Conditions For Global Asymptotic Stability Research Articles

Effects of stochastic perturbations on a phytoplankton allelopathy competitive system

In this paper, we investigate stochastic models of two species competitive phytoplankton system with allelopathic interaction. We construct stochastic models from deterministic models by introducing three different stochastic perturbations to the equations. For the first model, we prove that this model is asymptotically stable in probability at three different equilibrium points. It is also shown that the system is positively recurrent under some conditions. For the second model, we study the extinction, existence of positive recurrence and global asymptotic stability under some sufficient conditions. For the last system, almost sufficient and necessary conditions for global asymptotic stability, the existence of positive recurrence and extinction of each population are established. In addition, using a suitable stochastic Lyapunov method, we prove that there is a unique solution to three models, we also discuss the ultimate boundedness of the three stochastic systems. Moreover, some numerical simulations are introduced to illustrate our mathematical results. We show that stochastic models still retain the desirable stability of their deterministic counterparts if stochastic perturbations are relatively small.

Stability and Bifurcation Control for a Generalized Delayed Fractional Food Chain Model

In this paper, a generalized fractional three-species food chain model with delay is investigated. First, the existence of a positive equilibrium is discussed, and the sufficient conditions for global asymptotic stability are given. Second, through selecting the delay as the bifurcation parameter, we obtain the sufficient condition for this non-control system to generate Hopf bifurcation. Then, a nonlinear delayed feedback controller is skillfully applied to govern the system’s Hopf bifurcation. The results indicate that adjusting the control intensity or the control target’s age can effectively govern the bifurcation dynamics behavior of this system. Last, through application examples and numerical simulations, we confirm the validity and feasibility of the theoretical results, and find that the control strategy is also applicable to eco-epidemiological systems.

A study of integrated pest management models with instantaneous and non-instantaneous impulse effects.

The occurrence of pests and diseases during agricultural production affects the quality and quantity of agricultural products. It is important to evaluate the impact of various factors on pests to achieve optimal results of integrated pest management (IPM) during its implementation. In this paper, we considered the transient and non-transient effects of chemical control on pests and the effects on natural enemies at different times, and developed a corresponding pest control model. Detailed studies and comparisons were conducted for spraying pesticides either more or less frequently as compared to strategies for releasing natural enemies. The threshold conditions for global asymptotic stabilization of the pest extinction period solution was obtained. Using two-parameter and sensitivity analysis techniques, the parameters affecting the variation of the threshold were discussed. By comparing these two pest control strategies, we found the existence of optimal application and release frequencies. Finally, in order to control pests below the economic threshold level, the state-dependent pest model was numerically investigated. The results show that the presence or absence of chemical control of pests can depend on the values taken for the parameters in the model. Based on this information, pest control experts can make decisions about the best spraying time and the best release rate.

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>

Stabilization and asymptotic stability of the Caputo–Fabrizio fractional-order linear and semilinear evolution equations

We discuss the asymptotic stability of the fractional-order linear and semilinear evolution equations involving a Caputo–Fabrizio fractional derivative for fractional-order α∈(0,1) with a non-singular kernel. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo–Fabrizio system admits only a constant solution and a new concept of a global solution for the Caputo–Fabrizio system is introduced. Laplace transform and Grönwall inequality are used to derive the local and global asymptotic stability conditions. By constructing a suitable linear feedback control and using our main results, we stabilize the Caputo–Fabrizio fractional-order linear and semilinear evolution equations. At the end, by using the stabilization result, we stabilize a fractional-order chaotic system to support the obtained results.

Dynamical analysis on stochastic two-species models

In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results.

