On Gopalsamy and Liu's conjecture for global stability in a population model

For a logistic equation with a piecewise constant argument which models the dynamics of a population of a single species undergoing a density-dependent harvesting, Gopalsamy and Liu (J. Math. Anal. Appl. 224 (1998) 59–80), have offered a condition of the growth rate of species and conjectured that this is a necessary and sufficient condition of not only the asymptotic stability but also the global asymptotic stability for the positive equilibrium of the equation. But until now, there were no mathematical answers except computer simulations.In this paper, we establish a mathematically rigorous proof of a partial affirmative answer of this conjecture.     

An affirmative answer to Gopalsamy and Liu's conjecture in a population model

In this paper we study the global asymptotic stability for a class of delay logistic equations. A sufficient and necessary condition is derived. As a result, we obtain a full affirmative answer to the Gopalsamy and Liu' conjecture.     

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

In this paper, we consider an SIS epidemic reaction–diffusion model with spontaneous infection and logistic source in a heterogeneous environment. The uniform bounds of solutions are established, and the global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. This paper aims to analyze the asymptotic profile of endemic equilibria (when it exists) as the diffusion rate of the susceptible or infected population is small or large. Our results on this new model reveal that varying total population and spontaneous infection can enhance persistence of infectious disease, which may provide some implications on disease control and prediction.     

具有免疫接种且总人口规模变化的SIR传染病模型的稳定性

讨论一类具有预防免疫接种且有效接触率依赖于总人口的SIR传染病模型,给出了决定疾病灭绝和持续生存的基本再生数σ的表达式,在一定条件下证明了疾病消除平衡点的全局稳定性,得到了唯一地方病平衡点的存在性和局部渐近稳定性条件.最后研究了具有双线性传染率和标准传染率的两个具体模型,并证明了当σ>1时该模型地方病平衡点的全局渐近稳定性.     

Geometric approach for global asymptotic stability of three-dimensional Lotka–Volterra systems

In this paper, by applying the geometric criterion and time average property to Lotka–Volterra systems, some results for the global asymptotic stability of the systems are obtained. Furthermore, we consider Li–Wang Conjecture for a three-dimensional system which is transformed from a Lotka–Volterra system.     

ON THE GLOBAL STABILITY CONJECTURE OF THE GENOTYPE SELECTION MODEL

In 1994, Grove, Kocic, Ladas, and Levin conjectured that the local stability and global stability conditions of the fixed point y = 1/2 in the genotype selection model should be equivalent. In this article, we give an affirmative answer to this conjecture and prove that local stability implies global stability. Some illustrative examples are included to demonstrate the validity and applicability of the results.     

A generalized Degn–Harrison reaction–diffusion system: Asymptotic stability and non-existence results

In this paper we study the Degn–Harrison system with a generalized reaction term. Once proved the global existence and boundedness of a unique solution, we address the asymptotic behavior of the system. The conditions for the global asymptotic stability of the steady state solution are derived using the appropriate techniques based on the eigen-analysis, the Poincaré–Bendixson theorem and the direct Lyapunov method. Numerical simulations are also shown to corroborate the asymptotic stability predictions.Moreover, we determine the constraints on the size of the reactor and the diffusion coefficient such that the system does not admit non-constant positive steady state solutions.     

The Fourier–Stieltjes transform of Minkowskiʼs ?(x) function and an affirmative answer to Salemʼs problem

By using structural and asymptotic properties of the Kontorovich–Lebedev transform associated with Minkowski?s question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 439) whether its Fourier–Stieltjes transform vanishes at infinity.     

Stability analysis of Takagi–Sugeno fuzzy Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays

In this paper, the global asymptotic stability problem of Takagi–Sugeno (TS) fuzzy Cohen–Grossberg Bidirectional Associative Memory neural networks (FCGBAMNNs) with discrete and distributed time-varying delays is considered. A novel LMI-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of FCGBAMNNs which are represented by TS fuzzy models. Our results can be easily verified and are also less restrictive than previously known criteria and can be applied to Cohen–Grossberg neural networks, recurrent neural networks and cellular neural networks. Finally, the proposed stability conditions are demonstrated with a numerical example.     

Global stability analysis of epidemiological models based on Volterra–Lyapunov stable matrices

In this paper, we study the global dynamics of a class of mathematical epidemiological models formulated by systems of differential equations. These models involve both human population and environmental component(s) and constitute high-dimensional nonlinear autonomous systems, for which the global asymptotic stability of the endemic equilibria has been a major challenge in analyzing the dynamics. By incorporating the theory of Volterra–Lyapunov stable matrices into the classical method of Lyapunov functions, we present an approach for global stability analysis and obtain new results on some three- and four-dimensional model systems. In addition, we conduct numerical simulation to verify the analytical results.     

