Stability Of Equilibrium Point Research Articles

Pattern dynamics and Turing instability induced by self-super-cross-diffusive predator-prey model via amplitude equations

<abstract><p>Incorporating self-diffusion and super-cross diffusion factors into the modeling approach enhances efficiency and realism by having a substantial impact on the scenario of pattern formation. Accordingly, this work analyzes self and super-cross diffusion for a predator-prey model. First, the stability of equilibrium points is explored. Utilizing stability analysis of local equilibrium points, we stabilize the properties that guarantee the emergence of the Turing instability. Weakly nonlinear analysis is used to get the amplitude equations at the Turing bifurcation point (WNA). The stability analysis of the amplitude equations establishes the conditions for the formation of small spots, hexagons, huge spots, squares, labyrinthine, and stripe patterns. Analytical findings have been validated using numerical simulations. Extensive data that may be used analytically and numerically to assess the effect of self-super-cross diffusion on a variety of predator-prey systems.</p></abstract>

On stability analysis study and strategies for optimal control of a mathematical model of hepatitis HCV with the latent state

In this work, we analyze a viral hepatitis C model. This epidemic remains a major problem for global public health, in all communities, despite the efforts made. The model is analyzed using the stability theory of systems of nonlinear differential equations. Based on the results of the analysis, the proposed model has two equilibrium points: a disease-free equilibrium point E0 and an endemic equilibrium point E∗. We investigate the existence of equilibrium point of the model. Furthermore, based on the indirect Lyapunov method, we study the local stability of each equilibrium point of the model. Moreover, by constructing the appropriate Lyapunov function and by using LaSalle invariance principle, we get some information on the global stability of equilibrium points under certain conditions. The basic reproduction number R0 is calculated using the Next Generation method. The positivity of the solutions and their bornitude have been proven, the existence of the solutions has also been proven. Optimal control of the system was studied by proposing three types of intervention: awareness program, early detection, isolation and treatment. The maximum principle of Pontryagin was used to characterize the optimal controls found. Numerical simulations were carried out with a finite numerical difference diagram and using MATLAB to confirm acquired results.

Mathematical Modeling and Analysis of Corruption Dynamics in Kenya

This study presents a mathematical model that aims to study corruption in Kenya. The model is validated both epidemiologically and mathematically, with all solutions demonstrating positivity and boundedness within a meaningful set of initial conditions. By investigating unique corruption-free and endemic equilibrium points, as well as computing the basic reproduction number, we assess the system’s behavior. Our analysis reveals that a locally asymptotically stable corruption-free equilibrium point is achieved when the reproduction number is below one, while a locally asymptotically stable endemic equilibrium point is attained when the reproduction number exceeds one. Simulation results confirm the agreement with analytical findings. This research enhances our understanding of corruption dynamics and provides valuable insights for designing effective anti-corruption strategies in Kenya.

Generation of stochastic mixed-mode oscillations in a pair of VDP oscillators with direct-indirect coupling.

Environmental noise can lead to complex stochastic dynamical behavior in nonlinear systems. In this paper, we studied the phenomenon of a pair of Van der Pol (VDP) oscillators with direct-indirect coupling affected by Gaussian white noise. That is to say, a noise-induced equilibrium transition oscillation was observed in three types of different parameter regions, where the deterministic system had two kinds of stable equilibrium points. Meanwhile, with the noise intensity increasing, we found that the stochastic system will constantly switch between two stable equilibrium points. To analyze the stochastic behavior, we used the stochastic sensitivity equation and confidence ellipse method. When the confidence ellipsoid crossed the boundary of the attraction basin of the equilibrium point, the system entered into the state of stochastic mixed-mode oscillations, which was consistent with the simulation results.

