一类SI传染病模型的平衡点稳定性及行波解的存在性（英文）

本文考虑了一类SI传染病模型,并引入了扩散和时滞的影响,得到一类捕食型的反应扩散模型.运用线性化方法得到了该系统平衡点的稳定性,由此指出了控制传染病传播的有效措施.然后运用上下解单调迭代的方法证明了行波解的存在性.     

带饱和项的互惠交错扩散模型整体解的存在性和稳定性

应用能量估计方法和Gagliardo-Nirenberg型不等式证明了带饱和项的Shigesada-Kawasaki-Teramoto两种群互惠模型在齐次Neumann边值条件下整体解的存在唯一性和一致有界性.通过构造Lyapunov函数给出了该模型正平衡点全局渐近稳定的条件.     

具有周期传染率的SIR传染病模型的周期解

考虑了具有周期传染率的SIR流行病模型,定义了基本再生数^-R0=β／（μ＋γ）,分析了该模型的动力学性态,证明了当^-R0〈1时无病平衡点是全局稳定的;^-R0〉1时,无病平衡点是不稳定的,模型至少存在一个周期解。对小振幅的周期传染率模型,给出了模型周期解的近似表达式,证明了该周期解的稳定性,最后做了数值模拟,结果显示周期解可能是全局稳定的。     

一类具有分布时滞的扩散种群模型行波解的存在性

研究了一类具有分布时滞的扩散种群模型行波解的存在性,证明了当平均时滞充分小时,方程具有连接两个平衡点的单调行波解.     

具有性别结构的捕食者-食饵扩散模型的稳定性

讨论一类具有性别结构的捕食者-食饵扩散模型在齐次Neumann边界条件下解的存在唯一性和一致有界性,并由线性化方法和Lyapunov函数方法分别证明了该模型正平衡点的局部和全局渐近稳定.     

一类具有时空时滞的单种群模型行波解的存在性

通过单调迭代和上下解技术,研究了一类具有时空时滞的单物种种群模型行波解的存在性,证明了当时滞充分小时,方程具有连接两个平衡点的波前解,并得到了一些新的结果.     

一类具有性传播的HIV/AIDS模型全局稳定性分析（英文）

本文,我们研究了一类同时考虑异性之间传播和同性之间传播以及具有常数输入的艾滋病传播模型.首先讨论了系统解的正性,平衡点的存在性等基本性质.利用比较原理证明了无病平衡点E_0的全局渐近稳定性.证明地方病平衡点的全局渐近稳定性时,我们用到了几何方法.最后使用MATLAB和取自南昌市东湖区数据进行了数值模拟,验证了结论的正确性并预测了该地区艾滋病人数的变化趋势.     

一类具有时滞和基于比率的阶段结构捕食扩散模型

讨论了一类具有时滞和基于比率的阶段结构捕食扩散模型,其中捕食种群具有两个阶段结构,并且成年捕食种群可以在两斑块间扩散.利用比较原理证明了系统在适当的条件下是持续生存的;通过构造Lyapunov泛函,得到了系统存在唯一全局渐近稳定的正周期解的充分条件.     

一类包虫病传播动力学模型的研究

研究了一类具有终宿主产卵期和中间宿主虫卵成熟期两时滞的包虫病传播动力学模型,得到了决定系统动力学行为的阈值R_0,当R_0〈1时,证明了未感染平衡点是局部渐近稳定的;当R_0〉1时,得到了感染平衡点是局部渐近稳定的充分条件。通过数值仿真验证了理论结果并探讨了时滞对系统动力学行为的影响,且发现若时滞在一定的范围内系统存在周期解.     

具有功能性反应函数xn/n+1的捕食系统

讨论了具有功能性反应函数为x^n/n 1的捕食系统，其中n是任意正整数．证明了此系统如存在正平衡点必唯一，且必全局生态稳定．并讨论了此唯一正平衡点的Hopf分支．     

Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion

This paper is concerned with the existence of travelling waves to a SIR epidemic model with nonlinear incidence rate, spatial diffusion and time delay. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state to this system under homogeneous Neumann boundary conditions is discussed. By using the cross iteration method and the Schauder's fixed point theorem, we reduce the existence of travelling waves to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a travelling wave connecting the disease-free steady state and the endemic steady state. Numerical simulations are carried out to illustrate the main results.     

Synchronizing genetic oscillators by signaling molecules

The authors examine collective rhythms in a general multicell system with both linearly diffusive and nondiffusive couplings. The effect of coupling on synchronization through intercellular signaling in a population of Escherichia coli cells is studied. In particular, a synchronization solution is given through the auxiliary individual system for 2 types of couplings. The sufficient conditions for the global synchronization of such a coupled system are derived based on the Lyapunov function method. The authors show that an appropriate design of the coupling and the inner-linking matrix can ensure global synchronization of the coupled synthetic biological system. Moreover, they demonstrate that the dynamics of an individual cell with coupling and without coupling may be qualitatively different; one is oscillatory, and the other is steady state. The change from a nonoscillatory state to an oscillatory one is induced by appropriate coupling, which also entrains all cells to synchronization. These results establish not only a theoretical foundation but also a quantitative basis for understanding the essential cooperative dynamics, such as collective rhythms or synchronization, in a population of cells.     

