bites should be employed simultaneously during the time period gained by releasing Wolbachia-carrying males.

Dynamics and Control Strategies for SLBRS Model of Computer Viruses Based on Complex Networks

Dynamics and Control Strategies for SLBRS Model of Computer Viruses Based on Complex Networks

Optimal Dispatching of Microgrids with Development of Prosumers Sharing Energy Storage

The charge/discharge operation of the prosumer’s energy storage and the energy interaction between prosumers and MGs are chaotic from the overall point of the MG’s operation. It causes considerable resource waste and reduces the overall benefits of the MG with multi-prosumers. Therefore, a game theory-based optimal scheduling strategy for the MG with multi-prosumers combined into a PRCO is proposed in this paper. According to the prosumers’ complementary characteristics of ES utilization and energy production, prosumers can be integrated into the PRCO to obtain energy reciprocity by sharing ES with an ordered charge–discharge operation. Meanwhile, to improve the collaboration of prosumers and the overall efficiency of the MG, a game scheduling model is established with the MG as the leader and the PRCO as the follower. The ToU price incentive policy is implemented in the MG to maximize the operational benefits and reduce the difference between the valley and peak load. Meanwhile, the PRCO responds to the price policy and implements an ordered charge–discharge strategy of ES to optimize each member’s energy scheduling strategy and minimize the total costs. The PRCO revenues are distributed to prosumers based on the Shapley value method. The uniqueness and existence of Stackelberg equilibrium in the game model are proved. The simulations of a community MG show that the ordered charge–discharge operation of ES is achieved and the overall benefits of the system are improved.