КОНТРОЛИРУЕМАЯ БИСТАБИЛЬНОСТЬ ЗАРЯЖЕННЫХ ЧАСТИЦ В ПОВЕРХНОСТНЫХ ИОННЫХ ЛОВУШКАХ

In the present work, we numerically simulate the dynamics of porous charged microparticles localized in surface radio-frequency trap under atmospheric conditions, taking into account laser irradiation. The dynamic system transition from bistability to states characterized by either one or three stable equilibrium points is revealed. The number of stable equilibrium points and their spatial position appear to depend on the magnitude of the particle gravity and optical pressure. The phase portraits of the particle trajectories are calculated for the each dynamic system state. The obtained results are generalized and discussed from a practical point of view

Contribution of a Ca<sup>2+</sup>-activated K<sup>+</sup> channel to neuronal bursting activities in the Chay model

<abstract><p>The central nervous system extensively expresses Ca<sup>2+</sup>-stimulated K<sup>+</sup> channels, which serve to use Ca<sup>2+</sup> to control their opening and closing. In this study, we explore the numerical computation of Hopf bifurcation in the Chay model based on the equilibrium point's stability and the center manifold theorem to illustrate the emergence of complicated neuronal bursting induced by variation of the conductance of the Ca<sup>2+</sup>-sensitive K<sup>+</sup> channel. The results show that the formation and removal of various firing activities in this model are due to two subcritical Hopf bifurcations of equilibrium based on theoretical computation. Furthermore, the computational simulations are shown to support the validity of the conceptual approach. Consequently, the conclusion could be helpful to improve and deepen our understanding of the contribution of the Ca<sup>2+</sup>-sensitive K<sup>+</sup> channel.</p></abstract>

Analysis of HBV and COVID-19 Coinfection Model with Intervention Strategies.

Coinfection of hepatitis B virus (HBV) and COVID-19 is a common public health problem throughout some nations in the world. In this study, a mathematical model for hepatitis B virus (HBV) and COVID-19 coinfection is constructed to investigate the effect of protection and treatment mechanisms on its spread in the community. Necessary conditions of the proposed model nonnegativity and boundedness of solutions are analyzed. We calculated the model reproduction numbers and carried out the local stabilities of disease-free equilibrium points whenever the associated reproduction number is less than unity. Using the well-known Castillo-Chavez criteria, the disease-free equilibrium points are shown to be globally asymptotically stable whenever the associated reproduction number is less than unity. Sensitivity analysis proved that the most influential parameters are transmission rates. Moreover, we carried out numerical simulation and shown results: some parameters have high spreading effect on the disease transmission, single infections have great impact on the coinfection transmission, and using protections and treatments simultaneously is the most effective strategy to minimize and also to eradicate the HBV and COVID-19 coinfection spreading in the community. It is concluded that to control the transmission of both diseases in a population, efforts must be geared towards preventing incident infection with either or both diseases.

Dynamics of a nonlinear discrete predator-prey system with fear effect

<abstract><p>In this paper, we investigate a nonlinear discrete prey-predator system with fear effects. The existence, local stability and boundedness of positive equilibrium point are discussed. Using the center manifold theorem and bifurcation theory, the conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation in the interior of $ \mathbb{R}_{+}^{2} $ are established. Furthermore, the numerical simulations not only show complex dynamical behaviors, but also verify our analysis results. A feedback control strategy is employed to control bifurcation and chaos in the system.</p></abstract>

Under nonlinear prey-harvesting, effect of strong Allee effect on the dynamics of a modified Leslie-Gower predator-prey model.

In the present study, the effects of the strong Allee effect on the dynamics of the modified Leslie-Gower predator-prey model, in the presence of nonlinear prey-harvesting, have been investigated. In our findings, it is seen that the behaviors of the described mathematical model are positive and bounded for all future times. The conditions for the local stability and existence for various distinct equilibrium points have been determined. The present research concludes that system dynamics are vulnerable to initial conditions. In addition, the presence of several types of bifurcations (e.g., saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, homoclinic bifurcation) has been investigated. The first Lyapunov coefficient has been evaluated to study the stability of the limit cycle that results from Hopf bifurcation. The presence of a homoclinic loop has been demonstrated by numerical simulation. Finally, possible phase drawings and parametric figures have been depicted to validate the outcomes.

