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。     

Dynamical analysis of a diffusion plant-wrack model with delay

In this paper, in view of the senescence of plant and the decay of wrack, time delays are introduced into the plant-wrack model. The effects of wrack decay and time delay on the dynamical behaviors of the diffusive plant-wrack model are studied analytically and numerically. When the delay is zero, the wrack decay will induce the change of stability of the unique equilibrium point, further lead to the occurrence of the Hopf bifurcation and the Turing instability. When the delay is present, the conditions for the occurrence of the Hopf bifurcation are established. By comparing the results of the model without and with delay, it is found that the increases of delay may induce no stability switches, a single stability switch or multiple stability switches, when the value of wrack decay can stabilize model with zero delay. When the value of wrack decay can destabilize model with zero delay, numerical simulations show that the small delay may cause homogeneous distributions of vegetation, while the larger delay may cause the emergence of periodic oscillation of vegetation. The obtained results provide a basis for understanding the spatiotemporal evolution of such a plant-wrack model with delay.     

一类具分段常数变量的捕食-食饵系统的Neimark-Sacker分支

讨论了具时滞与分段常数变量的捕食-食饵生态模型的稳定性及Neimark-Sacker分支;通过计算得到连续模型对应的差分模型,基于特征值理论和Schur-Cohn判据得到正平衡态局部渐进稳定的充分条件;以食饵的内禀增长率为分支参数,运用分支理论和中心流形定理分析了Neimark-Sacker分支的存在性与稳定性条件;通过举例和数值模拟验证了理论的正确性。     

具有时滞的HBV病毒动力学模型稳定性分析

HBV（HePatitis B virus）是一种具有严重传染性的肝炎病毒,迄今为止,人们对它的免疫和慢性化的机制等方面还不甚了解。本文基于相关的病理知识,对应的建立了具有时滞的微分方程数学模型,系统地探讨了肝炎B病毒与宿主细胞之间的关系,利用Lyapunov函数方法研究了病毒动力学模型感染平衡点的局部稳定性和未感染平衡点全局稳定性,并利用数学模拟验证了理论分析。结果表明时滞的存在不会影响到感染平衡点的局部稳定性,但能影响平衡点到达的时间跨度,对于药物治疗的疗程和治疗时机的确定有参考意义。     

The effect of long time delays in predator-prey systems

Past studies have indicated that a time delay longer than the natural period of a system will generally cause instability; however here it is shown that including long maturational time delays in a general predator-prey model need not have this effect. In each of the three cases studied (a predator delay, a prey delay, and both), local stability can persevere despite the presence of arbitrarily long time delays. This perseverence depends upon an interaction between delayed and undelayed features of the model. Delayed processes always act to destabilize the model. For example, prey self-regulation, usually a source of stability, becomes destabilizing if subject to a long delay. However, the effect of such a delay is offset by undelayed regulatory processes, such as a stabilizing functional reponse. In addition, the adverse effects of delayed predator recruitment can be reduced by the nonreproductive component of the numerical reponse, a feature not usually involved in determining stability. Finally, it is shown that long time delays are not necessarily more disruptive than short delays; it cannot be assumed that lengthening a time delay progessively reduces stability.     

Hopf bifurcation in three-species food chain models with group defense.

Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.     

Stability analysis of the time delay in a host-parasitoid model

The effect of a time delay on the local stability of a host-parasitoid model is analyzed. The delay is between the time of parasitization of the host and the emergence of the parasitoid from the host. Both analytic methods and computer simulations are used in this study. By linearizing and transforming the original equations, sufficient conditions for the local stability are found. In the case of the parameters considered, the results illustrate the destabilizing effect of the time delay. As the lag increases the number of stable points decreases and the points become more scattered in the parameter space. Simulations of the original model are also produced. The region of stability indicated by the simulations is greater than that predicted by the use of the analytic technique. The analysis also reveals the impact of the population parameters upon the stability of the time delay model.The importance of understanding time lags is discussed with reference to population regulation.     

Role of reflex gain and reflex delay in spinal stability--a dynamic simulation

The goal of this study was to investigate the role of reflex and reflex time delay in muscle recruitment and spinal stability. A dynamic biomechanical model of the musculoskeletal spine with reflex response was implemented to investigate the relationship between reflex gain, co-contraction, and stability in the spine. The first aim of the study was to investigate how reflex gain affected co-contraction predicted in the model. It was found that reflexes allowed the model to stabilize with less antagonistic co-contraction and hence lower metabolic power than when limited to intrinsic stiffness alone. In fact, without reflexes there was no feasible recruitment pattern that could maintain spinal stability when the torso was loaded with 200N external load. Reflex delay is manifest in the paraspinal muscles and represents the time from a perturbation to the onset of reflex activation. The second aim of the study was to investigate the relationship between reflex delay and the maximum tolerable reflex gain. The maximum acceptable upper bound on reflex gain decreased logarithmically with reflex delay. Thus, increased reflex delay and reduced reflex gain requires greater antagonistic co-contraction to maintain spinal stability. Results of this study may help understanding of how patients with retarded reflex delay utilize reflex for stability, and may explain why some patients preferentially recruit more intrinsic stiffness than healthy subjects.     

