Global dynamics behaviors for a nonautonomous Lotka–Volterra almost periodic dispersal system with delays

A nonautonomous Lotka–Volterra dispersal system with continuous delays and discrete delays is considered. By using a comparison theorem and delay differential equation basic theory, we obtain sufficient conditions for the permanence of the population in every patch. By constructing a suitable Lyapunov functional, we prove that the system is globally asymptotically stable under some appropriate conditions. Using almost periodic functional hull theory, we get sufficient conditions for the existence, uniqueness and globally asymptotical stability for an almost periodic solution. This implies that the population in every patch exhibits stable almost periodic fluctuation. Furthermore, the results show that the permanence and global stability of system, and the existence and uniqueness of a positive almost periodic solution, depend on the delay; then we call it “profitless”.     

具有Holling功能反应的非自治捕食系统的持续性和周期解

研究了具有一般的Holling功能反应函数,且种群之间既有捕食关系,又有竞争关系的三种群混合模型,得到了该系统惟一存在全局渐近稳定正周期解的条件,推广了已有结论.     

Periodic solution of a chemostat model with variable yield and impulsive state feedback control

In this paper, a chemostat model with variable yield and impulsive state feedback control is considered. We obtain sufficient conditions of the globally asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable.     

脆弱斑块中植物种子的脉冲扩散

考虑了一类脆弱斑块中植物种子的脉冲漂移模型,得到了系统永久持续生存性.在此基础上,利用单调凸算子理论,得到了系统唯一全局渐进稳定的周期解.数值模拟也表明脉冲扩散能挽救脆弱斑块中的植物种群.     

Globally asymptotical stability and periodicity for a nonautonomous two-species system with diffusion and impulses

Existence and globally asymptotical stability of positive periodic solutions for a nonautonomous two-species competition system with diffusion and impulses are studied in this paper. By employing Mawhin continuation theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one positive periodic solution, and by means of a suitable Lyapunov functional, the globally asymptotical stability of positive periodic solution is presented. Finally, an illustrative example and simulations are given to show the effectiveness of the main results.     

Some new results for a logistic almost periodic system with infinite delay and discrete delay

A nonautonomous logistic almost periodic system with infinite delay and discrete delay is considered. Our result shows that the system is globally asymptotically stable under the condition for the boundedness of the system. By using almost periodic functional Hull theory and new computational techniques, we show that the almost periodic system has a unique globally asymptotically stable strictly positive almost periodic solution under the condition for the boundedness of the system. Some recent results are improved, and an open question is answered.     

Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay

In this paper, we studied a non-autonomous predator-prey system with discrete time-delay, where there is epidemic disease in the predator. By using some techniques of the differential inequalities and delay differential inequalities, we proved that the system is permanent under some appropriate conditions. When all the coefficients of the system is periodic, we obtained the existence and global attractivity of the positive periodic solution by Mawhin’s continuation theorem and constructing a suitable Lyapunov functional. Furthermore, when the coefficients of the system are not absolutely periodic but almost periodic, sufficient conditions are also derived for the existence and asymptotic stability of the almost periodic solution.     

具有巢寄生行为和阶段结构的两种群模型分析

研究具有巢寄生行为的两种群模型,两个种群都具有阶段结构.利用微分方程比较定理得到该系统持续生存的充分性条件,进一步利用Liapunov函数方法,证明了在适当的条件下,周期解及概周期解的全局渐近稳定性.     

Dynamics of product inhibition on lactic acid fermentation

In this paper, a mathematical model for the lactic acid fermentation in membrane bioreactor is investigated. Firstly, continuous input substrate is taken. The existence and local stability of two equilibria are studied. According to Poincare-Bendixson theorem, we obtain the condition for the globally asymptotical stability of the equilibria. Secondly, using the Floquet’s theorem and small-amplitude perturbation method, we obtain the biomass-free periodic solution is locally stable if R₂ < 1. The permanent conditions of the system are also given. Finally, our findings are confirmed by means of numerical simulations.     