Mathematical modeling of the gaming disorder model with media coverage: optimal control approach

In this article, we propose a PEARM mathematical model to depict the dynamic of a population that reacts in the spread of the gaming disorder with media coverage. The basic reproduction number and existence of free equilibrium point and endimec equilibrium point are obtained with same fundamental properties of the model including existence and positivity as well as boundedness of equilibria are investigated. By using Routh–Hurwitz criteria, the local stability of free equilibrium point and endimec equilibrium point are obtained. Also, we propose an optimal strategy to implement the optimal campaigns through directing children and adolescents to educational and entertaining alternative means, and creating centers to restore the rehabilitation of addicts to electronic games. The existence of the optimal control are obtained by Pontryagain's maximum principle. Finally, some numerical simulations are also performed to illustrate the theoretical analysis of our results, using Matlab software. Our results show that media coverage is an effective measure to quit electronic gaming disorder.

Finite-time stability of equilibrium solutions for the inertial neural networks via the figure analysis method.

The existence and finite-time stability (FTS) of equilibrium point (EP) for a kind of inertial neural networks (INNS) with varying-time delays is studied. Firstly, by adopting the degree theory and the maximum-valued method, a sufficient condition in the existence of EP is attained. Then by adopting the maximum-valued approach and the figure analysis approach, without adopting the matrix measure theory, linear matrix inequality (LMI), and FTS theorems, a sufficient condition in the FTS of EP for the discussed INNS is proposed.

SEARC Mathematical Modeling of Drug Addiction in South Sulawesi Province

This study aims to build a SEARC mathematical model of drug addiction in South Sulawesi. The purpose of this study was to analyze and predict the number of drug addiction cases in South Sulawesi. The data used in this study is secondary data from the Provincial National Narcotics Agency (BNNP) of South Sulawesi and the Rehabilitation Center of the National Narcotics Agency (BNN) Baddoka Makassar. The steps in this study are to make assumptions and determine the SEARC model for drug addiction, determine the equilibrium point of the model obtained, analyze the stability of the equilibrium point, perform simulations with the Maple program, and conclude. The results of this study obtained the SEARC mathematical model of drug addiction in the form of a five dimensional system of differential equations, and also obtained an addiction-free equilibrium point and a stable endemic equilibrium point, as well as the basic reproduction number R0= 0.05 which shows that the rate of drug addiction cases in South Sulawesi can be controlled but preventive measures still need to be taken. Based on the simulation, it is found that the rehabilitation program can reduce the number of drug addicts so that by improving rehabilitation facilities and services, it can reduce the number of drug addicts effectively.

Features of the interaction of stationary oscillating charges

Systems of stationary oscillators determine the structure of a large number of natural materials by the model of many physical processes. In particular, they are the main physical model of the energy transfer mechanism in communication lines and power transmission systems. The interaction of oscillators usually leads to a change in their mode of oscillation, in particular, to synchronization of oscillations. In the present work, the possibilities of another effect of the interaction of oscillators are considered, which, at a stable frequency of oscillations of each of the system of coherent oscillators, leads to their mutual displacement to the points of stable equilibrium. It has been established that the violation of the coherence of the radiation of oscillators leads to the motion of both individual oscillators and a system of oscillators. Conditions for the equilibrium state of a system of oscillators are obtained.

