Fitz-Hugh Nagumo系统的渐近稳定性

用无穷维动力系统的方法研究了Fitz-Hugh Nagumo神经系统中沿神经细胞轴突信号传递的长时间行为.在齐次边界条件与非齐次边界条件下证明了系统在其不变流形上的全局吸引子为系统在该不变流形内的唯一平衡点,从而证明了该系统的渐近稳定性.     

一类扩散的种内相食系统正平衡态的全局渐近稳定性

讨论一类扩散的种内相食的捕食者-食饵模型,该模型带有齐次Neumann边界条件.通过迭代的方法证明:在适当的条件下,该模型唯一正常数平衡解全局渐近稳定.     

带存放率的周期竞争扩散系统的稳定共存

应用上、下解方法和抛物型方程的极值原理,研究了带存放率的周期竞争系统ut-D1Δu=u（α-bu-cv）＋h,vt-D2Δv=v）d-eu-fv）＋k 在齐次Neumann边界条件下解的渐近性态,得到了该系统的全局渐近性．     

以亚硝氮为唯一氮源生长的海洋紫色硫细菌去除无机三态氮

【目的】揭示以亚硝氮为唯一氮源生长的海洋紫色硫细菌去除水体中无机三态氮的特征和规律。【方法】在光照厌氧环境下,以乙酸盐为唯一有机物,在分别以氨氮、亚硝态氮、硝态氮为唯一氮源和三氮共存的模拟水体中,采用Nessler’s试剂分光光度法、N-(1-萘基)-乙二胺分光光度法和紫外分光光度法分别测定水体中氨氮、亚硝态氮和硝态氮的含量,比浊法测定菌体生物量。【结果】随着时间的延长,海洋紫色硫细菌Marichromatium gracile YL28分别在氨氮、亚硝态氮和硝态氮为唯一氮源的水体中对三氮的去除量增加,生物量增大,水体pH升高,并逐渐趋于平衡;YL28对氨氮的最大去除量和最大耐受浓度分别为9.64 mmol/L和36.64 mmol/L,当氨氮浓度低于3.21 mmol/L时,去除率可达97.61%以上;与氨氮相比,以亚硝态氮和硝态氮为唯一氮源,菌体的生长速率、生物量和水体最终pH较低,但对亚硝态氮和硝态氮的去除速率和去除量仍然很高,当亚硝态氮和硝态氮浓度分别达13.50 mmol/L和22.90 mmol/L时,YL28仍能够完全去除。在三氮共存的水体中,YL28也能良好的去除无机三态氮,对亚硝态氮和硝态氮去除能力更强。【结论】在模拟水体中,海洋紫色硫细菌YL28能够分别以氨氮、亚硝态氮和硝态氮为唯一氮源生长,具有良好的耐受和去除无机三态氮的能力,尤其对亚硝态氮具有良好的去除能力。本研究为进一步开发高效脱氮,尤其是去除亚硝态氮的不产氧光合细菌水质调节剂奠定了基础,也为微生物制剂的合理应用提供参考。     

一类具时滞的禽流感模型

针对具扩散和时滞的SI-SIR传染病模型,用特征分析和Lyapunov泛函方法研究了相应的具齐次Neumann边界条件反应扩散方程组解的渐近性质.最后给出数值模拟来说明如果染病鸟类的接触率和染病人类的接触率小,那么全系统的无病平衡点是全局渐近稳定的;但当染病鸟类的接触率大或者和染病人类的接触率大时,变异的禽流感将在人类中扩散.     

双CSTR耦合系统中乙醇发酵过程中的分岔研究

本文首次运用现代分岔理论方法,研究了双CSTR耦合系统中乙醇发酵过程中的多定态和振荡现象．计算结果表明,当入口糖浓度大于某值时,多定态普遍存在．同时振荡受到底物浓度S_0、两耦合反应器的体积比V_(12)及稀释率D的影响.同单CSTR相比,双CSTR加强了反应深度,并通过适当改变V_(12),产率可大大提高.     

具性别和阶段结构的非自治二种群捕食者-食饵系统研究

讨论了一类食饵具有性别结构,捕食者具有阶段结构的非自治捕食者．食饵系统,运用Liapunov函数方法,得到了该系统一致持续生存的充分条件．对于该模型的周期系统,在适当条件下,存在唯一、全局渐近稳定的周期解．对更具普遍意义的概周期现象,也得出了概周期正解唯一存在且全局渐近稳定的充分条件．     

反应扩散模型的全局渐近稳定性和持续性

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.     

一类带存放率的竞争扩散系统解的渐近性态

应用上、下解方法和抛物型方程的极值原理，研究了带存放率的竞争扩散系统ut-D1△u=u(a-bu-cv) h1vt-D2△v=v(d-eu-fv) k在齐次Neumann边界条件下解的渐近性态。     

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.     

