A bioeconomic model of a ratio-dependent predator-prey system and optimal harvesting

This paper deals with the problem of a ratio-dependent prey-predator model with combined harvesting. The existence of steady states and their stability are studied using eigenvalue analysis. Boundedness of the exploited system is examined. We derive conditions for persistence and global stability of the system. The possibility of existence of bionomic equilibria has been considered. The problem of optimal harvest policy is then solved by using Pontryagin’s maximal principle.     

On the dynamical behavior of three species food web model

In this paper, a mathematical model consisting of two preys one predator with Beddington–DeAngelis functional response is proposed and analyzed. The local stability analysis of the system is carried out. The necessary and sufficient conditions for the persistence of three species food web model are obtained. For the biologically reasonable range of parameter values, the global dynamics of the system has been investigated numerically. Number of bifurcation diagrams has been obtained; Lyapunov exponents have been computed for different attractor sets. It is observed that the model has different types of attractors including chaos.     

一类具有时滞和比率依赖的竞争扩散系统

研究了一类具有时滞和基于比率的两种群非自治竞争扩散系统.利用比较原理证明了系统在适当条件下是一致持久的;利用B row er不动点原理和构造Lyapunov泛函,得到了系统存在唯一全局渐近稳定周期正解的充分条件.     

Delay model of glucose-insulin systems: Global stability and oscillated solutions conditional on delays

Recently P. Palumbo, S. Panunzi and A. De Gaetano analyzed a delay model of the glucose-insulin system. They proved its persistence, the existence of a unique positive equilibrium point, as well as the local stability of this point. In this paper we consider further the uniform persistence of such equilibrium solutions and their global stability. Using the omega limit set of a persistent solution and constructing a full time solution, we also investigate the effect of delays in connection with the behavior of oscillating solutions to the system. The model is shown to admit global stability under certain conditions of the parameters. It is also shown that the model admits slowly oscillating behavior, which demonstrates that the model is physiologically consistent and actually applicable to diabetological research.     

具有阶段结构且食饵有避难所的模型分析

研究了捕食者具有阶段结构且食饵有避难所的非自治捕食系统.利用Lyapunov函数方法得到了系统持续生存的条件,以及在一定条件下存在唯一全局渐进稳定的周期正解.对于更广泛的概周期现象,也得到了存在唯一全局渐进稳定的概周期正解的充分条件.     

Global Qualitative Analysis for a Ratio-Dependent Predator-Prey Model with Delay

A ratio-dependent predator-prey model with time lag for predator is proposed and analyzed. Mathematical analyses of the model equations with regard to boundedness of solutions, nature of equilibria, permanence, and stability are analyzed. We note that for a ratio-dependent system local asymptotic stability of the positive steady state does not even guarantee the so-called persistence of the system and, therefore, does not imply global asymptotic stability. It is found that an orbitally asymptotically stable periodic orbit exists in that model. Some sufficient conditions which guarantee the global stability of positive equilibrium are given.     

On a nonlinear nonautonomous predator–prey model with diffusion and distributed delay

In this paper, a nonlinear nonautonomous predator–prey model with diffusion and continuous distributed delay is studied, where all the parameters are time-dependent. The system, which is composed of two patches, has two species: the prey can diffuse between two patches, but the predator is confined to one patch. We first discuss the uniform persistence and global asymptotic stability of the model; after that, by constructing a suitable Lyapunov functional, some sufficient conditions for the existence of a unique almost periodic solution of the system are obtained. An example shows the feasibility of our main results.     

一类SARS传染病自治动力系统的稳定性分析

在K-M传染病模型的基础上,进一步考虑易感人群的密度制约以及患病者类的死亡与治愈率等因素,建立了描述SARS传染病的一个新的动力学模型,分析了该模型平衡点的稳定性态．证明了疾病消除平衡点在一定条件下是全局渐进稳定的,而地方病平衡点不是渐近稳定的．得到了该传染病系统在适当条件下为永久持续生存的结果．     

Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence

In this paper, an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence is proposed. In the model, we consider the infection age of infectious individuals and the biological age of pathogen in the environment. It is verified that the global dynamics of the model is completely determined by the basic reproduction number. Asymptotic smoothness is verified as a necessary argument. By analyzing corresponding characteristic equations, we discuss the local stability of each of feasible steady states. Uniform persistence is shown by using the persistence theory for infinite dimensional dynamical system. The global stability of each of feasible steady states is established by using suitable Lyapunov functionals and LaSalle’s invariance principle. Numerical simulations are carried out to illustrate the theoretical results.     

