Stability Of Equilibrium Solutions Research Articles

Geometric Approach to Predator-Prey Model with Carrying Capacity on Prey Population

In this paper, we explore a classical predator-prey model where the birth rate of the prey is significantly lower than the mortality rate of the predators, while also considering a limited prey population. We incorporate an environmental carrying capacity factor for the prey to account for this. Given the different timescales of the predator and prey populations, some system solutions may exhibit a fast-slow structure. We analyze this fastslow behavior using geometric singular perturbation theory (GSPT), which allows us to separate the system into fast and slow subsystems. Our research investigates the existence and stability of equilibrium solutions and the behavior of solutions near the critical manifold. Additionally, we use an entry-exit function to analytically establish the connection between the solutions of the slow subsystem and those of the fast subsystem.

Dynamics of Activation and Regulation of the Immune Response to Attack by Viral Pathogens Using Mathematical Modeling

In this paper, a mathematical model is developed to simulate the activation of regulatory T lymphocytes dynamics. The model considers the adaptive immune response and consists of epithelial cells, infected cells, free virus particles, helper and cytotoxic T lymphocytes, B lymphocytes, and regulatory T lymphocytes. A mathematical analysis was carried out to discuss the conditions of existence and stability of equilibrium solutions in terms of the basic reproductive number. In addition, the definitions and properties necessary to preserve the positivity and stability of the model are shown. The precision of these mathematical models can be affected by numerous sources of uncertainty, partly due to the balance between the complexity of the model and its predictive capacity to depict the biological process accurately. Nevertheless, these models can provide remarkably perspectives on the dynamics of infection and assist in identification specific immunological traits that improve our comprehension of immune mechanisms. The theoretical results are validated by numerical simulations using data reported in the literature. The construction, analysis, and simulation of the developed models demonstrate that the increased induced regulatory T lymphocytes effectively suppress the inflammatory response in contrast to similar cells at lower contents, playing a key role in maintaining self-tolerance and immune homeostasis.

Analysis of Labor Relations in Private Enterprises A Three-party Game Based on Overtime Work

At present, China has successively introduced multiple laws and regulations to prevent employees from being forced to work overtime. However, this situation is still serious in private enterprises, which has led to gradually tense labor relations. Starting from the issue of overtime, this article constructs a game model between government regulatory departments, enterprises, and employees, explores the impact of government regulation on the labor management relationship of private enterprises, and uses evolutionary games to analyze the stability of the strategies of the three parties. The result shows that there is a stable equilibrium solution in the tripartite game: private enterprises arrange employees to work overtime normally, employees take the initiative to work overtime, and the government's project management department strictly supervises them. Among them, strict supervision by labor supervision agencies plays an important role in balancing labor management relations, therefore, certain incentive measures should be given to labor supervision agencies to ensure their strict supervision; Private enterprises should strictly abide by market order, proactively reduce the burden on employees, and avoid forcing employees to work overtime excessively; Employees should maintain a good work attitude and promote the development of common interests with private enterprises.

A system dynamics-based model for the evolution of environmental group events

Based on the system dynamics theory, this paper establishes an environmental mass event evolution model and explores the evolution law of mass events caused by environmental problems. From a methodological point of view, the mixed-strategy evolutionary game principle and dynamic punishment measures are combined, and simulation analysis is carried out by Anylogic software, and the results show that there is no stable evolutionary equilibrium solution for the two sides of the game in the traditional asymmetric mixed-strategy game model, and after adjusting the game payoff matrix and incorporating the dynamic punishment strategy, stable evolutionary equilibrium solutions appear in the evolutionary game model, and the system begins to tend to be stabilized. The process and conclusions of the simulation experiment provide methodological reference and theoretical support for the analysis of the evolution of environmental mass events.

Confinement Effects in Droplet Formation on a Solid Particle.

Formation of a droplet around a spherical solid particle in supersaturated vapor is considered. The number and stability of equilibrium solutions in a closed small system are studied in the canonical ensemble in comparison to an open system in the grand canonical ensemble. Depending on the system's parameters, two modes exist in the canonical ensemble: the first one with only one solution and the second one with three solutions; the presence of the third solution is due to confinement. The analysis is conducted first on a macroscopic thermodynamic level of description, and then the results are supported by studies within two versions of classical density functional theory: the square-gradient approximation with the Carnahan-Starling equation of state for hard spheres on a completely wettable particle and the random-phase approximation with the fundamental measure theory on a poorly wettable particle. In the latter case, a solution breaking the spherical symmetry is observed at a small total number of molecules.

