Suitable Liapunov Function Research Articles

Dynamical Study of a Vector Host Epidemic Model With Non-Monotone Incidence

Epidemics of vector borne disease studied in this paper when disease incidence rate is non- monotone. Non monotonic behaviors of infected hosts as well as infected vectors are studied for influence of increase in disease on susceptible host population and effect of repellents and insecticides on vectors, respectively. Suitable Liapunov functions are constructed to discuss the global asymptotic stabilities at the equilibrium points. Results are verified by conducting numerical simulation. Mechanism of disease spread by vectors and hosts during the epidemic can be better understood with the help of this model.

Global properties of vector-host disease models with time delays.

Since there exist extrinsic and intrinsic incubation periods of pathogens in the feedback interactions between the vectors and hosts, it is necessary to consider the incubation delays in vector-host disease transmission dynamics. In this paper, we propose vector-host disease models with two time delays, one describing the incubation period in the vector population and another representing the incubation period in the host population. Both distributed and discrete delays are used. By constructing suitable Liapunov functions, we obtain sufficient conditions for the global stability of the endemic equilibria of these models. The analytic results reveal that the global dynamics of such vector-host disease models with time delays are completely determined by the basic reproduction number. Some specific cases with discrete delay are studied and the corresponding results are improved.

Global Stability of Almost Periodic Solution of a Discrete Multispecies Gilpin-Ayala Mutualism System

This paper discusses a discrete multispecies Gilpin-Ayala mutualism system. We first study the permanence and global attractivity of the system. Assume that the coefficients in the system are almost periodic sequences, we obtain the sufficient conditions for the existence of a unique almost periodic solution which is globally attractive and uniformly asymptotically stable by constructing a suitable Liapunov function, respectively. Two examples together with numerical simulation indicate the feasibility of the main results.

The abstract Cauchy problem for dissipative operators with respect to metric-like functionals

This paper is devoted to a characterization of semigroups of Lipschitz operators on a closed subset D of a Banach space X and the abstract Cauchy problem for an operator A in X satisfying the following condition: There exists a proper lower semicontinuous functional φ from X into [0,∞] such that the effective domain of φ is D(A) and such that limn→∞⁡Axn=Ax in X for any x∈D(A) and any sequence {xn} in D(A) satisfying two conditions limn→∞⁡xn=x in X and limsupn→∞φ(xn)≤φ(x). The main result asserts that a semigroup of Lipschitz operators on D can be generated by an operator A satisfying the above-mentioned condition, a dissipative condition with respect to a metric-like functional and a subtangential condition. The Kirchhoff equations with acoustic boundary conditions are solved by the method based on the abstract result with the construction of suitable Liapunov functionals and the use of a metric-like functional on a suitable set.

Non-asymptotic stability and integral stability trough a reduction principle

This paper concerns the analysis of transferring stability properties from an invariant manifold to the whole space for an ordinary differential system. In previous papers we already treated this problem in the case of asymptotic and total stability. Here we deal with the case of non-asymptotic stability. We generalize to differential systems depending on time a reduction principle (Kelley in J Math Anal Appl 18:336–344, 1967; Pliss in Izv Akad Nauk SSSR Mat Ser 28:1297–1324, 1964) relative to autonomous systems. Our procedure is very different from the fixed point theorem argument used in Kelley (J Math Anal Appl 18:336–344, 1967), and it is based on the use of a suitable Liapunov function. Some results concerning integral stability are also given.

Existence and Exponential Stability of Multiple Periodic Solutions for a Multidirectional Associative Memory Neural Network

The paper proposes several mathematical models of the multidirectional associative memory (MAM) neural network by analyzing its structure. A model of MAM with distributed delays is studied. Under some new assumptions on activation functions, $${2^{n_0[m/2]}}$$ invariant subsets of MAM are constructed. Then the existence and the exponential stability of $${2^{n_0[m/2]}}$$ periodic solutions located on invariant subsets are obtained by constructing a suitable Liapunov function and a Poincare mapping. An estimating method of the exponential convergence rate is given. The obtained results are new to MAM neural networks. An example is given to illustrate the effectiveness of the results.

On the dynamical behavior of some algal allelopathic competitions in chemostat-like environment

The dynamics of some algal allelopathic competitions taking place in a laboratory chemostat-like environment are analyzed. Our main results concern the global asymptotic stability of some biologically meaningful steady-state solutions and have been obtained by means of limiting systems theory and construction of suitable Liapunov functions. In particular one of these results completely explains the outcome of several experiments recently performed and illustrated in Fergola et al. (Ecol Model 208:205–214, 2007) on the competition between C. vulgaris and P. subcapitata. Numerical simulations confirm the analytical results and show that saddle-node bifurcation phenomena can appear.

Exponential stability of periodic neural networks with impulsive effects and time-varying delays

By employing Young inequality and constructing suitable Liapunov functions, we investigate the existence and globally exponential stability of periodic neural networks with impulses and time-varying delays. The results extend and improve some earlier ones [1,5,12]. An illustrative example and simulations are given to show the validity of the main results.

Periodic solution and global stability for a nonautonomous competitive Lotka–Volterra diffusion system

The periodic solution and global stability for a nonautonomous competitive Lotka–Volterra diffusion system is considered in this paper. By using of Brouwer fixed point theorem and constructing a suitable Liapunov function, under some appropriate conditions, the system has a unique periodic solution which is globally stable.

