DYNAMICS OF A NONLINEAR NON-AUTONOMOUS n-PATCHES PREDATOR-PREY DISPERSION-DELAY MODEL

In this paper, a nonlinear nonautonomous predator-prey dispersion model with continuous distributed delay is studied, where all parameters are time-dependent. In this system consisting of n-patches the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. It is proved that the system is uniformly persistent under any dispersion rate effect. Furthermore~ some sufficient conditions are established for the existence of a unique almost periodic solution of the system. The example shows that the criteria in the paper are new, general and easily verifiable.

The Lotka-Volterra Competition System with Discrete and Distributed Time Delays

In this paper, the Lotka-Volterra competition system with discrete and distributed time delays is considered. By analyzing the characteristic equation of the linearized system, the local asymptotic stability of the positive equilibrium is investigated. Moreover, we discover the delays don＇t effect the stability of the equilibrium in the delay system. Finally, we can conclude that the positive equilibrium is global asymptotically stable in the delay system.

Mean Square Stability of the Composite Milstein Method for Nonlinear Stochastic Differential Delay Equations

In this paper, we construct a composite Milstein method for nonlinear stochastic differential delay equations. Then we analyze the mean square stability for this method and obtain the step size condition under which the composite Milstein method is mean square stable. Moreover, we get the step size condition under which the composite Milstein method is global mean square stable. A nonlinear test stochastic differential delay equation is given for numerical tests. The results of numerical tests verify the theoretical results proposed.

Discrete Time Optimal Adaptive Control for Linear Stochastic Systems

The least-squares （LS） algorithm has been used for system modeling for a long time. Without any excitation conditions, only the convergence rate of the common LS algorithm can be obtained. This paper analyzed the weighted least-squares （WLS） algorithm and described the good properties of the WLS algorithm. The WLS algorithm was then used for adaptive control of linear stochastic systems to show that the linear closed-loop system was globally stable and that the system identification was consistent. Compared to the past optimal adaptive controller, this controller does not impose restricted conditions on the coefficients of the system, such as knowing the first coefficient before the controller. Without any persistent excitation conditions, the analysis shows that, with the regulation of the adaptive control, the closed-loop system was globally stable and the adaptive controller converged to the one-step-ahead optimal controller in some sense.

A Mathematical Model with Delays for Schistosomiasis Japonicum Transmission

A dynamic model of schistosoma japonicum transmission is presented that incorporates effects of the prepatent periods of the different stages of schistosoma into Baxbour＇s model. The model consists of four delay differential equations. Stability of the disease free equilibrium and the existence of an endemic equilibrium for this model are stated in terms of a key threshold parameter. The study of dynamics for the model shows that the endemic equilibrium is globally stable in an open region if it exists and there is no delays, and for some nonzero delays the endemic equilibrium undergoes Hopf bifurcation and a periodic orbit emerges. Some numerical results are provided to support the theoretic results in this paper. These results suggest that prepatent periods in infection affect the prevalence of schistosomiasis, and it is an effective strategy on schistosomiasis control to lengthen in prepatent period on infected definitive hosts by drug treatment （or lengthen in prepatent period on infected intermediate snails by lower water temperature）.

ALMOST PERIODIC SOLUTION OF TWO－SPECIES PATCHTYPE NONAUTONOMOUS DIFFUSIVE LOTKA－VOLTERRA COMPETITIVE MODELS

In this paper, we consider two-species n-patch almost periodic competitive systemswith diffusion, where one species can diffuse between any two of n patches, but the otheris confined to one of the patches and cannot diffuse. We prove that the systems can have aunique positive almost periodic solution, which is globally uniformly asymptotically stableunder some appropriate conditions. In particular, if the system is a periodic system of period ω, it can have a positive globally uniformly asymptotically stable periodic solution ofperiod ω, which is a generalization of Theorem 4 in paper [6].

Global stability of a SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment strategy

We consider a SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment.The treatment rate has at most a finite number of jump discontinuities in every compact interval.By using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis,the basic reproductive number Ro is proved to be a sharp threshold value which completely determines the dynamics of the model.If Ro<1,then there exists a disease-free equilibrium which is globally stable.If Ro>1,the disease-free equilibrium becomes unstable and there exists an endemic equilibrium which is globally stable.We discuss that the disease will die out in a finite time which is impossible for the corresponding SEIR model with continuous treatment.Furthermore,the numerical simulations indicate that strengthening treatment measure after infective individuals reach some level is beneficial to disease control.