Research on the evolution of tripartite data protection strategy based on game theory

In recent years, data leakage incidents have been frequently exposed, and personal privacy protection has become a major issue. To explore the laws of data leakage and solve practical problems, the reasons for data leakage and the evolution of players’ strategy in the game are discussed. After sorting out the interest relationship of data leakage incidents, a three-party evolution model of invaders, operators, and regulators is established to analyze the evolution path from the perspective of a single party. The Lyapunov theory is used to analyze the stability of equilibrium points of the system and then verified by MATLAB simulation. The influence relationship of various parameters on the system is analyzed. Finally, the ideal state is defined and the system is attempted to be modified. The results demonstrate that under the current system, the evolution direction cannot be changed by the initial intention of players. The game result will eventually form a situation with a strong attack from invaders, advanced defense from operators, and extensive management from regulators. The attack level can be reduced by adjusting the controllable parameters, modifying the evolution path, and calculating the parameter range.

Stability of equilibrium points of equation of regulatory mechanisms of cardiac activity

The article presents a mathematical model of regulatory mechanisms of cardiac activity in the form of system of functional-differential equations with delay arguments. With the use of reduction and scaling methods, the system of equations is reduced to the form of a functional-differential equation with delay argument. The equation was qualitatively analyzed. Equilibrium points are revealed and their stability is analyzed. As a result of a qualitative analysis, it was revealed that the mathematical model can reflect various modes of regulatory mechanisms of cardiac activity in normal conditions and in case of anomalies, the modes such as stationary, auto-oscillating, dynamic chaos, “black hole” and falling state.

Generalized Defensive Modeling of Malware Propagation in WSNs Using Atangana–Baleanu– Caputo (ABC) Fractional Derivative

The malware spreading in Wireless Sensor Network (WSN) has lately attracted the attention of many researchers as a hot problem in nonlinear systems. WSN is a collection of sensor nodes that communicate with each other wirelessly. These nodes are linked in a decentralised and distributed structure, allowing for efficient data collection and communication. Due to their decentralised architecture and limited resources, WSN is vulnerable to security risks, including malware attacks. Malware can attack sensor nodes, causing them to malfunction and consume more energy. These attacks can spread from one infected node to others in the network, making it essential to protect WSN against malware attacks. In this paper, we focus on the analysis of a novel fractional epidemiology model, specifically the fractional order SEIVR epidemic model in the sense of Caputo’s fractional derivative of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0{ &lt; {\alpha }{\leq }}1$ </tex-math></inline-formula> with the goal of examining the efficacy of vaccination strategies and the heterogeneity of a scale-free network on epidemic spreading. First, using the next-generation technique and obtain the basic reproduction number of the proposed epidemic model, which is essential for determining both the locally asymptotically stable equilibrium point of the worm-free system and the unique existence of the endemic equilibrium point. To numerically solve the model, the Adam-Bashforth-Moulton predictor-corrector (ABM) method is applied. The fractional calculus enables us to deal directly with the “memory effect” of numerous phenomena, taking into account the system’s dependence on previous stages. This method provides the results of a complex system. Additionally, research demonstrates that vaccine treatments are quite effective at preventing the spread of malware. The outcome of the study reveals that the applied ABM predictor-corrector method is computationally strong and effective to analyse fractional order dynamical systems in the SEIVR epidemic model for malware propagation in WSN. The results show that the order of the fractional derivative has a significant effect on the dynamic process. Also, from the result, it is obvious that the memory effect is zero for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> = 1. When the fractional order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> is decreased from 1, the memory effect appears, and its dynamics vary according to the time. This memory effect points out the difference between derivatives of fractional and integer orders. The theorems and their proofs are presented to validate the validity of the proposed model. To validate the proposed model, extensive theoretical study and computational analysis have also been applied.