N-种群Lotka-Volterra扩散竞争反馈控制生态系统的持久性和全局渐近性

本文研究了一类具有ai类功能性反应函数的n种群非自治Lotka-Volterra扩散竞争反馈控制生态系统的持久性和全局渐近性．利用比较原理和构造Lyapunov函数分别得到系统的持久生存与全局渐近稳定的充分条件．     

The Asymptotic Behaviour of a Stochastic 3D LANS-α Model

The long-time behaviour of a stochastic 3D LANS-α model on a bounded domain is analysed. First, we reformulate the model as an abstract problem. Next, we establish sufficient conditions ensuring the existence of stationary (steady state) solutions of this abstract nonlinear stochastic evolution equation, and study the stability properties of the model. Finally, we analyse the effects produced by stochastic perturbations in the deterministic version of the system (persistence of exponential stability as well as possible stabilisation effects produced by the noise). The general results are applied to our stochastic LANS-α system throughout the paper.     

Epidemics in two competing species

An SIS epidemic model in two competing species with the mass action incidence is formulated and analysed. Thresholds for the existence of boundary equilibria are identified and conditions for their local asymptotic stability or instability are found. By persistence theory, conditions for the persistence of either hosts or pathogens are proved. Using Hopf bifurcation theory and numerical simulations, some aspects of the complicated dynamic behaviours of the model are shown: the system may have zero up to three internal equilibria, may have a stable limit cycle, may have three stable attractors. Through the results on persistence and stability of the boundary equilibria, some important interactions between infection and competition are revealed: (1) a species that would become extinct without the infection, may persist in presence of the infection; (2) a species that would coexist with its competitor without the infection, is driven to extinction by the infection; (3) an infection that would die out in either species without the interinfection of disease, may persist in both species in presence of this factor.     

Asymptotic properties of an HIV/AIDS model with a time delay

A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe.     

THE EFFECT OF A PREDATOR RESOURCE ON A DIFFUSIVE PREDATOR PREY SYSTEM

A model of a predator-prey system with diffusion and predator resource is studied. Both constant and variable resources are considered. In the absence of diffusion, criteria for local stability, instability, and global stability of equilibria, as well as persistence and extinction, are obtained. It is shown that an otherwise unstable uniform equilibrium state may be stabilized by diffusion.     

Analysis of a diffusive Leslie–Gower predator–prey model with nonmonotonic functional response

In this paper, a diffusive Leslie–Gower predator–prey system with nonmonotonic functional respond is studied. We obtain the persistence of this model and show the local asymptotic stability of positive constant equilibrium by linearized analysis and the global stability by constructing Liapunov function. Besides, Turing instability of this equilibrium is obtained. The existence and nonexistence of positive nonconstant steady states of this model are established. Furthermore, by numerical simulations we illustrate the patterns of prey and predator.     

THE PERSISTENCE AND STABILITY FOR A PREDATOR-PARASITE-PEST SYSTEM

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一类具有离散时滞的非自治扩散的两种群模型的一致持续生存

持续生存概念是种群生态系统稳定性的一个重要描述,而研究竞争种群共存的问题是种群生态学的一个重要问题.进一步考虑具有离散时滞的非自治的两种群竞争扩散摸型,使模型更符合其生态意义,通过微分不等式获得了其一致持续生存的充分条件.     

具有连续时滞的非自治扩散Lotka-volterra模型的渐近性

持续生存概念是种群生态系统稳定性的一个重要描述,而研究竞争种群共存的问题是种群生态学的一个重要问题,考虑非自治的两种群L otka-vo lterra周期系数的时滞扩散摸型,通过构造李亚普诺夫泛函,微分不等式等获得了其一致持续生存及正周期解存在与全局渐近稳定的充分条件.     

考虑媒体播报效应的双时滞传染病模型

在疾病控制过程中, 媒体的重要性举足轻重.该文旨在建立并分析一个含有媒体效应的多时滞传染病模型, 研究模型的稳定性, 并通过分析相应特征方程根, 分别研究在时滞不同的5种情况下, 系统的稳定性发生变化, 以及产生Hopf分支的条件.再利用持续性理论, 证明模型的持续生存性.最后将时滞模型研究结果应用于苏格兰小儿肺炎中, 验证媒体效应对疫情控制起到的重要作用以及时滞大小对模型稳定性的影响.     

PERSISTENCE AND STABILITY OF A DISCRETE PERIODIC PREDATOR-PREY MODEL

In this paper,we investigate the persistence and global asymptotic stability of a discrete predator-prey system.Based on Lyapunov functions,several crite- ria are established concerning the persistence and globally asymptotic stability of difference systems.