具有IV类功能反应的非自治捕食系统的持续性和周期解

本文讨论了一类具有HollingIV类功能反应的非自治捕食系统,利用Brouwer不动点定理和构造Liaponov函数的方法,得到该系统永久持续生存和存在唯一全局稳定的周期解的充分条件.     

一类具有脉冲效应的浮游生物模型的持久性以及概周期解

利用脉冲微分方程的对比定理以及李雅普诺夫函数法,我们研究了一类具有脉冲效应的浮游生物模型的持久性以及概周期解.文中所得结论改进了以往的研究成果.文中所用的研究方法可以用来研究其他带有脉冲的生物数学模型的持久性以及概周期解.最后,我们总结阐述了脉冲如何影响模型的持久性,概周期解以及一致渐进稳定性.     

Global attractivity and almost periodic solution of a discrete mutualism model with delays

In this paper, we consider an almost periodic discrete Lotka–Volterra mutualism model with delays. We first obtain the permanence and global attractivity of the system. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution, which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

一类Lotka-Volterra捕食-竞争扩散系统的概周期解

研究了一类带扩散项的n种群Lotka-volterra非自治捕食-竞争系统，应用Liapunov泛函方法得到系统持久生存和存在唯一全局渐近稳定正概周期解的新的充分条件，并举例说明定理的应用．     

Existence and stability of almost periodic solution for Cohen–Grossberg neural networks with variable coefficients

This paper is devoted to the existence and globally exponential stability of almost periodic solution for a class of Cohen–Grossberg neural networks with variable coefficients. By using Banach fixed point theorem and applying inequality technique, we give some sufficient conditions ensuring the existence and globally exponential stability of almost periodic solution. These results have important leading significance in designs and applications of Cohen–Grossberg neural networks. Finally, two examples with their numerical simulations are provided to show the correctness of our analysis.     

Existence and exponential stability of almost periodic solutions for a neutral multi-species Logarithmic population model

In this paper, we study the almost periodic solution for a neutral multi-species Logarithmic population model. By employing Banach’s fixed point theorem and using differential inequality technique, we present some sufficient conditions ensuring the existence, uniqueness and globally exponential stability of almost periodic solution for the model. The results obtained extend and improve the earlier publications. Finally, two examples are provided to show the correctness of our analysis.     

Uniqueness and exponential stability of almost periodic positive solution for Lasota‐Wazewska model with impulse and infinite delay

In this paper, impulsive Lasota‐Wazewska model with infinite delay is studied. By using fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of unique almost periodic positive solution. Particularly, we give iterative sequence, which converges to the almost periodic positive solution. Moreover, we investigate exponential stability of the almost periodic positive solution by Liapunov functional. Copyright © 2014 John Wiley & Sons, Ltd.     

An impulsive predator–prey system with modified Leslie–Gower and Holling type II schemes

An impulsive predator–prey system with modified Leslie–Gower and Holling-type II schemes is presented. By using the Floquet theory of impulsive equation and small amplitude perturbation method, the globally asymptotical stability of prey-free positive periodic solution and the permanence of system are discussed. The corresponding threshold conditions are obtained respectively. Finally, numerical simulations are given.     

Dynamical analysis of a new microbial pesticide model with the Monod growth rate

In this paper, a new mathematical model for the entomopathogenic nematode with the Monod growth rate is formulated. Firstly, continuous release of the entomopathogenic nematode is considered. The existence of limit cycles, the Hopf bifurcation and the stability of the periodic solution created by the bifurcation are proved. The sufficient conditions for the globally asymptotical stability of system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. By using the Floquet’s theorem and the small amplitude perturbations, we show that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations. Thus, we provide mathematical evidence on how to release the entomopathogenic nematode in order to control pests at acceptably low levels.     

比率型-捕食者-两竞争食饵模型的动力学行为

本文研究比率型非自治的捕食者 -食饵模型 .该系统是两个具有竞争关系的食饵种群被一个捕食种群捕食 .我们研究其动力学行为 ,包括持久性 ,全局渐近稳定性 ,周期解 ,概周期解的存在唯一性