Global stability and bifurcation analysis of an infochemical induced three species discrete-time phytoplankton–zooplankton model

This article analyzes a discrete-time three-species model of phytoplankton and zooplankton with infochemical producing, non-infochemical producing, and zooplankton. It is assumed that the grazing of infochemical producing phytoplankton indirectly increases the mortality rate of zooplankton. The fixed points of the system and their respective local stability has been examined. The conditions for Neimark–Sacker bifurcation along with period doubling bifurcation at the positive fixed point are outlined. The sufficient conditions for global asymptotic stability of the positive fixed point are obtained. The application of a hybrid control strategy effectively stabilizes the chaotic behavior of the system. Lastly, our theoretical discussion is solidified through a numerical simulation which enables us to witness the complex behavior of the system.

Sufficient conditions for uniform global asymptotic stabilization of affine discrete-time systems with periodic coefficients

Sufficient conditions for uniform global asymptotic stabilization of affine discrete-time systems with periodic coefficients

Bifurcation and global stability of a discrete prey–predator model with saturated prey refuge

This article aims to describe the qualitative behavior of a discrete‐time prey–predator model including prey refuge. It is assumed that the fraction of prey in refuge is a monotonically increasing but self‐limiting function of prey population. The fixed points of the system and their local stability are investigated. The criteria for Neimark–Sacker bifurcation and period‐doubling bifurcation are presented. The sufficient conditions for global asymptotic stability of the interior fixed point are derived. The chaotic behavior of the system is stabilized by applying a hybrid control strategy. We conclude with a numerical simulation that strengthens our theoretical discussion and provides a way to observe the complex behavior of the system.

Global asymptotic stability of switched boolean networks with missing data

This paper investigates the global asymptotic stability of a switched Boolean network (SBN) with missing data. Due to the physical constraints of the communication media, data loss often occurs during the transmission. In this paper, the data loss is described as a Bernoulli distribution sequence, and the switching signal sequence is determined by a logical system. Using a semi-tensor product, the dynamics of a SBN with missing data are converted to an algebraic form. Based on this, a necessary and sufficient condition for global asymptotic stability of SBNs with missing data and auxiliary theorems are provided. Two illustrative examples are presented to show the applicability of the developed theorems.

A Fractional-Order Eco-Epidemiological Leslie–Gower Model with Double Allee Effect and Disease in Predator

In this paper, a fractional order of a modified Leslie–Gower predator-prey model with disease and the double Allee effect in predator population is proposed. Then, we analyze the important mathematical features of the proposed model such as the existence and uniqueness as well as the non-negativity and boundedness of solutions to the fractional-order system. Moreover, the local and global asymptotic stability conditions of all possible equilibrium points are investigated using Matignon’s condition and by constructing a suitable Lyapunov function, respectively. Finally, numerical simulations are presented to verify the theoretical results. We show numerically the occurrence of two limit cycles simultaneously driven by the order of the derivative, the bistability phenomenon for both the weak and strong Allee effect cases, and more dynamic behaviors such as the forward, backward, and saddle-node bifurcations which are driven by the transmission rate. We have found that the risk of extinction for the predator with a strong Allee effect is much higher when the spread of disease is relatively high.

Novel Global Asymptotic Stability Conditions for Discrete-Time Systems With Time-Varying Delay and Generalized Overflow Arithmetic

Novel global asymptotic stability (GAS) criterion for discrete-time (DT) systems with generalized overflow arithmetic and time-varying delay is presented in this brief. The criterion is based on two unique lemmas that allow for a more precise characterization of nonlinearities by utilizing system information in greater detail. The approach leads to significantly relaxed stability conditions as compared to a recently reported criterion.

Mathematical Modelling of the Spatial Distribution of a COVID-19 Outbreak with Vaccination Using Diffusion Equation.

