一类病毒自身发生变异的传染病模型的全局分析

针对病毒变异前和变异后传染病患者具有不同的传染率情形,建立了一类分阶段传播的SIS模型,通过构造Liapunov函数和定性分析,得到病毒变异前和变异后传染病患者平衡点的存在条件以及它们的全局渐近稳定性。     

Lotka-Volterra equations: Decomposition,stability, and structure

Summary The major objective of this paper is to propose a new decomposition-aggregation framework for stability analysis of Lotka-Volterra equations employing the concept of vector Liapunov functions. Both the disjoint and the overlapping decompositions are introduced to increase flexibility in constructing Liapunov functions for the overall system. Our second objective is to consider the Lotka-Volterra equations under structural perturbations, and derive conditions under which a positive equilibrium is connectively stable. Both objectives of this paper are directed towards a better understanding of the intricate interplay between stability and complexity in the context of robustness of model ecosystems represented by Lotka-Volterra equations. Only stability of equilibria in models with constant parameters is considered here. Nonequilibrium analysis of models with nonlinear time-varying parameters is the subject of a companion paper.Research supported by U.S. Department of Energy under the Contract EC-77-S-03-1493.On leave from Kobe University, Kobe, Japan.     

Stability analysis of predator-prey models via the liapunov method

As is well-known from the classical applications in the electrical and mechanical sciences, energy is a suitable Liapunov function: thus, by analogy, all energy functions proposed in ecology are potential Liapunov functions. In this paper, a generalized Lotka-Volterra model is considered and the stability properties of its non-trivial equilibrium are studied by means of an energy function first proposed by Volterra in the context of conservative ecosystems. The advantage of this Liapunov function with respect to the one that can be induced through linearization is also illustrated.     

具有三个年龄阶段的单种群自食模型

建立并研究了两个具有三个年龄阶段的单种群自食模型．这篇文章的主要目的是研究时滞对种群生长的作用,对于没有时滞的的模型,我们利用Liapunov函数,得到了系统平衡点全局渐近稳定的充分条件;而具有时滞的的模型,我们得到,随着时滞T增加,当系数满足一定条件时,正平衡点的稳定性可以改变有限次,最后变成不稳定;否则,时滞模型的正平衡点的稳定性不改变。     

具有时滞的Hopfield人工神经网络动力系统模型的全局渐近稳定性

Hopfield人工神经网络动力系统模型平衡点的全局渐近稳定性在网络记忆以及最优化等领域具有广泛的应用。本文中,作者研究了一类具有时滞的Hopfield人工神经网络动力系统,通过构造Liapunov泛函的方法,获得了其平衡点全局渐近稳定和局部渐近稳定的充分判定条件。所给出的判定条件只依赖于系统本身的拳数参数和传递函数以及系统中出现的部分时滞。同时,当系统的自身反馈项为负时,此自身反馈项对于系统的稳定性起到稳定化的作用。此外,数值模拟表明时滞的变化对于系统的稳定性具有重要的影响。可破坏系统的稳定性。进而产生周期振动或更为复杂的非线性现象。     

捕食者有无限时滞效应的两种群系统的全局渐近稳定性

本文用Liapunov泛函方法研究捕食者有无限时滞效应的捕食-被捕食系统的平衡状态的稳定性．文章提供了判定系统的平衡状态全局渐近稳定的简单条件,不要求积分核指数衰减.     

具有脉冲效应的非自治Lotka-Valterra竞争系统的持久性和渐近行为

本文考虑了两种群具有脉冲效应的非自治Lotka-Valterra竞争系统．运用比较定理和李雅普诺夫函数法,建立了系统的持久性和渐近行为的充分条件．     

一类具有广义Logistic增长的比率型捕食模型的定性分析

对具有广义Logistic增长的比率型捕食模型进行了分析,得到了系统永久持续生存的充分条件．通过对系统进行无量纲代换和应用Liapunov函数的方法,进一步得到了正平衡点全局渐近稳定的充分条件．     

On global stability of discrete population models

A necessary and sufficient condition for the global stability of a large class of discrete population models is provided which does not require the construction of a Liapunov function. The general result is applied to difference equations defined in terms of “two hump” functions and to an example of frequency dependent selection.     

反应扩散模型的全局渐近稳定性和持续性（英文）

本文主要应用反应扩散方程的不变原理和构造适当的Liapunov泛函方法研究了一类反应扩散模型的唯一齐次正平衡态的全局渐近稳定性.进一步,根据Dunbar et al.（1986）和Hutson与Moran（1987）得出的动力学系统结果,我们得到了判别具有持续性的充分条件.     