The elastica sling

The nonlinear mechanics of a flexible elastic rod constrained at its edges by a pair of sliding sleeves is analyzed. The planar equilibrium configurations of this variable-length elastica are found to have shape defined only by the inclination of the two constraints, while their distance is responsible only for scaling the size. By extending the theoretical stability criterion available for systems under isoperimetric constraints to the case of variable domains, the existence of no more than one stable equilibrium solution is revealed. The set of sliding sleeves’ inclination pairs for which the stability is lost are identified. Such critical conditions allow the indefinite ejection of the flexible rod from the sliding sleeves, thus realizing an elastica sling. Finally, the theoretical findings are validated by experiments on a physical prototype. The present results lead to a novel actuation principle that may find application as a mechanism in energy harvesting, wave mitigation devices, and soft robotic locomotion.

Multistable states in a predator–prey model with generalized Holling type III functional response and a strong Allee effect

In this paper, we present a complete parametric analysis on a nonlinear predator–prey system with the generalized Holling type III functional response and a strong Allee effect. We apply the hierarchical parametric analysis to derive explicit conditions for the existence and stability of equilibrium solutions in a 5-dimensional parameter space. Specifically, a detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles, and complex dynamics of multistable states including bistable, tristable and tetrastable phenomena. We also conduct simulations and provide a biological interpretation of the multistable states under various conditions.

A neural network finite element method for contact mechanics

Computational biomechanical models of organ systems utilizing the finite element (FE) method have been extensively applied to the study of normal and pathophysiological function. While a robust method and providing for many unique pathophysiological insights, traditional FE methods remain prohibitively slow for real-time many-query clinical applications. To meet these demanding computational requirements, we have developed a general neural network finite element (NNFE) approach for hyperelastic soft tissue organ simulations that can produce equivalent accuracy within clinically relevant time frames. However, many organ systems involve contact between structural elements (e.g. heart valve leaflets), which has not been previously addressed in the NNFE and related approaches. In the present work, we developed a NNFE-based method for contact between elastic deformable bodies. We exploited the fact that stable equilibrium solutions in nonlinear hyperelasticity minimize the potential energy to train the neural network, with contact incorporated through an potential energy penalty. This penalty was discretized and computationally implemented in JAX (Google, Inc.) in a way that could be statically allocated and jit compiled for rapid computation on modern GPU hardware. Following our previous NNFE formulation, we represented the problem domain geometry and enforced the necessary boundary conditions using a Non-Uniform Rational B-Splines (NURBS) based geometry scheme. NURBS were chosen for their ability to smoothly represent body and surface geometries. To demonstrate the method, we developed and verified two basic contact problems using a hyperelastic material model: indentation and flap. Both problems were simulated with a fully connected neural network with a parallel linear layer. Training continued until the gradient of the energy with respect to network parameters dropped below 10−2, taking less than six hours in each case studied. Once trained, the NNFE approach was capable of accurately predicting the equilibrium configurations of representative contact problems. Moreover, the NNFE contact approach required nearly six order of magnitude less time to evaluate than an equivalent FEniCSx implementation that utilized a similar mesh of cubic order Lagrange elements (0.01 s compared to 1715 seconds, respectively). Moreover, the solutions were trained over full ranges of loading and were thus able to faithfully represent families of solutions without requiring retraining. While addressing basic problems in contact, this study serves as crucial stepping stone to generate NNFE organ simulations for rapid prediction of patient specific solutions that involve contact.

How does stakeholder loss aversion affect the promotion of green housing?

Incorporating the principles of loss aversion psychology within the context of green housing stakeholders – the government, realtors, and residents – this study extends its purview into an evolutionary game model. This model takes into account the consequences of inaction in the absence of government oversight, the additional loss incurred by realtors due to the unmarketable development of green housing, and the psychological loss experienced by residents when their green housing acquisitions fall short of their expectations. To further enrich our analysis, three distinct scenarios are considered for the government: exclusive rewards without penalties, exclusive penalties without rewards, and a combination of both rewards and penalties. The simulation results show that: (1) The initial probability values assigned to the three parties do not significantly impact the long-term stable equilibrium. Interestingly, a higher initial probability value tends to steer all three parties towards selecting the stable equilibrium solution. (2) Within the context of the three reward–penalty​ scenarios, it becomes evident that an increase in the magnitude of government rewards and penalties inclines realtors towards a greater inclination to engage in green housing development. Nonetheless, a balanced approach appears to yield the most favorable results. (3) A sensitivity analysis reveals that a heightened aversion to loss within the government’s decision-making process leads to a higher likelihood of regulatory intervention. Conversely, a diminished loss aversion mindset among realtors and residents correlates with an increased propensity for realtors to invest in green housing development, while residents are more inclined to purchase green housing. Finally, corresponding policy implications are given according to the conclusions.

