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双语推荐：平衡的稳定性条件

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运用常微分方程定性与稳定性理论研究了一类带收获和毒素项捕食扩散系统的局部稳定性.找出了正平衡点存在的条件,并得出当系统满足一定的条件时,系统的正平衡点是局部渐近稳定的,补充和完善了前人的结果.
The local behavior of a class of a predator-prey diffusive model with gain and toxicity is studied by using qualitative and stability theory of differential equation in this paper. The conditions that the positive equilibrium exists are found. That the positive equilibrium of the system is local asymptotical stable is given when certain conditions are satisfied in the model,then the conclusion is renewed.

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文章考虑捕食者具有阶段结构的时滞捕食模型，通过研究模型雅克比矩阵的特征方程得到所有非负平衡点的局部稳定性，并得到 Hopf分支存在性的充分条件；利用比较原理并构造Lyapunov函数得到边界平衡点的全局稳定性；运用迭代方法与比较原理得到正平衡点的全局渐近稳定的充分条件。
A delayed predator‐prey model with stage structure for predator is considered .By studying the characteristic equation of the Jacobian matrix ,the local stability of all the non‐negative equilibria is obtained as well as the sufficient conditions for the existence of Hopf bifurcation .And by using comparison arguments and Lyapunov functions ,the global stability of the boundary equilibrium is in‐vestigated .Moreover ,the sufficient conditions of the global asymptotic stability of the positive equi‐librium are derived by using iteration technique and comparison arguments .

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从吉布斯函数判据出发，推导出系统的平衡条件和平衡的稳定性条件，并应用吉布斯函数判据讨论了范德瓦尔斯等温气液相变．
On the basis of Gibbs function criterion, the authors of this paper deduced the equilibrium condition and the balance stable condition of homogeneous system, and discussed the van der Waals isothermal gas liquid phase transition by means of Gibbs function criterion.

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许多传染病流行时间远超过物种的生命周期，基于此我们建立和研究了一类具logis-tic出生率的SIS传染病模型。利用微分方程稳定性理论，研究了平衡点的存在性及其稳定性的条件，并用Dulac函数证明了闭轨线和奇异闭轨线的不存在性，证明了各平衡点稳定的条件。
The transmission time of many infectious diseases is longer than the lifetime of spe-cies .In this paper ,we consider an SIS epidemic model with the logistic birth rate .By using the qualitative theory of ordinary differential equations and Dulac functions ,we obtain the ex-istence of equilibrium and stability conditions and the nonexistence of closed trajectory and singular closed trajectory .In addition ,we prove the conditional factors for the stability of e-quilibrium .

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研究一类具有时滞和捕食者、食饵均具有阶段结构的捕食模型的全局稳定性.首先通过分析特征方程,运用Hurwitz判定定理,讨论了该模型非负平衡点的局部稳定性,并得到了Hopf分支存在的充分条件;其次运用单调迭代方法和比较定理,讨论了该模型的非负平衡点的全局稳定性,从而得到了保证该生态系统永久持续生存与灭绝的充分条件.
A predator-prey model with time delay and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each feasible equilibria of the system is discussed. Further, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium. By using an iteration technique and comparison arguments, su?cient conditions are obtained for the global stability of each feasible equilibria.

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分析一类带有边界控制和迁移的非线性尺度结构种群模型。讨论了平衡态的存在唯一性,给出了平衡态的表达式,导出了系统的特征方程,获得了平衡解的稳定性条件。
This paper is concerned with a class of nonlinear size-structured population model with boundary control and immigration. We discuss the existence and uniqueness, derive the characteristic equation. Some results on the stability of the steady state of the system are established.

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本文在齐次Neumann边界条件下研究了一类捕食-食饵模型正平衡解的稳定性与存在性.首先,我们利用算子谱理论得到了正常数平衡解的一致渐近稳定性,其次,运用最大值原理和Harnack不等式,我们给出了正平衡解的先验估计,再次,利用积分的性质并结合ε-Young不等式和Poincar′e不等式,文中证明了非常数正平衡解的不存在性,最后,利用Leray-Schauder度理论证明了非常数正平衡解的存在性,并且给出了正平衡解存在的充分条件.研究结果表明,当参数满足一定条件时,两物种可以共存.
The stability and existence of positive steady-state solutions for a predator-prey model are studied under homogeneous Neumann boundary condition. Firstly, the global asy-mptotic stability of positive constant steady-state solution is obtained by means of spectrum theory. Secondly, the priori estimates of positive steady-state solutions are given by applying the maximum principle and the Harnack inequality. Thirdly, the non-existence of the non-constant positive steady-state solutions is proved through the integral property,ε-Young inequality and Poincar′e inequality. Lastly, the existence of non-constant positive steady-state solutions is investigated with the help of the priori estimates and Leray-Schauder degree theory. Moreover, the su?cient conditions for the existence of positive steady-state solutions are obtained. The results show that when the parameters satisfy certain conditions, two species will coexist.

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研究扩散的先锋－顶级种群竞争模型的平衡态线性稳定性。所讨论的系统有4～6个平衡点，其中最多有2个正平衡点，其动力学性态十分丰富。运用线性化方法和特征根分析技巧等，对系统可能的各个常数平衡点的线性稳定性进行分析并给出每个平衡点稳定的充分条件或充分必要条件，最后举例说明结论的应用并进行了数值模拟和讨论。
The linear stability of equilibria for a diffused pioneer and climax species competition model is dis-cussed.This system has at least 4 and at most 6 equilibria with two possible positive equilibria, and thus the dy-namical behaviors are very rich.All the constant equilibria of the model are given in the second section, and then the linear stabilities of these equilibria are analyzed in the third section.The sufficient condition or sufficient and necessary condition of the stability for every equilibrium is given.The linearization method is used accompanied with the eigenvalue skill.At last, an example is given to illustrate the application of the results with numerical sim-ulation and conclusion discussion.

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研究了一类四维超混沌Liu系统的基本动力学特性,求得了该系统的平衡点并分析了平衡点的稳定性,对平衡点进行了Hopf分岔分析,得出Hopf分岔的参数条件.运用范式的方法求得了系统发生Hopf分岔时极限环的方向和稳定性.对Liu系统进行的数值仿真结果验证了理论推导的正确性.
A four‐dimensional hyperchaos Liu system is proposed in this paper .Its basic dynamic characteristics are studied ,the equilibria of the system is obtained through the simple calculation , then the stability of equilibria is analyzed .T hrough the analysis of Hopf bifurcation of equilibria , the parameter condition of Hopf bifurcation is derived .The stability and direction of limit cycle are derived by form theory .Simulation is given to illustrate the theoretical analysis with the aid of the computer .It is shown that the numerical results quite well with our theoretical analysis .

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分析时变时滞的细胞神经网络全局渐近稳定性问题,确定时变时滞细胞神经网络平衡点全局渐近稳定的新充分判定准则。通过研究时滞细胞神经网络模型、系统激励函数所需满足的条件及需要的引理,讨论时滞细胞神经网络的全局渐近稳定性,所得条件与时滞相关,并用实例验证所得的稳定性条件是有效的.
Analysis of global asymptotic stability of cellular neural network with time-varying and time-delay determines the new sufficient criteria of the global asymptotic stability of cellular neural network balance point with time-varying and time-delay. By researching the time-delay cellular neural network model and the conditions and lemma the system excitation function needs, it discusses the global asymptotic stability of the time-delay cellular neural network, the result is delay dependent, finally, an illustrative example is proposed to validate its stability condition feasibility and effectiveness.

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