A new criterion to global exponential periodicity for discrete-time BAM neural network with infinite delays

The discrete-time bidirectional associative memory neural network with periodic coefficients and infinite delays is studied. And not by employing the continuation theorem of coincidence degree theory as other literatures, but by constructing suitable Liapunov function, using fixed point theorem and some analysis techniques, a sufficient criterion is obtained which ensures the existence and global exponential stability of periodic solution for the type of discrete-time BAM neural network. The obtained result is less restrictive to the BAM neural networks than previously known criteria. Furthermore, it can be applied to the BAM neural network which signal transfer functions are neither bounded nor differentiable. In addition, an example and its numerical simulation are given to illustrate the effectiveness of the obtained result.

Global Attractivity of Positive Periodic Solutions of Delay Differential Equations with Feedback Control

By constructing suitable Liapunov functionals and estimating uniform upper and lower bounds of solutions, sufficient conditions are obtained for the global attractivity of positive periodic solutions of the delay differential system with feedback controldy/dt=y(t)F(t,y(t−τ1(t)),…,y(t−τn(t)),u(t−δ(t))),du/dt=−η(t)u(t)+a(t)y(t−σ(t)). When these results are applied to the periodic logistic equation with several delays and feedback control, some new results are obtained.

Stability with respect to partial variables for Birkhoff systems

In this paper, the stability with respect to partial variables for the Birkhoff system is studied. By transplanting the results of the partial stability for general systems to the Birkhoff system and constructing a suitable Liapunov function, the partial stability of the system can be achieved. Finally, two examples are given to illustrate the application of the results.

Global stability and asymptotically periodic solution for nonautonomous cooperative Lotka–Volterra diffusion system

Two asymptotically cooperative Lotka–Volterra populations in the two-patch-system with diffusion were considered in this paper. Each population could diffuse independently and discretely between its intrapatch and interpatch. By means of constructing a suitable Liapunov function, we obtained that the systems had a unique asymptotically periodic solution which was globally asymptotically stable.

LONG-TIME PROPERTIES OF PREY-PREDATOR SYSTEM WITH CROSS-DIFFUSION

Using calculus inequalities and embedding theorems in <TEX>$R^1$</TEX>, we establish <TEX>$W^1_2$</TEX>-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) <TEX>$d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$</TEX>, and (ii) <TEX>$0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$</TEX>. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when <TEX>$t\;{\to}\;{\infty}$</TEX> via suitable Liapunov functionals.

Asymptotic behaviour of the energy for electromagnetic systems with memory

AbstractSome linear evolution problems arising in the theory of hereditary electromagnetism are considered here. Making use of suitable Liapunov functionals, existence of solutions as well as asymptotic behaviour, are determined for rigid conductors with electric memory. In particular, we show the polynomially decay of the solutions, when the memory kernel decays exponentially or polynomially. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Global analysis of some epidemic models with general contact rate and constant immigration

An epidemic models of SIR type and SIRS type with general contact rate and constant immigration of each class were discussed by means of theory of limit system and suitable Liapunov functions. In the absence of input of infectious individuals, the threshold of existence of endemic equilibrium is found. For the disease-free equilibrium and the endemic equilibrium of corresponding SIR model, the sufficient and necessary conditions of global asymptotical stabilities are all obtained. For corresponding SIRS model, the sufficient conditions of global asymptotical stabilities of the disease-free equilibrium and the endemic equilibrium are obtained. In the existence of input of infectious individuals, the models have no disease-free equilibrium. For corresponding SIR model, the endemic equilibrium is globally asymptotically stable; for corresponding SIRS model, the sufficient conditions of global asymptotical stability of the endemic equilibrium are obtained.

Asymptotic behavior of the energy for a class of weakly dissipative second-order systems with memory

A class of second-order abstract systems with memory and Dirichlet boundary conditions is investigated. By suitable Liapunov functionals, existence of solutions as well as asymptotic behavior, are determined. In particular, when the memory kernel decays exponentially, the polynomially decay of the solutions is proved.

Global attractivity and global exponential stability for delayed Hopfield neural network models

Some global properties such as global attractivity and global exponential stability for delayed Hopfield neural networks model, under the weaker assumptions on nonlinear activation functions, are concerned. By constructing suitable Liapunov function, some simpler criteria for global attractivity and global exponential stability for Hopfield continuous neural networks with time delays are presented.

Existence and global attractivity of a positive periodic solution of a class of delay differential equation

The existence and the global attractivity of a positive periodic solution of the delay differential equationy(t)=y(t) F[t, y](t-τ1(t)),⋯,y(t−τn(t))] are studied by using some techniques of the Mawhin coincidence degree theory and the constructing suitable Liapunov functionals. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved.

A degenerate reaction diffusion system modelling an optical biosensor

We investigate qualitative properties of a global semiflow associated with a degenerate reaction diffusion system with nonlinear boundary conditions. The system models an optical biosensor. By defining a suitable Liapunov function it is possible to give a complete description of the long time behaviour of the solutions. Indeed, we are able to prove that positive solutions converge to a single spatially homogeneous equilibrium state.