Bifurcation analysis and optimal control of COVID-19 with exogenous reinfection and media coverages

In this paper,a SEIR epidemic model related to media coverage and exogenous reinfections is established to explore the transmission dynamics of COVID-19.The basic reproduction number is calculated using the next generation matrix method.First,the existence of equilibrium points is investigated,and different kinds of equilibrium points indicate that the disease may disappear,or exist that result in different quantity of susceptible individuals,pre-symptomatic infected individuals and symptomatic infected individuals.The stability of the equilibria is discussed by a geometric approach,and it is found that controlling reproduction number to be lower than 1 is not suficient for eradication of COVID-19.Second,transcritical bifurcation is explored,and it is found that improving the ratio of exogenous reinfection may lead to backward bifurcation under poor medical conditions,which indicates that two endemic equilibrium points appear.Third,to investigate the infuence of parameters on the basic reproduction,sensitivity analysis is done to choose relatively sensitive parameters,and the parameters for treatment and media coverage are selected.An optimal control model is established to balance the treatment and media awareness.By exploring the existence and the uniqueness of the optimal control solution,the optimal control strategies are given.Finally,we run numerical simulations to verify the theoretical analysis on actual data of China,and the data from the four different states of India is used for forecasting the situation of infected individuals in a short period.It is found by the simulation that the co-function of treatment and media coverage results in the reduced number of infectious individuals.

EXISTENCE AND STABILITY OF PERIODIC SOLUTIONS OF DELAYED BAM NEURAL NETWORKS WITH PERIODIC COEFFICIENTS

In this paper, we study the BAM neural networks with variable coefficients and delays. By using the Banach fixed point theorem and constructing suitable Lyapunov function, we obtain some sufficient conditions ensuring the existence, uniqueness and global stability of periodic solution. These results are helpful to design global exponential stable BAM networks and periodic oscillatory BAM networks.

SOME PROPERTIES OF A KIND OF DIFFERENCE EQUATION

This paper proves that difference equation(1.1)is permanent and every solution of equationoscillates about the eauilibrium JSufficient conditions of which the equilibrium x is blobally asymptotically stable are alsoobtained.

Study of two-nutrient and two-micro-organism chemostat model with pulsed input in a polluted environment

In this paper, a model of Beddington-DeAngelies chemostat involving two species of micro-organism competing for two perfectly complementary growth-limiting nutrients and pulsed input of toxicant in the polluted environment was studied. Using Floquet theory and small amplitude perturbation method, a conclusion was that there exists twomicro-organism eradication periodic solution and which is global asymptotical stability. At the same time, the condition of the permanence for system was obtained. From the biological point of view, the method for protecting species is to improve the amount of impulsive period, and control the amount of toxicant input to the chemostat. Finally, our results are illustrated by numerical simulations.

Robust H_(∞) Control of Switched Nonlinear Systems Under Sampled Data

This paper investigates the globally asymptotically stable and L_(2)-gain of robust H_(∞)control for switched nonlinear systems under sampled data.By considering the relationship between the sampling period and the dwell time,the non-switching and one switching are discussed in the sampling interval,respectively.Firstly,a state feedback sampled-data controller is constructed by the back-stepping method,and the switching converts to asynchronous switching if it happens within the sampling interval.Then,under the limiting conditions of the sampling period,which are obtained by the average dwell time method,the closed-loop system is globally asymptotically stable and has L_(2)-gain.Finally,two numerical examples are provided to demonstrate the effectiveness of the proposed method.

Dynamical analysis for delayed virus infection models with cell-to-cell transmission and density-dependent diffusion

In this paper,certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated.For the viral model with a single strain,we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number R_(0) and structuring proper Lyapunov functional.Moreover,we found that the infection-free equilibrium is globally asymptotically stable if R_(0)<1,and the infection equilibrium is globally asymptotically stable if R_(0)>1.For the multi-strain model,we found that all viral strains coexist if the corresponding basic reproductive number R^(e)_(j)>1,while virus will extinct if R^(e)_(j)<1.As a result,we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions.

DynamicalSystem for Computing the Elliptic Boundary Problem

A dynamical system for computing the elliptic boundary problem is presented.It is shown that the proposed system is globally asymptotically stable.The system can be simulated by a neural network.Simulation results are valid and satisfactory.

An SIS Epidemic Model with Stage Structure and a Delay

A disease transmission model of SIS type with stage structure and a delay is formulated. Stability of the disease free equilibrium, and existence, uniqueness, and stability of an endemic equilibrium, are investigated for the model. The stability results arc stated in terms of a key threshold parameter. The effects of stage structure and time delay on dynamical behavior of the infectious disease are analyzed. It is shown that stage structure has no effect on the epidemic model and Hopf bifurcation can occur as the time delay increases.

Global properties within-host model for an age-structured with Crowley-Martin functional response

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley-Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotieal property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.

THE VOLTERRA MODEL FOR THREE SPACIES PREDATOR-PREY SYSTEMS IN LOOPS

We consider the three species predator-prey model with the same intrinsic growth rates, where species 3 feeds on species 2, species 2 feeds on species 1, species 1 feeds on species 3. An open question raised by Nishan Krikorian is answered: We obtain the necessary and sufficient conditions for all the orbits to be unbounded. We also obtain the necessary and sufficient conditions for the positive equilibrium to be globally stable. It is shown that there exists a family of neutrally stable periodic orbits, in which we extend Darboux method to three-species models for the first time.