Threshold policies control for predator-prey systems using a control Liapunov function approach

The stability of predator-prey models, in the context of exploitation of renewable resources, subject to threshold policies (TP) is studied in this paper using the idea of backstepping and control Liapunov functions (CLF) well known in control theory, as well as the concept of virtual equilibria. TPs are defined and analysed for different types of one and two species predator-prey models. The models studied are the single species Noy-Meir herbivore-vegetation model, in a grazing management context, as well as the Rosenzweig-MacArthur two species predator-prey model, in a fishery management context. TPs are shown to be versatile and useful in managing renewable resources, being simple to design and implement, and also yielding advantages in situations of overexploitation.     

变时滞区间细胞神经网络的全局鲁棒稳定性

利用拓扑度理论和Liapunov泛函方法讨论了变时滞区间细胞神经网络的全局鲁棒稳定性．给出了实用有效的判定条件,推广了有关文献中的结果．     

具有扩散的非自治两种群Lotka-Volterra模型的概周期问题

研究非自治两种群竞争系统,其中一种群可以在两个斑块之间扩散。而另一种群在一个斑块中,不能扩散。本文结合运用Liapunov函数,得到该系统唯一存在全局渐近稳定的正概周期解的条件．     

一类带有隔离项和分布时滞的SIQS模型的稳定性分析

研究了一类带有隔离项和分布时滞的SIQS模型,利用局部线性化方法并构造了适当的Liapunov函数,获得了该系统平衡点的稳定性结论.     

Estimating the effect of continual disturbances on discrete-time population models

In this paper, methods for estimating the effect of continual (but bounded) disturbances on ecosystems modelled by difference equations are discussed. The approach adopted is to estimate the region of state space (called a reachable set) which can be reached by the disturbed system from an initial healthy state in a given time period. Liapunov stability methods for estimating these reachable sets are presented and applied to two specific population models.     

Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control

In this paper, one investigates the dynamic behaviors of one-prey multi-predator model with Holling type II functional response by introducing impulsive biological control strategy (periodic releasing natural enemies at different fixed time). By using Floquet theorem and small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that multi-predator impulsive control strategy is more effective than the classical and single one.     

Metrics from nonlinear dynamics adapted for characterizing the behavior of nonequilibrium enzymatic rate functions

Several metrics from nonlinear dynamics and statistical mechanics have been characterized on computer-generated number series with various signal-to-noise ratios, demonstrating their individual reliability as a function of sample size and their relationships to each other. The root mean square (RMS) evaluates amplitude, and the power spectral density (PSD) provides a visual display of the frequency spectrum; both measures have very high reliability even for an N as low as 50. The Fractal Dimension (D) is shown to converge rapidly and also to be reliable when N is as low as 50. These three measures (RMS, PSD, and D) have been applied to the complex kinetics of tyrosine hydroxylase time courses (50-point curves) at various BH4 concentrations (near physiological, but far from equilibrium levels). Recently developed measures of spectral entropy and the Liapunov Exponent, -lambda are also characterized.     

Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points

This paper considers the dynamics of a discrete-time Kolmogorov system for two-species populations. In particular, permanence of the system is considered. Permanence is one of the concepts to describe the species coexistence. By using the method of an average Liapunov function, we have found a simple sufficient condition for permanence of the system. That is, nonexistence of saturated boundary fixed points is enough for permanence of the system under some appropriate convexity or concavity properties for the population growth rate functions. Numerical investigations show that for the system with population growth rate functions without such properties, the nonexistence of saturated boundary fixed points is not sufficient for permanence, actually a boundary periodic orbit or a chaotic orbit can be attractive despite the existence of a stable coexistence fixed point. This result implies, in particular, that existence of a stable coexistence fixed point is not sufficient for permanence.     

Global stability of population models

Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.     

Indistinguishability and identifiability analysis of linear compartmental models.

Two compartmental model structures are said to be indistinguishable if they have the same input-output properties. In cases in which available a priori information is not sufficient to specify a unique compartmental model structure, indistinguishable model structures may have to be generated and their attributes examined for relevance. An algorithm is developed that, for a given compartmental model, investigates the complete set of models with the same number of compartments and the same input-output structure as the original model, applies geometrical rules necessary for indistinguishable models, and test models meeting the geometrical criteria for equality of transfer functions. Identifiability is also checked in the algorithm. The software consists of three programs. Program 1 determines the number of locally identifiable parameters. Program 2 applies several geometrical rules that eliminate many (generally most) of the candidate models. Program 3 checks the equality between system transfer functions of the original model and models being tested. Ranks of Jacobian matrices and submatrices and other criteria are used to check patterns of moment invariants and local identifiability. Structural controllability and structural observability are checked throughout the programs. The approach was successfully used to corroborate results from examples investigated by others.