Stability and Sommerfeld effect in a multi-resonant types vibrating system with isolated rigid frame driven by four exciters

The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange’s equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It’s shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power.

Multi-player evolutionary game of federated learning incentive mechanism based on system dynamics

Federated learning has emerged as a new way of data sharing. The participants in the federation tend to choose different strategies based on their benefits, which is formalized into an evolutionary game model. Existing techniques can limit the malicious behavior of participants by detecting betrayers or weakening their influence. The problem that whether there is an incentive mechanism which makes participants spontaneously choose to cooperate honestly and maintains the stability of the federated learning system is urgent. In this paper, we develop a multi-player evolutionary game model in federated learning. We model the federated learning process by evaluating the payoffs of the central server, internal clients, and external clients. The stability of the federated learning system in the long-term dynamics process is assessed by seeking the evolutionarily stable equilibrium solutions. In this paper, mathematical reasoning and computer simulation are combined to analyze the impact of reward and punishment strategies in incentive mechanisms on the game process and game equilibrium. An incentive mechanism is designed to achieve evolutionarily stable equilibrium while make most clients join the federation spontaneously and cooperate honestly. Finally, the effectiveness, stability, and generalizability of this incentive mechanism are verified by sensitivity analysis and Lyapunov stability theory.

Evolutionary Game Analysis of Low-Carbon Transformation of Construction Enterprises Under the "Dual Carbon" Target

Purpose: This study aims to investigate the impacts of different choices made by construction enterprises, government, and the public on the low-carbon transformation of Chinese construction enterprises in the context of the global climate crisis. Theoretical framework: The main theoretical foundation of this research is the theory of dynamic evolutionary games, it can aptly reflects the dynamic evolutionary process of strategy choices by various stakeholders during the promotion of low-carbon construction. Method/design/approach: This study adopts a mixed-method approach involving literature review, qualitative analysis, and quantitative analysis. Firstly, a payoff matrix is constructed for the three stakeholders under different strategy choices. Then, the conditions under which different strategies tend to reach equilibrium are discussed. Finally, the stability of equilibrium solutions in evolutionary game is analyzed. Results and conclusion: There are 7 equilibrium solutions that can be evolutionary stable strategies in specific situations. It is challenging for the system to achieve a progressively stable strategy when only one or two entities participate in low-carbon transformation. Research implications: This study focuses on the background of the Chinese government's "dual carbon" target. It takes an innovative approach by considering the stakeholders in the low-carbon building supply chain, including the government and the general public, and develops a tripartite evolutionary game model for the low-carbon transformation of construction enterprises. This research makes a significant contribution to the future development of low-carbon buildings in China. Originality/value:The results obtained in this study are unprecedented, innovative and relevant to the scientific community, in the context of sustainable development of low-carbon buildings.

In-Plane Libration Suppression of a Two-Segment Tethered Towing System

A tethered towing system provides an effective method for capturing pieces of space debris and dragging them out of orbit. This paper focuses on the in-plane stability analysis and libration control of a two-segment tethered towing system. The first segment is the same as the traditional single-tether towing system. The second segment is similar to a simplified space tether net. The dynamic equations are established in the orbit frame. Considering the elasticity of the tethers, the equilibrium solutions are obtained and the stability of equilibrium solutions is proved. An in-plane libration controller based on the sliding mode control scheme is designed to ensure the safety of the towing mission and save fuel. The controller suppressed the librations of the in-plane angles in the desired state by applying two external torques. Finally, simulation results are provided to validate the effectiveness of the proposed controller.

The role of supporting and disruptive mechanisms of FT3 homeostasis in regulating the hypothalamic-pituitary-thyroid axis.