Evolutionary game theory analysis of supply chain with fairness concerns of retailers

&lt;p style='text-indent:20px;'&gt;This study focuses on evolutionary game analysis of supply chain (SC) strategies with fairness concerns. First, the equilibrium solutions under the eight combined strategies are obtained through the Stackelberg game. Subsequently, evolutionary game theory is applied to analyze the stability of these equilibrium solutions and to discuss the evolutionary stability strategy of the tripartite game. The study suggests that the effect of distributional fairness concerns on SC pricing differs from that of peer-induced fairness concerns in the short term. Implanting fairness concerns into SC pricing decisions is always detrimental to the supplier and not always beneficial to retailers. From a long-term perspective, only one evolutionarily stable equilibrium point exists. Retailers do not pay attention to the fairness of the channel profit distribution, and the supplier does not consider the fairness of retailers. The results help enterprises understand the influence of fairness preferences on pricing decisions and provide a reference for the efficient operation of SCs.&lt;/p&gt;

The Effect of Consumer Sentiment on Manufacturers’ Green Technology Innovation: A RDEU Evolutionary Game Model

In the information era, the fluctuation of consumer sentiments plays a key role in the green technology innovation of manufacturers. This paper introduces RDEU theory to the evolutionary game model to analyze the existence of equilibrium under different sentiment states. Then, the model is numerically simulated to study the influence of sentiments on the participants’ strategies. The results indicate that under different sentiment states green technology innovation and green purchasing behavior present different evolutionary trajectories. The main conclusions are as follows: (1) When both parties have no sentiments, there is a stable equilibrium point, suggesting customers are willing to purchase green products and manufacturers choose green technology innovation strategies. (2) When both parties have sentiments, the rising consumer boycott sentiment will hinder optimistic manufacturers from choosing green technology innovation strategies. Furthermore, the rising support sentiment of the consumer promotes optimistic manufacturers’ green technology innovations, and the more manufacturers deviate from the rational state, the more likely they are to maintain the current production mode. (3) When only one party has a sentiment, the manufacturer’s rationality plays a more important role in promoting green technology innovation than the consumer’s rationality. Based on the above conclusions, this paper proposes some sentiment guidance strategies that are conducive to green production and consumption. This study provides a new perspective and theoretical guidance for studying the behavior of green supply chain members to promote the development of green economy circulation.

Complicated Boundaries of the Attraction Basin in a Class of Three-Dimensional Polynomial Systems

Even if a system has only one stable equilibrium point, its dynamics can still be complicated. In the case of the equilibrium point with a complicated boundary of the attraction basin, it is probably difficult to predict the long-term behavior of the trajectory starting from a point outside but near the boundary. The exploration of such systems is helpful for deepening our understanding of the dynamics of complex systems. This paper studies a class of three-dimensional polynomial systems with a nonelementary singularity. With parameters satisfying some conditions, the asymptotic stability of the origin is proved and the complexity of the attraction basin is investigated. It is demonstrated that the boundary of the attraction basin of the origin has a fractal structure in the following sense: An invariant set homeomorphic to the well-known Lorenz strange attractor is contained in the basin boundary. Based on the non-negative function we constructed, how fast a trajectory of the system tends to the asymptotically stable nonelementary singularity is measured.

Global asymptotic stability of constant equilibrium point in attraction-repulsion chemotaxis model with logistic source term

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for an attraction-repulsion chemotaxis model with logistic source term of Equation 1 in bounded convex domains \(\Omega\subset\mathbb{R}^{n},~ n\geq1\), with smooth boundary. It is shown that if the ratio \(\frac{\mu}{\chi \alpha-\xi \gamma}\) is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by \((u_{1},u_{2},u_{3})=(1,~\frac{\alpha}{\beta},~\frac{\gamma}{\eta})\) is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data \((u_{10},u_{20},u_{30})\) such that \(u_{10}\not\equiv0\), the above problem possesses uniquely determined global classical solution \((u_{1},u_{2},u_{3})\) with \((u_{1},u_{2},u_{3})|_{t=0}=(u_{10},u_{20},u_{30})\) which satisfies $$\|u_{1}(\cdot,t)-1\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\|_{L^{\infty}(\Omega)}\rightarrow{0}.$$ \(\mathrm{as}~t\rightarrow{\infty}\).