Hopf Bifurcation and Stability Analysis for a Predator-prey Model with Time-delay

In this paper, a predator-prey model of three species is investigated, the necessary and sucient of the stable equilibrium point for this model is studied. Further, by introduc-ing a delay as a bifurcation parameter, it is found that Hopf bifurcation occurs when τ cross some critical values. And, the stability and direction of hopf bifurcation are determined by applying the normal form theory and center manifold theory. numerical simulation results are given to support the theoretical predictions. At last, the periodic solution of this system is computed.

Dynamical Analysis of Nonautonomous Trophic Cascade Chemostat Model with Regime Switching and Nonlinear Perturbations in a Polluted Environment

This paper investigates the stochastic dynamics of trophic cascade chemostat model perturbed by regime switching, Gaussian white noise and impulsive toxicant input. For the system with only white noise interference, sufficient conditions for stochastically ultimate boundedness and stochastically permanence are obtained, and we demonstrate that the stochastic system has at least one nontrivial positive periodic solution. For the system with Markov regime switching, sufficient conditions for extinction of the microorganisms are established. Then we prove the system is ergodic and has a stationary distribution. The results show that both impulsive toxins input and stochastic noise have great effects on the survival and extinction of the microorganisms. Finally, a series of numerical simulations are presented to illustrate the theoretical analysis.

Existence and finite-time stability of a unique almost periodic positive solution for fractional-order Lasota Wazewska red blood cell models

In this paper,we are concerned with a class of fractional-order Lasota-Wazewska red blood ccll modcls.By applying a fixed point theorem on a normal cone,we first obtain the sufficient conditions for the existence of a unique almost periodic positive solution of the considered models.Then,considering that all of the red blood cells in animals survive in a finite-time interval,we study the finite-time stability of the almost periodic positive solution by using some inequality techniques.Our results and methods of this paper are new.Finally,we give numerical examples to show the feasibility of the obtained results.

Existence and exponential stability of almost periodic positive solution for host-macroparasite difference model

This paper is concerned with a host-macroparasite difference model. By applying the con- traction mapping fixed point theorem, we prove the existence of unique almost periodic positive solution. Moreover, we investigate the exponential stability of Mmost periodic solution by means of Lyapunov functional.

Stability and Neimark-Sacker bifurcation analysis of a food-limited population model with a time delay

In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.

Stationary distribution and periodic solution of stochastic chemostat models with single-species growth on two nutrients

In this paper,we consider two chemostat models w th random perturbation,in which single species depends on two perfectly substitutable resources for growth.For the autonomous system,we first prove that the solution of the system is positive and global.Then we establish sufficient conditions for the existence of an ergodic stationary distribu tion by constructing appropriate Lyapunov functions-For the non-autonomous system,by using Mas'minskii theory on periodic Markov processes,we derive it admits a nontriv ial positive periodic solution.Finally,numerical simulations are carried out to illustrate our main results.

POSITIVE PERIODIC SOLUTION FOR A NONAUTONOMOUS LOGISTIC MODEL WITH LINEAR FEEDBACK REGULATION

A nonautonomous delayed logistic model with linear feedback regulation is proposed in this paper. Sufficient conditions are derived for the existence, uniqueness and global asymptotic stability of positive periodic solution of the model

BIFURCATION AND COMPLEXITY IN A RATIO-DEPENDENT PREDATOR-PREY CHEMOSTAT WITH PULSED INPUT

In this paper, a three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the discrete dynamical system determined by the stroboscopic map and Floquet theorem, an exact periodic solution with positive concentrations of substrate and predator in the absence of prey is obtained. When β is less than some critical value the boundary periodic solution （xs（t）, O, zs（t）） is locally stable, and when β is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period T the system undergoes a series of period-doubling bifurcation leading to chaos, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.

Cross-diffusion induced Turing instability of Hopf bifurcating periodic solutions in the reaction-diffusion enzyme reaction model

Aiming at the spatial pattern phenomenon in biochemical reactions,an enzyme-reaction Sporns-Seelig model with cross-diffusion is chosen as study object.Applying the central manifold theory,normal form method,local Hopf bifurcation theorem and perturbation theory,we study Turing instability of the spatially homogeneous Hopf bifurcation periodic solutions.At last,the theoretical results are verified by numerical simulations.

