疾病在食饵中流行的捕食与被捕食模型的分析

分析并建立了疾病在食饵中传播的生态-传染病模型,同时考虑到两种群都受密度制约因素的影响,讨论了模型解的有界性和各平衡点的存在性,利用Routh-Hurwitz判据证明了各平衡点的局部渐进稳定性,通过构造Lyapunov函数分析了各平衡点的全局渐进稳定性,得到了疾病存在与否的充分性条件.     

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.     

Stability preserving NSFD scheme for a delayed viral infection model with cell-to-cell transmission and general nonlinear incidence

In this paper, we designed and analysed a discrete model to solve a delayed within-host viral infection model by using non-standard finite difference scheme. The original model that we considered was a delayed viral infection model with cell-to-cell transmission, cell-mediated immune response and general nonlinear incidence. We show that the discrete model has equilibria which are exactly the same as those of the original continuous model and the conditions for those equilibria to be globally asymptotically stable are consistent with the original continuous model with no restriction on the time step size. The results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for corresponding continuous model.     

一类疾病在食饵中传播的生态-传染病模型分析

分析并建立疾病在食饵中传播的生态-传染病模型,且考虑易感食饵具有常数输入,捕食者种群以Logistic模型增长,讨论了系统解的有界性和各平衡点的存在性,以及局部渐近稳定性,通过构造适当的Lyapunov函数分析了各平衡点的全局渐近稳定性,并运用比较定理证明了系统的持久性.     

Analysis of a delayed HIV/AIDS epidemic model with saturation incidence

In this paper, a delayed HIV/AIDS epidemic model with saturation incidence is proposed and analyzed. The equilibria and their stability are investigated. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium. It is found that if the threshold R ₀<1, then the disease-free equilibrium is globally asymptotically stable, and if the threshold R ₀>1, the system is permanent and the endemic equilibrium is asymptotically stable under certain conditions.     

A predator–prey–disease model with immune response in infected prey

In this paper, a predator–prey–disease model with immune response in the infected prey is formulated. The basic reproduction number of the within-host model is defined and it is found that there are three equilibria: extinction equilibrium, infection-free equilibrium and infection-persistent equilibrium. The stabilities of these equilibria are completely determined by the reproduction number of the within-host model. Furthermore, we define a basic reproduction number of the between-host model and two predator invasion numbers: predator invasion number in the absence of disease and predator invasion number in the presence of disease. We have predator and infection-free equilibrium, infection-free equilibrium, predator-free equilibrium and a co-existence equilibrium. We determine the local stabilities of these equilibria with conditions on the reproduction and invasion reproduction numbers. Finally, we show that the predator-free equilibrium is globally stable.     

Dynamics of SIR epidemic models with horizontal and vertical transmissions and constant treatment rates

We investigate the dynamics and bifurcations of SIR epidemic model with horizontal and vertical transmissions and constant treatment rates. It is proved that such SIR epidemic model have up to two positive epidemic equilibria and has no positive disease-free equilibria. We find all the ranges of the parameters involved in the model under which the equilibria of the model are positive. By using the qualitative theory of planar systems and the normal form theory, the phase portraits of each equilibria are obtained. We show that the equilibria of the epidemic system can be saddles, stable nodes, stable or unstable focuses, weak centers or cusps. We prove that the system has the Bogdanov-Takens bifurcations, which exhibit saddle-node bifurcations, Hopf bifurcations and homoclinic bifurcations.     

具有食饵避难的Leslie-Gower捕食系统最优收获分析

研究了一类具有食饵避难的Leslie-Gower捕食与被捕食系统收获模型,利用Hurwitz判据,得到了正平衡点局部渐近稳定,进一步构造了适当的Lyapunov函数,证明了正平衡点的全局渐近稳定性.并且在捕获努力量假说下,对发生食饵避难的两种群同时捕获,考虑了生态经济平衡点的存在性和利用Pontryagin最大值原理对两种群进行最优收获,得到当贴现率为零时,既保持了生态平衡,又使得在渔业开发过程中取得最大经济利益.     

POPULATION AND HARVESTING THEORY FOR NONLINEAR SEX AND AGE-STRUCTURED RESOURCES

A sex-age-structured population model with density dependence in the conversion of reproductive potentials into zygotes and in first year survivorship is described. The model has two equilibria; the smallest is mathematically unstable, and the origin and the larger equilibrium are locally stable. The population can thus go extinct for certain initial states, or if the two equilibria coincide. The ratio between the two equilibria can be regarded as a measure of the risk of extinction, since it is related to the chance that detrimental environmental conditions will cause the population to enter the region of attraction of the origin. In simple monoecious models, recovery to former levels is only possible provided that the population is not driven to extinction before harvesting effort is reduced. Ratios between the two unexploited equilibria, and between the stable unexploited equilibrium and the recruitment level at which the two equilibria coincide are given solely in terms of the degree of density dependence in the model. I show that the harvesting strategy which maximizes the equilibrium yield has a four age form, involving harvesting of at most two male and two female age classes. Out of ten commercial Pacific groundfish species, knife-edge selectivity sustainable yields of eight are at least 90% of ultimate sustainable yield (USY). With no effort restrictions, the range of lengths at first capture which achieve more than 60% of USY is narrow. When one of the sexes is not harvested, sustainable yield is between 20% and 80% of USY, but lowest when females are not harvested.     