Thyroid hormones are controlled by the hypothalamic-pituitary-thyroid (HPT) axis through a complex network of regulatory loops, involving the hormones TRH, TSH, FT4, and FT3. The relationship between TSH and FT4 is widely used for diagnosing thyroid diseases. However, mechanisms of FT3 homeostasis are not well understood. We used mathematical modelling to further examine mechanisms that exist in the HPT axis regulation for protecting circulating FT3 levels. A mathematical model consisting of a system of four coupled first-order parameterized non-linear ordinary differential equations (ODEs) was developed, accounting for the interdependencies between the hormones in the HPT axis regulation. While TRH and TSH feed forward to the pituitary and thyroid, respectively, FT4 and FT3 feed backward to both the pituitary and hypothalamus. Stable equilibrium solutions of the ODE system express homeostasis for a particular variable, such as FT3, if this variable stays in a narrow range while certain other parameter(s) and system variable(s) may vary substantially. The model predicts that (1) TSH-feedforward protects FT3 levels if the FT4 production rate declines and (2) combined negative feedback by FT4 and FT3 on both TSH and TRH production rates keeps FT3 levels insensitive to moderate changes in FT4 production rates and FT4 levels. The optimum FT4 and FT3 feedback and TRH and TSH-feedforward ranges that preserve FT3 homeostasis were found by numerical continuation analysis. Model predictions were in close agreement with clinical studies and individual patient examples of hypothyroidism and hyperthyroidism. These findings further extend the concept of HPT axis regulation beyond TSH and FT4 to integrate the more active sister hormone FT3 and mechanisms of FT3 homeostasis. Disruption of homeostatic mechanisms leads to disease. This provides a perspective for novel testable concepts in clinical studies to therapeutically target the disruptive mechanisms.

Two Approaches to Modeling Phytoplankton Biomass Dynamics Based on the Droop Model

This work continues the study of the Droop model based on the concept of cell quota. Description of the photosynthetic processes in phytoplankton includes in the model structure. The concept of chlorophyll quota is used. It is the proportion of photosynthetic substances in plant cells. In addition to the chlorophyll quota, the photosynthetic activity of phytoplankton is determined by external conditions, primarily by the level of photosynthetically active radiation (PAR). The model is based on separating the dependence of phytoplankton reproduction on external conditions according to the stages of photosynthesis. The light stage is largely determined by the PAR, and the dark stage is limited by the nutrient resource under the controlling influence of the temperature of the aquatic environment. In order to develop the model, the storage of energy in the light stage of photosynthesis is described in detail. Energy is stored in the form of energy-intensive substances in macroergic molecules (macroergs). The most common cell macroerg is adenosine triphosphate (ATP). The proportion of ATP in phytoplankton varies depending on the light regime and on the energy amount stored in the dark stage. The model includes the Droop kinetics and equations for the dynamics of the chlorophyll quota and the ATP pool. The conditions for the existence and stability of equilibrium solutions are compared for the same values of parameters common to both models. The greatest influence on the dynamic modes of the minimum value of the cell quota has been established. The proportion of biomass associated with the light period of photosynthesis is also significant. For the first model that is the biomass produced during daylight hours. And in terms of the second model, it is the biomass formed due to the energy of ATP stored in the light phase. The influence of the structure of dynamic models on the daily and annual dynamics of phytoplankton was revealed. Scenarios of behavior of models under various lighting conditions, including constant and periodically changing lighting, have been studied.

A Coalition Formation Game-Based Multi-User Grouping Approach in the Jamming Environment

Due to the broadcast nature of wireless communication, it is vulnerable to malicious jamming attacks. Meanwhile, multiple users need to share the limited spectrum resources to enhance the spectrum utilization efficiency because of the scarcity of wireless spectrum resources. Therefore, wireless communication systems need to deal with the malicious jamming attacks and mutual interference among different users. In this paper, a multi-user anti-jamming grouping approach based on a coalition formation game is studied. The proposed approach can maximize users’ transmission rate while avoiding channel jamming through optimization of the coalition formation strategy under the influence of malicious jamming. With the help of the exact potential energy game, we demonstrate that the proposed game model can obtain stable Nash equilibrium solutions. Lastly, the effectiveness of the proposed algorithm is demonstrated through numerical.