Hopf Bifurcation of a Nonresident Computer Virus Model with Delay

In this paper, a delayed nonresident computer virus model with graded infection rate is considered in which the following assumption is imposed: latent computers have lower infection ability than infectious computers. With the aid of the bifurcation theory, sufficient conditions for stability of the infected equilibrium of the model and existence of the Hopf bifurcation are established. In particular, explicit formulae which determine direction and stability of the Hopf bifurcation are derived by means of the normal form theory and the center manifold reduction for functional differential equations. Finally, a numerical example is given in order to show the feasibility of the obtained theoretical findings.

Local Hopf bifurcation and global existence of periodic solutions in TCP system

This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu （Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350（12）, 4799-4838 （1998））.

Hopf Bifurcation Analysis for a Delayed SIRS Epidemic Model with a Nonlinear Incidence Rate

This paper is concerned with a delayed SIRS epidemic model with a nonlinear incidence rate. The main results are given in terms of local stability and Hopf bifurcation. Sufficient conditions for the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained by regarding the time delay as the bifurcation parameter. Further,the properties of Hopf bifurcation such as the direction and stability are investigated by using the normal form theory and center manifold argument. Finally,some numerical simulations are presented to verify the theoretical analysis.

THE EXISTENCE,UNIQUENESS AND STABILITY OF POSITIVE PERIODIC SOLUTION FOR PERIODIC REACTION-DIFFUSION SYSTEM

The existence of periodic solutions for periodic reaction-diffusion systems with time delay by the periodic upper-lower solution method is investigated. Some methods for proving the uniqueness and the stability of the periodic solution are also given. Two examples are used to show how to use our methods.

Chemostat系统中的Hopf分支

讨论了一类微生物连续培养模型的 Hopf分支问题 .该模型的消耗率参数中含有一个关于营养基的二次函数 ,功能反应函数则为具有内代谢的 Monod类型 ,以更好地模拟实际问题 .利用 Friedrich方法得到了该系统存在 Hopf分支的条件 ,并判定了周期解的稳定性 .

OPTIMAL HARVESTING POLICY FOR INSHORE-OFFSHORE FISHERY MODEL WITH IMPULSIVE DIFFUSION

This article studies the inshore-offshore fishery model with impulsive diffusion. The existence and global asymptotic stability of both the trivial periodic solution and the positive periodic solution are obtained. The complexity of this system is also analyzed. Moreover, the optimal harvesting policy are given for the inshore subpopulation, which includes the maximum sustainable yield and the corresponding harvesting effort.

非均匀Chemostat竞争模型的共存态

本文讨论了一类质粒载体的微生物与质粒自由的微生物竞争的非均匀Chemostat模型.利用极值原理以及Hopf边界引理给出了正平衡解的先验估计,然后利用锥映象不动点指数理论、算子谱分析以及局部分歧理论得到了正平衡解存在的充分条件,最后运用线性算子的扰动理论和分歧解的稳定性理论判定了局部分歧解的稳定性.研究结果表明,当参数满足一定条件时,两竞争物种能够共存.

非均匀Chemostat竞争模型的周期解

讨论非均匀Chemostat竞争模型半平凡周期解的存在性、稳定性及其正周期解的存在性。通过运用抛物型方程比较原理、稳定性理论、极值原理以及Leray-Schauder度理论,证明了该系统半平凡周期解的存在性和稳定性,得到了该系统正周期解存在的充分条件。

具有季节影响的登革热病毒传播动力学分析

登革热是非常典型的虫媒传染病.考虑到季节的变化会影响蚊子的生存,因此提出了一类具有季节性影响的登革热病毒传播模型.首先利用下一代矩阵方法得到了基本再生数R_(0),其次进行适定性分析,判断该模型无病周期解的全局渐近稳定性以及持久性.

一类Chemostat捕食模型正周期解的存在性

讨论了一类Chemostat捕食模型在一定条件下正周期解的存在性问题.运用周期抛物型算子理论、Schauder估计和分歧理论得到了该模型正周期解存在的充分必要条件.

The Effect of State-Dependent Control for an SIRS Epidemic Model with Varying Total Population

Based on the mechanism of prevention and control of infectious disease, we propose, in this paper, an SIRS epidemic model with varying total population size and state-dependent control, where the fraction of susceptible individuals in population is as the detection threshold value. By the Poincaré map, theory of differential inequalities and differential equation geometry, the existence and orbital stability of the disease-free periodic solution are discussed. Theoretical results show that by state-dependent pulse vaccination we can make the proportion of infected individuals tend to zero, and control the transmission of disease in population.