A CLASS OF VOLTERRA INTEGRAL EQUATIONS ARISING IN DELAYED-RECRUITMENT POPULATION MODELS

We formulate a Volterra integral equation which contains as special cases the differential-difference equation model of Blythe, Gurney and Nisbet for populations with delayed recruitment and a differential-difference equation with two delays related to the epidemic model of Wilson and Burke. We establish upper and lower bounds for positive solutions and give a classification of equilibria with conditions to determine whether an equilibrium is stable for all delays (absolutely stable), unstable for all delays, or switches from stable to unstable as the delay increases.     

SOURCES,SINKS AND SELECTIVITY

ABSTRACT. A model, which consists of a system of several nonlinear ordinary differential equations, is developed to describe a metapopulation. Results regarding the location and stability of fixed population vectors are obtained, including conditions for the existence of multiple stable equilibria. The convergence of all solutions is established.     

Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with a general contact rate. The model exhibits the traditional threshold behavior. We prove that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is globally asymptotically stable and when the basic reproduction ratio is great than unity, a unique endemic equilibrium exists and is globally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Lyapunov stability theory and LaSalle’s invariant set theorem. Numerical simulations are provided to illustrate the theoretical results.     

一类具有阶段结构的捕食-被捕食模型的全局稳定性（英文）

A Holling type III predator-prey model with stage structure for prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is discussed. By using the uniformly persistence theory, the system is proven to be permanent if the coexistence equilibrium exists. By using Lyapunov functionals and La Salle's invariance principle, it is shown that the two boundary equilibria is globally asymptotically stable when the coexistence equilibrium is not feasible. By using compound matrix theory, the sufficient conditions are obtained for the global stability of the coexistence equilibrium. At last, numerical simulations are carried out to illustrate the main results.     

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.     

一类带有不育控制的离散单种群模型

建立一类不育控制下的害鼠种群的离散模型.首先利用三个Jury条件,得到平衡点的局部渐近稳定性的充分条件.其次利用李雅普诺夫函数和细致分析法分别给出了零平衡点全局稳定及持续生存的充分条件.最后给出了平衡点全局稳定的数值模拟.     

一类具有Allee影响的捕食与被捕食模型

分析并建立了具有Allee影响的捕食与被捕食模型,被捕食者由于自身繁殖或是被捕食而具有了Allee效应,分别讨论了强Allee和弱Allee对被捕食种群的影响,讨论了解的有界性和各平衡点的存在性,并证明了各平衡点的局部渐近稳定性,进一步通过构造适当的Lyapunov函数分析了正平衡点E*的全局渐近稳定性.     

Study on a HIV/AIDS model with application to Yunnan province,China

This paper presents an epidemic model aiming at the prevalence of HIV/AIDS in Yunnan, China. The total population in the model is restricted within high risk population. By the epidemic characteristics of HIV/AIDS in Yunnan province, the population is divided into two groups: injecting drug users (IDUs) and people engaged in commercial sex (PECS) which includes female sex workers (FSWs), and clients of female sex workers (C). For a better understanding of HIV/AIDS transmission dynamics, we do some necessary mathematical analysis. The conditions and thresholds for the existence of four equilibria are established. We compute the reproduction number for each group independently, and show that when both the reproduction numbers are less than unity, the disease-free equilibrium is globally stable. The local stabilities for other equilibria including two boundary equilibria and one positive equilibrium are figured out. When we omit the infectivity of AIDS patients, global stability of these equilibria are obtained. For the simulation, parameters are chosen to fit as much as possible prevalence data publicly available for Yunnan. Increasing strength of the control measure on high risk population is necessary to reduce the HIV/AIDS in Yunnan.     

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.     

Qualitative analysis of a SIR epidemic model with saturated treatment rate

On account of the effect of limited treatment resources on the control of epidemic disease, a saturated removal rate is incorporated into Hethcote’s SIR epidemiological model (Hethcote, SIAM Rev. 42:599–653, 2000). Unlike the original model, the model has two endemic equilibria when R ₀<1. Furthermore, under some conditions, both the disease free equilibrium and one of the two endemic equilibria are asymptotically stable, i.e., the model has bistable equilibria. Therefore, disease eradication not only depends on R ₀ but also on the initial sizes of all sub-populations. By the Poincaré-Bendixson theorem, Poincaré index, center manifold theorem, Hopf bifurcation theorem and Lyapunov-Lasalle theorem, etc., the existence and asymptotical stability of the equilibria, the existence, stability and direction of Hopf bifurcation are discussed, respectively.     

GLOBAL DYNAMICS OF A PREDATOR-PREY MODEL WITH PREY REFUGE AND DISEASE

In this paper, we study a predator-prey model with prey refuge and disease. We study the local asymptotic stability of the equilibriums of the system. Further, we show that the equilibria are globally asymptotically stable if the equilibria are locally asymptotically stable. Some examples are presented to verify our main results. Finally, we give a brief discussion.