Complicated Dynamics of a Delayed Photonic Reservoir Computing System

In this paper, we consider the complicated dynamics of a delay-based photonic reservoir computing system. Since conventional computer architectures are approaching their limit, it is imperative to find new, efficient and fast ways of data processing. Photonic reservoir computing (RC) is a promising way which combines the computational capabilities of recurrent neural networks with high processing speed and energy efficiency of photonics. As the RC system is very promising, we analyze its dynamics so that we can make better use of it. In this paper, we mainly focus on its double Hopf bifurcation. We first analyze the existence of double Hopf bifurcation points. Then we use DDE-BIFTOOL to draw the bifurcation diagrams with respect to two bifurcation parameters, i.e. feedback strength [Formula: see text] and delay [Formula: see text], and give a clear picture of the double Hopf bifurcation points of the system. These figures show stability switches and the existence of double Hopf bifurcation points. Finally, we employ the method of multiple scales to obtain their normal forms, use the method of normal form to unfold and classify their local dynamics. The classification and unfolding of these double Hopf bifurcation points are obtained. Three types of double Hopf bifurcations are found. We verify the results by numerical simulations and find its complicated behavioral dynamics. For example, there exist stable equilibrium, stable periodic and quasi-periodic solutions in distinct regions. The discovered rich dynamical phenomena can help us to choose suitable values of parameters to achieve excellent performance of RC.

Using evolutionary game theory to study construction safety supervisory mechanism in China

PurposeIn China, external supervision on construction safety mainly comes from the government and supervision engineers (SEs). However, the construction safety supervisory mechanism (CSSM) contains some dilemmas affecting the improvement of safety performance, such as the declining impact of SEs, the increasing rent-seeking behaviors of contractor and excessive government interference. This study aims to depict and analyze the CSSM in China from an evolutionary game view. The objectives are to understand the supervision strategy and evolutionary behaviors of different stakeholders, propose suggestions for improving safety performance and help the key safety supervision stakeholders, especially the government, formulate a suitable safety supervision strategy.Design/methodology/approachThis research uses tripartite dynamic evolutionary game theory to study the CSSM in China and solve the stable equilibrium solution using system dynamics.FindingsThis study has revealed the game relationship of construction safety supervision mechanisms in China and solved the stable equilibrium solution. The results prove that a supervision engineer (SE) plays a crucial role in the CSSM, and “supervision engineer useless” is an unreasonable assertion. For government supervision agency (GSA), excessive inspection and free-market regulation are neither wise strategies. GSA can reduce the inspection frequency when general contractors (GCs) input high safety investments and SEs implement responsible supervision. But keeping proper government supervision to avoid GC's unlawful behaviors and SE's rent-seeking is indispensable. In addition, excessive governmental supervision will weaken SE's role, so the government should transfer some supervision powers to SE.Originality/valueThis study focuses on the dynamic evolution process between GSA, GC and SE. This method is different from most research that neglected the dynamic characteristic of system and game solution stability. The research methods not only contribute to construction safety supervision policy-making in China but also help to improve supervision efficiency in other countries and other fields.

Energy relaxation approximation for compressible multicomponent flows in thermal nonequilibrium

This work concerns the numerical approximation with an explicit first-order finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy (Coquel and Perthame in SIAM J. Numer. Anal. 35:2223–2249, 1998) to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver, and the HLL and pressure relaxation based approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the schemes.

Bubbling, Bistable Limit Cycles and Quasi-Periodic Oscillations in Queues with Delayed Information

We consider a model describing the length of two queues that incorporates customer choice behavior based on delayed queue length information. The symmetric case, where the values of the time-delay parameter in each queue are the same, was recently studied. It was shown that under some conditions, the stable equilibrium solution becomes unstable as the common time delay passes a threshold value. This one-time stability switch occurs only at a symmetry-breaking Hopf bifurcation where a family of stable asynchronous limit-cycle solutions arise. In this paper, we examine the non-symmetric case, wherein the values of the time-delay parameter in each queue are different. We show that, in contrast to the symmetric case, the non-symmetric case allows bubbling, multiple stability switches and coexistence of distinct families of stable limit cycles. An investigation of the dynamical behavior of the non-symmetric system in a neighborhood of a double-Hopf bifurcation using numerical continuation explains the occurrence of the bistable limit cycles. Quasi-periodic oscillations were also observed due to the presence of torus bifurcations near the double-Hopf bifurcation. These identifications of the underlying mechanisms that cause unwanted oscillations in the system give a better understanding of the effects of providing delayed information and consequently help in better management of queues.