Dynamics of a discrete food-limited population model with time delay

In this paper, we consider a discrete food-limited population model with time delay. Firstly, the stability of the equilibrium of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark-Sacker bifurcations occur when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction of the Neimark-Sacker bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are given to verify the theoretical analysis.     

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.     

Positive periodic solution for a two-species ratio-dependent predator-prey system with time delay and impulse

A two-species ratio-dependent predator-prey model with time delay and impulse is investigated. By using the continuation theorem of coincidence degree theory, the existence of a positive periodic solution for this system is established.     

Periodic solutions in periodic state-dependent delay equations and population models

Sufficient and realistic conditions are obtained for the existence of positive periodic solutions in periodic equations with state-dependent delay. The method involves the application of the coincidence degree theorem and estimations of uniform upper bounds on solutions. Applications of these results to some population models are presented. These application results indicate that seasonal effects on population models often lead to synchronous solutions. In addition, we may conclude that when both seasonality and time delay are present and deserve consideration, the seasonality is often the generating force for the often observed oscillatory behavior in population densities.

     

Permanence and extinction for a nonautonomous SIRS epidemic model with time delay

In this paper, a nonautonomous SIRS epidemic model with time delay is studied. We introduce some new threshold values _R∗$R_{\ast }$ and R^∗$R^{\ast }$ and further obtain the disease will be permanent when _R∗>1$R_{\ast }>1$ and the disease extinct when R^∗<1$R^{\ast }<1$. Using the method of Liapunov functional, some sufficient conditions are derived for the global attractivity of the system. The known results are extended.     

Stability of singularly perturbed switched systems with time delay and impulsive effects

This paper studies the stability properties of singularly perturbed switched systems with time delay and impulsive effects. Such systems are assumed to consist of both unstable and stable subsystems. By using the multiple Lyapunov functions technique and the dwell time approach, some stability criteria are established. Our results show that impulses do contribute in order to obtain stability properties even when the system consists of only unstable subsystems. Numerical examples are presented to verify our theoretical results.     

一类基于比率的且具有收获率和时滞的捕食系统的周期解

研究一类基于比率且具有收获和时滞的捕食系统．证明了系统正周期解的存在性,并通过构造适当的Lyapunov泛函,给出了正周期解全局稳定的充分条件．     

Global behavior and permanence of SIRS epidemic model with time delay

In this paper an autonomous SIRS epidemic model with time delay is studied. The basic reproductive number R₀ is obtained which determines whether the disease is extinct or not. When the basic reproductive number is greater than 1, it is proved that the disease is permanent in the population, and explicit formula are obtained by which the eventual lower bound of the fraction of infectious individuals can be computed. Throughout the total paper, we mainly use the technique of Lyapunov functional to establish the global stability of the infection-free equilibrium and the local stability of the endemic equilibrium but need another sufficient condition.     

Asymptotic properties of a HIV-1 infection model with time delay

Based on some important biological meanings, a class of more general HIV-1 infection models with time delay is proposed in the paper. In the HIV-1 infection model, time delay is used to describe the time between infection of uninfected target cells and the emission of viral particles on a cellular level as proposed by Herz et al. [A.V.M. Herz, S. Bonhoeffer, R.M. Anderson, R.M. May, M.A. Nowak, Viral dynamics in vivo: Limitations on estimates of intracellular delay and virus decay, Proc. Natl. Acad. Sci. USA 93 (1996) 7247-7251]. Then, the effect of time delay on stability of the equilibria of the HIV-1 infection model has been studied and sufficient criteria for local asymptotic stability of the infected equilibrium and global asymptotic stability of the viral free equilibrium are given.     

A nonlinear AIDS epidemic model with screening and time delay

A nonlinear mathematical model to study the effect of time delay in the recruitment of infected persons on the transmission dynamics of HIV/AIDS is proposed and analyzed. In modeling the dynamics, the population is divided into four subclasses: the susceptibles, the HIV positives or infectives that do not know they are infected, the HIV positives that know they are infected and the AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives. The model is analyzed using stability theory of delay differential equations. Both the disease-free and the endemic equilibria are found and their stability is investigated. It is shown that the introduction of time delay in the model has a destabilizing effect on the system and periodic solutions can arise by Hopf bifurcation. Numerical simulations are also carried out to investigate the influence of key parameters on the spread of the disease, to support the analytical conclusion and to illustrate possible behavioral scenario of the model.     

Optimal harvesting for a stochastic logistic model with S-type distributed time delay

In this paper, the optimization problem of harvesting for a stochastic logistic model with S-type distributed time delay (which contains both discrete time delay and continuously distributed time delay) is studied by using ergodic method.     

Positive periodic solutions for impulsive predator-prey model with dispersion and time delays

In this paper, we study the existence and global attractivity of positive periodic solutions for impulsive predator-prey systems with dispersion and time delays. By using coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solutions, and by means of a suitable Lyapunov functional, the uniqueness and global attractivity of positive periodic solution is presented. Some known results subject to the underlying systems without impulses are improved and generalized.     

Properties of stability and Hopf bifurcation for a HIV infection model with time delay

In this paper, we consider the classical mathematical model with saturation response of the infection rate and time delay. By stability analysis we obtain sufficient conditions for the global stability of the infection-free steady state and the permanence of the infected steady state. Numerical simulations are carried out to explain the mathematical conclusions.     

The principal eigenvalue for periodic nonlocal dispersal systems with time delay

The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic nonlocal dispersal cooperative system with time delay. Then we apply it to a Nicholson's blowflies population model and obtain a threshold type result on its global dynamics.     

Global dynamics of a discretized SIRS epidemic model with time delay

We derive a discretized SIRS epidemic model with time delay by applying a nonstandard finite difference scheme. Sufficient conditions for the global dynamics of the solution are obtained by improvements in discretization and applying proofs for continuous epidemic models. These conditions for our discretized model are the same as for the original continuous model.     

Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay

In this paper, we investigate the permanence of an SIR epidemic model with a density-dependent birth rate and a distributed time delay. We first consider the attractivity of the disease-free equilibrium and then show that for any time delay, the delayed SIR epidemic model is permanent if and only if an endemic equilibrium exists. Numerical examples are given to illustrate the theoretical analysis. The results obtained are also compared with those from the analog system with a discrete time delay.     

Global stabilization of switched control systems with time delay

In this paper, the stabilization problem of switched control systems with time delay is investigated for both linear and nonlinear cases. First, a new global stabilizability concept with respect to state feedback and switching law is given. Then, based on multiple Lyapunov functions and delay inequalities, the state feedback controller and the switching law are devised to make sure that the resulting closed-loop switched control systems with time delay are globally asymptotically stable and exponentially stable.     

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.