The formulation of mathematical models using differential equations has become crucial in predicting the evolution of viral diseases in a population in order to take preventive and curative measures. In December 2019, a novel variety of Coronavirus (SARS-CoV-2) was identified in Wuhan, Hubei Province, China, which causes a severe and potentially fatal respiratory syndrome. Since then, it has been declared a pandemic by the World Health Organization and has spread around the globe. A reaction−diffusion system is a mathematical model that describes the evolution of a phenomenon subjected to two processes: a reaction process, in which different substances are transformed, and a diffusion process, which causes their distribution in space. This article provides a mathematical study of the Susceptible, Exposed, Infected, Recovered, and Vaccinated population model of the COVID-19 pandemic using the bias of reaction−diffusion equations. Both local and global asymptotic stability conditions for the equilibria were determined using a Lyapunov function, and the nature of the stability was determined using the Routh−Hurwitz criterion. Furthermore, we consider the conditions for the existence and uniqueness of the model solution and show the spatial distribution of the model compartments when the basic reproduction rate R0<1 and R0>1. Thereafter, we conducted a sensitivity analysis to determine the most sensitive parameters in the proposed model. We demonstrate the model’s effectiveness by performing numerical simulations and investigating the impact of vaccination, together with the significance of spatial distribution parameters in the spread of COVID-19. The findings indicate that reducing contact with an infected person and increasing the proportion of susceptible people who receive high-efficacy vaccination will lessen the burden of COVID-19 in the population. Therefore, we offer to the public health policymakers a better understanding of COVID-19 management.

Optimal control strategy analysis for an human-animal brucellosis infection model with multiple delays

Firstly, we consider an animal-human infection model of brucellosis with three distributed delays, representing the latent period of brucellosis in infected animal and human population and the survival time of brucella in the environment, respectively. The equilibrium points and basic reproduction number R0 are calculated. By building appropriate Lyapunov functionals and applying LaSalle's invariance principle, the sufficient conditions for global asymptotic stability of two equilibria are given. Secondly, by introducing four control variables, we set the corresponding optimal control model and drive the first order necessary conditions for the existence of optimal control solution. Finally, we perform several numerical simulations to validate our theoretical results and show effects of different control strategies.

A Detailed Study of a Fractal-Fractional Transmission Dynamical Model of Viral Infectious Disease with Vaccination

This article is devoted to investigate a mathematical model consisting on susceptible, exposed, infected, quarantined, vaccinated, and recovered compartments of COVID-19. The concerned model describes the transmission mechanism of the disease dynamics with therapeutic measures of vaccination of susceptible people along with the cure of the infected population. In the said study, we use the fractal-fractional order derivative to understand the dynamics of all compartments of the proposed model in more detail. Therefore, the first model is formulated. Then, two equilibrium points disease-free (DF) and endemic are computed. Furthermore, the basic threshold number is also derived. Some sufficient conditions for global asymptotical stability are also established. By using the next-generation matrix method, local stability analysis is developed. We also attempt the sensitivity analysis of the parameters of the proposed model. Finally, for the numerical simulations, the Adams–Bashforth method is used. Using some available data, the results are displayed graphically using various fractal-fractional orders to understand the mechanism of the dynamics. In addition, we compare our numerical simulation with real data in the case of reported infected cases.

Impulsive Boundedness for Nonautonomous Dynamic Complex Networks with Constraint Nonlinearity

This paper discusses time-scale type global boundedness of nonautonomous complex networks (NCNs) under impulsive control. By using constraint nonlinearity of NCNs, properties of exponential function and impulsive inequalities, we establish new sufficient conditions for global boundedness and asymptotic stability of NCNs on time scales, and the upper bound of the system can be calculated easily. It is shown that the constraint nonlinearity and impulse interval length will affect the estimation of state bound of NCNs under different network size. It provides a unified criteria for global boundedness of NCNs on continuous, discrete and mixed-time domains and hence improves corresponding existing ones. The nonlinear constraints studied in this paper are widely applied to Lurie systems, ball motion models and so on. Numerical examples are given to illustrate the correctness of the results. Finally, the theoretical results are applied to Lurie systems and bounded driving of unmanned vehicles.