Analysis of a nonautonomous dynamical model of diseases through droplet infection and direct contact

In this paper, we have considered a nonautonomous dynamical model of diseases that spread by droplet infection and also through direct contact (with a lower risk) with varying total population size and distributed time delay to become infectious. It is assumed that there is a time lag due to incubation period of pathogens, i.e. the development of an infection from the time the pathogen enters the body until signs or symptoms first appear. Here, we have established some sufficient conditions on the permanence and extinction of the disease by using inequality analytical technique. We have obtained the explicit formula of the eventual lower bounds of infected persons. We have introduced some new threshold values. By Lyapunov functional method, we have also obtained some sufficient conditions for global asymptotic stability of this model. Computer simulations are carried out to explain the analytical findings. The aim of the analysis of this model is to trace the parameters of interest for further study, with a view to informing and assisting policy-maker in targeting prevention and treatment resources for maximum effectiveness.     

Stability analysis of impulsive delayed switched systems and applications

In this paper, a general class of impulsive delayed switched systems is considered. By employing the Lyapunov–Razumikhin method and some analysis techniques, we established several global asymptotic stability and global exponential stability criteria for the considered impulsive delayed switched systems, which improve and extend some recent works. As an application, the result of global exponential stability are used to study a class of uncertain linear switched systems with time‐varying delays. Several LMI‐based conditions are proposed to guarantee the global robust stability and global exponential stabilization. The designed memoryless state feedback controller can be easily checked by the LMI toolbox in Matlab. Moreover, the dwell time constraint is imposed for the switching law. Finally, two numerical examples and their simulations are given to show the effectiveness of our proposed results. Copyright © 2012 John Wiley & Sons, Ltd.     

Dynamics of SIS Epidemic Model with Varying Total Population and Multivaccination Control Strategies

In this paper, two susceptible‐infected‐susceptible epidemic models with varying total population size, continuous vaccination, and state‐dependent pulse vaccination are formulated to describe the transmission of infectious diseases, such as diphtheria, measles, rubella, pertussis, and so on. The first model incorporates the proportion of infected individuals in population as monitoring threshold value; we analytically show the existence and orbital asymptotical stability of positive order‐1 periodic solution for this control model. The other model determines control strategy by monitoring the proportion of susceptible individuals in population; we also investigate the existence and global orbital asymptotical stability of the disease‐free periodic solution. Theoretical results imply that the disease dies out in the second case. Finally, using realistic parameter values, we carry out some numerical simulations to illustrate the main theoretical results and the feasibility of state‐dependent pulse control strategy.     

Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple discrete and distributed time-varying delays

In this paper, the Takagi–Sugeno (T–S) fuzzy model representation is extended to the stability analysis for stochastic cellular neural networks with multiple discrete and distributed time varying delays. A novel linear matrix inequality (LMI) based stability criterion is derived to guarantee the asymptotic stability of stochastic cellular neural networks with multiple discrete and distributed time varying delays which are represented by T–S fuzzy models. The derived delay-dependent stability conditions are based on free-weighting matrices method, Lyapunov stability theory and LMI technique. In fact, these techniques lead to generalized and less conservative stability condition that guarantee the wide stability region. The delay-dependent stability condition is formulated, in which the restriction of the derivative of the time-varying delay is removed. Our results can be specialized to several cases including those studied extensively in the literature. Finally, numerical examples are given to demonstrate the effectiveness and conservativeness of our results.     

Stability analysis of uncertain fuzzy Hopfield neural networks with time delays

In this paper, the global stability problem of uncertain Takagi–Sugeno (T–S) fuzzy Hopfield neural networks with time delays (TSFHNNs) is considered. A novel LMI-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of TSFHNNs. Here, we choose a generalized Lyapunov functional and introduce a parameterized model transformation with free weighting matrices to it, in order to obtain generalized stability region. In fact, these techniques lead to generalized and less conservative stability condition that guarantee the wide stability region. The proposed stability conditions are demonstrated with four numerical examples. Comparison with other stability conditions in the literature shows our conditions are the more powerful ones to guarantee the widest stability region.     

Polytopes passing through circles

Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.     

Stability properties for Hopfield neural networks with delays and impulsive perturbations

In this paper, we consider the uniform asymptotic stability, global asymptotic stability and global exponential stability of the equilibrium point of Hopfield neural networks with delays and impulsive perturbation. Some new stability criteria for such system are derived by using the Lyapunov functional method and the linear matrix inequality approach. The results are related to the size of delays and impulses. Our results are less restrictive and conservative than that given in some earlier references. Finally, two numerical examples showing the effectiveness of the present criteria are given.     

Global dynamics of a nonlocal population model with age structure in a bounded domain:A non-monotone case

We study the global dynamics of a nonlocal population model with age structure in a bounded domain. We mainly concern with the case where the birth rate decreases as the mature population size become large. The analysis is rather subtle and it is inadequate to apply the powerful theory of monotone dynamical systems. By using the method of super-sub solutions, combined with the careful analysis of the kernel function in the nonlocal term, we prove nonexistence, existence and uniqueness of positive steady states of the model.Moreover, due to the mature individuals do not diffuse, the solution semiflow to the model is not compact. To overcome the difficulty of non-compactness in describing the global asymptotic stability of the unique positive steady state, we first establish an appropriate comparison principle. With the help of the comparison principle,we can employ the theory of dissipative systems to obtain the global asymptotic stability of the unique positive steady state. The main results are illustrated with the nonlocal Nicholson's blowflies equation and the nonlocal Mackey-Glass equation.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.