Stability and bifurcation of a two-dimension discrete neural network model with multi-delays

A two-dimension discrete neural network model with multi-delays is obtained using Euler method. Furthermore, the linear stability of the model is studied. It is found that there exists Hopf bifurcations when the delay passes a sequence of critical values. Using the normal form method and the center manifold theorem, the explicit formulas which determine the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions are derived. Finally, computer simulations are performed to support the theoretical predictions.     

Hopf bifurcation and stability for a neural network model with mixed delays

A generalized model of the two-neuron network with mixed delays is studied. The main purpose of this paper is to explore the linear stability of the trivial solution and Hopf bifurcation of a two-neuron network with continuous and discrete delays. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solutions are also studied. Finally, the numerical simulations are given to illustrate the theoretical analysis.     

Dynamic behaviors of the periodic predator–prey system with distributed time delays and impulsive effect

In this paper, a periodic predator–prey system with distributed time delays and impulsive effect is investigated. By using the Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We improve some results in Guo and Chen (2009) [1].     

具传输时滞和变系数的模糊细胞神经网络的指数稳定性及周期解

介绍一类具传输时滞和变系数的模糊细胞神经网络(FCNN),通过使用线性矩阵不等式(LMI)和Lyapunov-Krasovskii泛函,研究它的周期解的存在性及全局指数稳定性,并获得一些充分条件。此外,给出一个实例说明结果是可行的。     

Almost Periodic Solutions of Nonautonomous Linear Difference Equations

Explicit almost periodic solutions are obtained for a class of almost periodic linear difference equations. The stability characteristics of the almost periodic solution are investigated. The results are applied to a nonautonomous hyperbolic difference equation modelling the dynamics of a single species population in temporally varying environments.     

Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect

In this paper, a predator–prey system which based on a modified version of the Leslie–Gower scheme and Holling-type II scheme with impulsive effect are investigated, where all the parameters of the system are time-dependent periodic functions. By using Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We use standard bifurcation theory to show the existence of nontrivial periodic solutions which arise near the semi-trivial periodic solution. As an application, we also examine some special case of the system to confirm our main results.     

Periodic solutions for a type of neutral system with variable parameters

In this paper, we firstly analyze some properties of the linear difference operator $[Ax](t)=x(t)-C(t)x(t-\tau),$ where $C(t)$ is a $n\times n$ matrix function, and then using Mawhin''s continuation theorem, a first-order neutral functional differential system is studied. Some new results on the existence and stability of periodic solutions are obtained. The results are related to the deviating arguments $\tau$ and $\mu$. Meanwhile, the approaches to estimate a $prior$ bounds of periodic solutions are different from the corresponding ones of the known literature.     

On the qualitative behavior of periodic solutions of differential systems

This article deals with the reflective function of differential systems. The obtained results are applied to studying the existence and stability of the periodic solutions of some linear and nonlinear periodic differential systems.     

Dynamics of prey threshold harvesting and refuge

The dynamics of a predator-prey model with continuous threshold prey harvesting and prey refuge is studied. One central question is how harvesting and refuge could directly affect the dynamics of the ecosystem, such as the stability properties of some coexistence equilibria and periodic solutions. Theoretical and numerical methods are used to investigate boundedness of solutions, existence of bionomic equilibria, as well as the existence and stability properties of equilibrium points and periodic solutions. Several bifurcations are also studied.     

Hopf bifurcation and stability for a delayed tri-neuron network model

A neural network model with three neurons and a single time delay is considered. Its linear stability is investigated and Hopf bifurcations are demonstrated by analyzing the corresponding characteristic equation. In particular, the explicit formulae determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are obtained by applying the normal form theory and the center manifold theorem. In order to illustrate our theoretical analysis, some numerical simulations are also included in the end.     

Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy

This paper is concerned with the existence and asymptotic behavior of periodic solutions for a periodic reaction diffusion system of a planktonic competition model under Dirichlet boundary conditions. The approach to the problem is by the method of upper and lower solutions and the bootstrap argument of Ahmad and Lazer. It is shown under certain conditions that this system has positive or semi-positive periodic solutions. A sufficient condition is obtained to ensure the stability and global attractivity of positive periodic solutions.     

两自由度非线性振动系统周期运动及其稳定性研究

运用Liapunov函数方法,对一类两自由度非线性振动系统周期运动及其稳定性进行了研究,得到了存在唯一渐近稳定的周期解的充分条件.     

Hopf bifurcation and analysis of equilibrium for a third-order differential equation in a model of competition

1. IntroductionOne of the more challenging aspects of mathematical biology is competition modelling.Although the mathematical idea is simple[1], this type of modelling is so difficult to carryout in any generality since there are so maily ways for a population to compete.The simplest form of competition is called eXPloitative competition. This occurs whenone or more populations compete for the same resources such as a common food supplyor a growth-limiting nutriellt. A simple example of this…     

On the global dynamic behaviour for a generalized haematopoiesis model with almost periodic coefficients and oscillating circulation loss rate

A generalized non‐linear nonautonomous model for the haematopoiesis (cell production) with several delays and an oscillating circulation loss rate is studied. We prove a fixed point theorem in abstract cones, from which different results on existence and uniqueness of positive almost periodic solutions are deduced. Moreover, some criteria are given to guarantee that the obtained positive almost periodic solution is globally exponentially stable.     

Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments

In this paper, a class of delayed predator-prey model of prey dispersal in two-patch environments is considered. By analyzing the associated characteristic transcendental equation, its linear stability is investigated and Hopf bifurcation is demonstrated. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulation for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

A periodic predator–prey-chain system with impulsive perturbation

A periodic predator–prey-chain system with impulsive effects is considered. By using the global results of Rabinowitz and standard techniques of bifurcation theory, the existence of its trivial, semi-trivial and nontrivial positive periodic solutions is obtained. It is shown that the nontrivial positive periodic solution for such a system may be bifurcated from an unstable semi-trivial periodic solution. Furthermore, the stability of these periodic solutions is studied.     

带有Leslie-Gower功能性反应的三维食物链模型研究

研究了离散时间上带有Leslie-Gower功能性反应的三维食物链模型.首先给出保证系统永久持续生成的条件,由于系统系数的周期性,正周期解是存在的,最后通过在正的周期解领域内线性化系统,并通过构造Lyapunov函数,我们得出了保证系统正周期解全局稳定的条件.     

具有时滞及B-D功能反应的食物链系统周期解的存在性与全局渐近稳定性

研究了具有时滞和Beddington-DeAngelis功能反应的一类食物链系统,通过运用Mawhin重合度理论中的延拓定理,得到了系统至少存在一个周期解的充分条件,并通过构造Lyapunov泛函的方法得到了系统周期解的全局稳定性.     

Stability and bifurcation analysis of a nutrient-phytoplankton model with time delay

We proposed a nutrient-phytoplankton interaction model with a discrete and distributed time delay to provide a better understanding of phytoplankton growth dynamics and nutrient-phytoplankton oscillations induced by delay. Standard linear analysis indicated that delay can induce instability of a positive equilibrium via Hopf bifurcation. We derived the conditions guaranteeing the existence of Hopf bifurcation and tracked its direction and the stability of the bifurcating periodic solutions. We also obtained the sufficient conditions for the global asymptotic stability of the unique positive steady state. Numerical analysis in the fully nonlinear regime showed that the stability of the positive equilibrium is sensitive to changes in delay values under select conditions. Numerical results were consistent with results predicted by linear analysis.     

Marchuk模型周期解和概周期解的存在性与稳定性(英文)

本文讨论具有时滞和扩散的Marchuk模型.根据模型参数求出了模型方程的上、下解,利用单调方法和Liapunov函数分别证明了周期解和概周期解的存在性及其稳定性.