Global behaviour of quaternion Riccati rational difference equation

Since non-commutative algebraic structure of quaternions, it is difficult to study the forbidden sets and asymptotic stability of quaternion rational difference equations. In this paper, by generalized Niven's algorithm and its related properties, the representations of general solutions for the third order linear quaternion difference equation are given through the roots of the corresponding quaternion characteristic equation, based on which, the corresponding forbidden sets of the Riccati quaternion rational difference equation are computed. Moreover, we obtain the quaternion curves of initial points located in two-dimensional quaternion space outside which we determine the convergence, divergence and global asymptotic stability of the solutions for the second order Riccati quaternion rational difference equations, some sufficient and necessary conditions for global asymptotic stability are established. Finally, several examples are illustrated to show the feasibility of our obtained results.

STABILITY ANALYSIS AND NUMERICAL SIMULATIONS OF IVGTT GLUCOSE-INSULIN INTERACTION MODELS WITH TWO TIME DELAYS

This paper presents a systematic study of a mathematical model of glucose and insulin interaction with two time delays, with a focus on analytical studies, bifurcation analysis, and very well numerical simulations. This model based on the Intra-Venous Glucose Tolerance Test (IVGTT) and is presented with two time delays. One delay is the insulin response time to an increase in glucose concentration, and the hepatic glucose production time delay is the other. Then, we establish results on positivity, boundedness, and persistence. We also provide sufficient stability analysis conditions for both local and global asymptotic stability of the proposed models. For the latter, two different strategies are used: stability bifurcation analysis and Lyapunov-Krasovskii functionals. We investigate different regions of parameter space using two approaches, that yield different sets of sufficient conditions for global stability. The bifurcation graphs generated from our extensive and carefully designed simulations complement and confirm these analytical results. The insulin concentration level peaks after the glucose concentration level, according to the numerical simulations.

Sampled-Data Stabilization of Fractional Linear System under Arbitrary Sampling Periods

The sampled-data stabilization of a fractional continuous linear system under arbitrary sampling periods was first investigated in this paper wherein novel co-designed sampled-data controllers were constructed based on the compensation of scaling gains. With the help of fractional difference approximation, sufficient and necessary conditions for global asymptotic stability were first presented in the discrete-time domain, and then co-designed sampled-data controllers were constructed with only the “newest” or “oldest” state information available for controller design. Due to the compensation scheme between scaling gains and sampling periods, much more flexibility on selecting different sampling periods was provided in the sampled-data stabilization of the fractional continuous linear system which is significantly preferred for digital implementation. Numerical studies are also presented to illustrate the effectiveness of our co-designed sampled-data controllers under different sampling periods.

Nonlinear Modeling and Global Stability Condition of Single-Phase Grid-Tied Inverter Considering SRF-PLL and Duty-Cycle Saturation

Considering that the traditional linearization method cannot accurately analyze the grid-tied inverter (GTI) system with nonlinear nature, in the single-phase case, a novel nonlinear model for the GTI system considering nonlinear factors, including synchronous reference frame phase-locked loop (PLL) and duty-cycle saturation in pulsewidth modulation, is proposed, and the corresponding global asymptotic stability condition is given. Specifically, some system nonlinear factors, e.g., duty-cycle saturation phenomenon and Park ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dq</i> ) transformation and cosine operation in second-order generalized integrator-based PLL (SOGI-PLL), are investigated, and their sector-bounded conditions are derived. In order to seek less conservative results, the system state of the SOGI-PLL is restructured such that the linear part of the PLL system can be stabilized. Then, the nonlinear state-space model with sector bounded for the whole GTI system is formulated. Furthermore, the deduced sector conditions are introduced into the derivation of the Lyapunov function by using the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S-procedure</i> lemma, and the global asymptotic stability condition of the inverter system is, thus, obtained. The system stability regions under different grid impedances, current references, and control parameters are further analyzed. Finally, the effectiveness of the proposed theories is fully supported by transient experimental results.