A mathematical model for Chagas disease with infection-age-dependent infectivity

In this paper we develop a mathematical model for Chagas disease with infection-age-dependent infectivity. The effects of vector and blood transfusion transmission are considered, and the infected population is structured by the infection age (the time elapsed from infection). The authors identify the basic reproduction ratio R0 and show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We show that depending on parameters, backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. We also discuss local and global stability of steady states.     

Dependence of stability on inter-enzyme distance in cooperative systems

A formulation in terms of quasilinear parabolic equations is given for a two-component cooperative system, with components free to diffuse in one dimension and cooperative reactive sites fixed at the boundaries of the interval. Localization of enzymes introduces Dirac delta functions into the equations. A unique, space-independent steady state is found, which is unstable with respect to diffusion and inter-enzyme distance. The biological implications of the results are formulated as a mechanism for “off-on” switching, or coupling of structure and function, in mitochondria.     

Evolution of dispersal in open advective environments

We consider a two-species competition model in a one-dimensional advective environment, where individuals are exposed to unidirectional flow. The two species follow the same population dynamics but have different random dispersal rates and are subject to a net loss of individuals from the habitat at the downstream end. In the case of non-advective environments, it is well known that lower diffusion rates are favored by selection in spatially varying but temporally constant environments, with or without net loss at the boundary. We consider several different biological scenarios that give rise to different boundary conditions, in particular hostile and “free-flow” conditions. We establish the existence of a critical advection speed for the persistence of a single species. We derive a formula for the invasion exponent and perform a linear stability analysis of the semi-trivial steady state under free-flow boundary conditions for constant and linear growth rate. For homogeneous advective environments with free-flow boundary conditions, we show that populations with higher dispersal rate will always displace populations with slower dispersal rate. In contrast, our analysis of a spatially implicit model suggest that for hostile boundary conditions, there is a unique dispersal rate that is evolutionarily stable. Nevertheless, both scenarios show that unidirectional flow can put slow dispersers at a disadvantage and higher dispersal rate can evolve.     

Analysis of unstable behavior in a mathematical model for erythropoiesis

We consider an age-structured model that describes the regulation of erythropoiesis through the negative feedback loop between erythropoietin and hemoglobin. This model is reduced to a system of two ordinary differential equations with two constant delays for which we show existence of a unique steady state. We determine all instances at which this steady state loses stability via a Hopf bifurcation through a theoretical bifurcation analysis establishing analytical expressions for the scenarios in which they arise. We show examples of supercritical Hopf bifurcations for parameter values estimated according to physiological values for humans found in the literature and present numerical simulations in agreement with the theoretical analysis. We provide a strategy for parameter estimation to match empirical measurements and predict dynamics in experimental settings, and compare existing data on hemoglobin oscillation in rabbits with predictions of our model.     

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.     

Mathematical analysis of a metapopulation model with space-limited recruitment

We investigate a mathematical aspect of a multi-species' sessile metapopulation model with space-limited recruitment proposed by Iwasa et al. in 1986. We define some basic reproduction numbers to show the threshold condition for the stability of trivial steady state and the existence of coexistent steady state. We show the existence of steady state where all species exist when some reproduction numbers are greater than one by the fixed point theorem. And we construct the Lyapunov function to show the global stability of trivial steady state when some basic reproduction numbers are not greater than one.     

Behavior of chemical reaction catalyzed by allosteric enzyme: threshold phenomena and discontinuous transition to oscillatory state

A theoretical study is made on a chemical reaction system catalyzed by an allosteric protein, especially on its behavior in far-from-equilibrium situations. The reaction system, which was introduced in a previous paper, consists of two chemical species, S and P, and an allosteric enzyme, E, which catalyzes the reaction of interconversion between them. This system is kept far-from-equilibrium by an interaction with its environment. This interaction is characterized by four parameters. For certain values of the parameters, the system was previously shown to have multiple steady states. In the present paper it is shown that a sustained oscillation takes place in a certain region of the control-parameter space. On one part of the boundary of this region, the system undergoes a discontinuous transition from a steady state to a state oscillating with finite amplitude, while on the other part of the boundary the amplitude of oscillation is vanishingly small right after the transition. It is also shown that this system exhibits a threshold phenomenon. A few possible mechanisms are discussed by which the assumed interaction of the system with its environment can be realized.     

Mathematical analysis of an age-structured population model with space-limited recruitment

In this paper, we investigate structured population model of marine invertebrate whose life stage is composed of sessile adults and pelagic larvae, such as barnacles contained in a local habitat. First we formulate the basic model as an Cauchy problem on a Banach space to discuss the existence and uniqueness of non-negative solution. Next we define the basic reproduction number R0 to formulate the invasion condition under which the larvae can successfully settle down in the completely vacant habitat. Subsequently we examine existence and stability of steady states. We show that the trivial steady state is globally asymptotically stable if R0 < or = 1, whereas it is unstable if R0 > 1. Furthermore, we show that a positive (non-trivial) steady state uniquely exists if R0 > 1 and it is locally asymptotically stable as far as absolute value of R0 - 1 is small enough.