Steady-state solution of a two-species biofilm problem

Through a thorough investigation of the boundary conditions for a general two-species biofilm model, a simple and fast method for solving the steady-state case is developed and presented. The methods used may be extended to biofilm models in which more than two species are considered. Four different sets of boundary conditions are possible for the two-species biofilm model. Each set is shown to be asymptotically stable. A biofilm model describing the competition between autotrophic and heterotrophic bacteria and a biofilm model considering only Nitrosomonas and Nitrobacter are used for illustration. A parameter L(crit), critical film thickness for bacterial coexistence, is introduced from which criteria on the bulk concentrations for coexistence are derived. From these criteria it is seen that the thinner the biofilm, the more restrictive the conditions are for steady-state coexistence. For thin biofilms there may, in many cases, be no point in considering more than one species in the biofilm model. Furthermore, the gradients of the bacterial concentrations are in many cases negligible in thin biofilms, and the biofilm may then be assumed to be homogeneous. The criteria on the bulk concentrations together with the four sets of boundary conditions provide the necessary information for a direct solution of the steady-state two-species biofilm model by means of an ordinary differential and algebraic equation solver. (c) 1996 John Wiley & Sons, Inc.     

Coexistence in MacArthur-style consumer-resource models

The nature of and conditions for permanent coexistence of consumers and resources are characterized in a family of models that generalize MacArthur's consumer-resource model. The generalization is of the resource dynamics, which need not be of Lotka-Volterra form but are subject only to certain restrictions loose enough to admit many resource dynamics of biological interest. For any such model, (1) if there is an interior equilibrium, then it is globally attracting, else some boundary equilibrium is globally attracting-thus permanent coexistence is coexistence at a globally attracting equilibrium; (2) there is an interior equilibrium if and only if for any species, the equilibrium approached in the absence of that species and the presence of the others is invasible by that species--thus permanent coexistence is equivalent to mutual invasibility; (3) for resources without direct interactions, the conditions for permanent coexistence of the consumers admit an instructive formulation in terms of regression statistics. The significance and limitations of the models and results are discussed.     

Coexistence of multiple propagating wave-fronts in a regulated enzyme reaction model: link with birhythmicity and multi-threshold excitability

We analyze the spatial propagation of wave-fronts in a biochemical model for a product-activated enzyme reaction with non-linear recycling of product into substrate. This model was previously studied as a prototype for the coexistence of two distinct types of periodic oscillations (birhythmicity). The system is initially in a stable steady state characterized by the property of multi-threshold excitability, by which it is capable of amplifying in a pulsatory manner perturbations exceeding two distinct thresholds. In such conditions, when the effect of diffusion is taken into account, two distinct wave-fronts are shown to propagate in space, with distinct amplitudes and velocities, for the same set of parameter values, depending on the magnitude of the initial perturbation. Such a multiplicity of propagating wave-fronts represents a new type of coexistence of multiple modes of dynamic behavior, besides the coexistence involving, under spatially homogeneous conditions, multiple steady states, multiple periodic regimes, or a combination of steady and periodic regimes.     

Coexistence of Multiple Periodic and Chaotic Regimes in Biochemical Oscillations with Phase Shifts

The numerical study of a glycolytic model formed by a system of three delay differential equations reveals a multiplicity of stable coexisting states: birhythmicity, trirhythmicity, hard excitation and quasiperiodic with chaotic regimes. For different initial functions in the phase space one may observe the coexistence of two different quasiperiodic motions, the existence of a stable steady state with a stable torus, and the existence of a strange attractor with different stable regimes (chaos with torus, chaos with bursting motion, and chaos with different periodic regimes). For a single range of the control parameter values our system may exhibit different bifurcation diagrams: in one case a Feigenbaum route to chaos coexists with a finite number of successive periodic bifurcations, in other conditions it is possible to observe the coexistence of two quasiperiodicity routes to chaos. These studies were obtained both at constant input flux and under forcing conditions.     

Global positive coexistence of a nonlinear elliptic biological interacting model

The purpose of this note is to give a necessary and sufficient condition for the coexistence of positive solutions to a rather general type of elliptic predator-prey system of the Dirichlet problem on the bounded domain omega when omega is a subset of Rn is large. The result is that the partial differential equation system possesses positive coexistence if and only if the corresponding ordinary differential equation system has positive equilibrium, the positive constant states. This result thus yields an algebraically computable criterion for the positive coexistence of predator and prey in many biological models.     

Competitive exclusion and coexistence of pathogens in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population has been studied completely by Jiang and Chai in (J Math Biol 56:373-390, 2008). In this paper, we give a analysis for a SIS STD with two competing strains, where populations are divided into three differential groups based on their susceptibility to two distinct pathogenic strains. We investigate the existence and stability of the boundary equilibria that characterizes competitive exclusion of the two competing strains; we also investigate the existence and stability of the positive coexistence equilibrium, which characterizes the possibility of coexistence of the two strains. We obtain sufficient and necessary conditions for the existence and global stability about these equilibria under some assumptions. We verify that there is a strong connection between the stability of the boundary equilibria and the existence of the coexistence equilibrium, that is, there exists a unique coexistence equilibrium if and only if the boundary equilibria both exist and have the same stability, the coexistence equilibrium is globally stable or unstable if and only if the two boundary equilibria are both unstable or both stable.     

Large amplification in stage-structured models: Arnol’d tongues revisited

The coexistence of periodic and point attractors has been confirmed for a range of stage-structured discrete time models. The periodic attractor cycles have large amplitude, with the populations cycling between extremely low and surprisingly high values when compared to the equilibrium level. In this situation a stable state can be shocked by noise of sufficient strength into a state of high volatility. We found that the source of these large amplitude cycles are Arnold tongues, special regions of parameter space where the system exhibits periodic behaviour. Most of these tongues lie entirely in that part of parameter space where the system is unstable, but there are exceptions and these exceptions are the tongues that lead to attractor coexistence. Similarity in the geometry of Arnold tongues over the range of models considered might suggest that this is a common feature of stage-structured models but in the absence of proof this can only be a useful working hypothesis. The analysis shows that although large amplitude cycles might exist mathematically they might not be accessible biologically if biological constraints, such as non-negativity of population densities and vital rates, are imposed. Accessibility is found to be highly sensitive to model structure even though the mathematical structure is not. This highlights the danger of drawing biological conclusions from particular models. Having a comprehensive view of the different mechanisms by which periodic states can arise in families of discrete time models is important in the debate on whether the causes of periodicity in particular ecological systems are intrinsic, environmental or trophic. This paper is a contribution to that continuing debate.     

Competitive coexistence of self-reproducing macromolecules.

Several model systems of self-reproducing molecular species subject to the constraint of constant organization are investigated to see under what conditions more than one species may coexist over extended periods of time. It is found that while coexistence is not possible for systems of simple autocatalytic competitors, it can indeed occur if the nature of the reproductive process gives rise to “catalytic niches”. For systems catalyzed by n Michaelis-Menten type enzymes, it is shown that no more than n species may coexist. A precise characterization is provided of “how different” two such enzymes must be in order to allow coexistence of two species. Under certain conditions, one observes a transition from one dominant species to another which may occur either via a coexistent state or directly, but with hysteresis.     

The Role of Space in the Exploitation of Resources

In order to understand the role of space in ecological communities where each species produces a certain type of resource and has varying abilities to exploit the resources produced by its own species and by the other species, we carry out a comparative study of an interacting particle system and its mean-field approximation. For a wide range of parameter values, we show both analytically and numerically that the spatial model results in predictions that significantly differ from its nonspatial counterpart, indicating that the use of the mean-field approach to describe the evolution of communities in which individuals only interact locally is invalid. In two-species communities, the disagreements between the models appear either when both species compete by producing resources that are more beneficial for their own species or when both species cooperate by producing resources that are more beneficial for the other species. In particular, while both species coexist if and only if they cooperate in the mean-field approximation, the inclusion of space in the form of local interactions may prevent coexistence even in cooperative communities. Introducing additional species, cooperation is no longer the only mechanism that promotes coexistence. We prove that, in three-species communities, coexistence results either from a global cooperative behavior, or from rock-paper-scissors type interactions, or from a mixture of these dynamics, which excludes in particular all cases in which two species compete. Finally, and more importantly, we show numerically that the inclusion of space has antagonistic effects on coexistence depending on the mechanism involved, preventing coexistence in the presence of cooperation but promoting coexistence in the presence of rock-paper-scissors interactions. Although these results are partly proved analytically for both models, we also provide somewhat more explicit heuristic arguments to explain the reason why the models result in different predictions.     

Structural identification of large systems by reduction to subsystems: VLDL triglycerides

The problem of determining whether or not a particular parameter, or the entire compartmental system, is structural identifiable can be extremely difficult if the number of unknown parameters and the number of compartments is large. However, the problem can be considerably simplified if the system can be decomposed into smaller subsystems in such a way that a parameter is structural identifiable with respect to the large system if and only if it is structural identifiable with respect to the subsystem in which it is contained. The paper offers sufficient conditions under which the desired decomposition can be achieved. The conditions are expressed in terms of the digraph of the system so that they are not difficult to verify. Illustrative examples are provided in terms of application to lipoprotein kinetics.