一类具有时滞的传染病模型的稳定性分析

研究了一类具有时滞的传染病生物模型．首先研究了该模型的线性稳定性,并给出了一列Hopf分支值,然后利用中心流形定理和正规型方法,给出了确定分支周期解的分支方向与稳定性的计算公式．     

一类具有时滞和治愈率的HIV病理模型的稳定性

研究一类具有饱和感染率、治愈率和细胞内时滞的HIV病理模型.首先分析平衡态的存在性与稳定性,然后给出染病平衡态对于任意时滞保持稳定（不稳定）的充分条件,并利用Nyquist准则度量染病平衡点保持稳定的时滞长度.     

Absolute stability in predator-prey models

An equilibrium of a time-lagged population model is said to be absolutely stable if it remains locally stable regardless of the length of the time delay, and it is argued that the criteria for absolute stability provide a valuable guide to the behavior of population models. For example, it is sometimes assumed that time delays have a limited impact until they exceed the natural time scale of a system; here it is stressed that under some conditions very short time delays can have a marked (and often maximal) destabilizing effect. Consequently it is important that our understanding of population dynamics is robust to the inclusion of the short time delays present in all biological systems. The absolute stability criteria are ideally suited for this role. Another important reason for using the criteria for absolute stability rather than using criteria which depend upon the details of a time delay is that biological time delays are unlikely to be constant. For example, a time delay due to maturation inevitably varies between individuals and the mean may itself vary over time. Here it is shown that the criteria for absolute stability are generally robust in the presence of distributed delays and of varying delays. The analysis presented is based upon a general predator-prey model and it is shown that absolute stability can be expected under a broad range of parameter values whenever the time delay is due to the maturation time of either the predator or the prey or of both. This stability occurs because of the interaction between delayed and undelayed dynamic features of the model. A time-delayed process, when viewed across all possible delays, always reduces stability and this effect occurs regardless of whether the process would act to stabilize or destabilize an undelayed system. Opposing the destabilization due to a time delay and making absolute stability a possibility are a number of processes which act without delay. Some of these processes can be identified as stabilizing from the analysis of undelayed models (for example, the type 3 functional response) but other cannot (for example, the nonreproductive numerical response of predators).     

Role of gestation delay in a plankton-fish model under stochastic fluctuations

The present paper studies a minimal prey-predator model in the context of marine plankton interaction together with predation by planktivorous fish. The time lag required for gestation of the predator is incorporated and the resulting delayed model is analyzed for stability and bifurcation phenomena. A stochastic extension of the model is considered by perturbing the growth process of phytoplankton using colored noise process known to be more appropriate for the marine environment. The stochastic models with and without gestation delay are analyzed for stability aspects and a threshold value of gestation delay is obtained; this threshold is then compared with that of the deterministic model.     

Modeling and analysis of a predator-prey model with disease in the prey

A system of retarded functional differential equations is proposed as a predator-prey model with disease in the prey. Mathematical analyses of the model equations with regard to invariance of non-negativity, boundedness of solutions, nature of equilibria, permanence and global stability are analyzed. If the coefficient in conversing prey into predator k=k(0) is constant (independent of delay tau;, gestation period), we show that positive equilibrium is locally asymptotically stable when time delay tau; is suitable small, while a loss of stability by a Hopf bifurcation can occur as the delay increases. If k=k(0)e(-dtau;) (d is the death rate of predator), numerical simulation suggests that time delay has both destabilizing and stabilizing effects, that is, positive equilibrium, if it exists, will become stable again for large time delay. A concluding discussion is then presented.     

具免疫时滞的HIV感染模型动力学性质分析

讨论了一类具免疫时滞的HIV感染模型.分析了未感染平衡点的全局渐近稳定性,给出了感染无免疫平衡点及感染免疫平衡点局部渐近稳定的充分条件.数值模拟结果表明,当易感细胞生成率的取值使得基本再生数满足平衡存在的条件且低于某一临界值时,时滞对平衡点的稳定性没有影响;若大于该临界值,随着时滞增大,稳定性开关发生,平衡点不稳定,出现一系列Hopf分支,最终表现为周期波动模式.     

Analysis of tumour-immune evasion with chemo-immuno therapeutic treatment with quadratic optimal control

A simple mathematical model for the growth of tumour with discrete time delay in the immune system is considered. The dynamical behaviour of our system by analysing the existence and stability of our system at various equilibria is discussed elaborately. We set up an optimal control problem relative to the model so as to minimize the number of tumour cells and the chemo-immunotherapeutic drug administration. Sensitivity analysis of tumour model reveals that parameter value has a major impact on the model dynamics. We numerically illustrate how does these delay can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumour. Finally, epidemiological implications of our analytical findings are addressed critically.     

Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.     

Effects of delay,truncations and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models

Understanding effects of hypotheses about reproductive influences, reproductive schedules and the model mechanisms that lead to a loss of stability in a structured model population might provide information about the dynamics of natural population. To demonstrate characteristics of a discrete time, nonlinear, age structured population model, the transition from stability to instability is investigated. Questions about the stability, oscillations and delay processes within the model framework are posed. The relevant processes include delay of reproduction and truncation of lifetime, reproductive classes, and density dependent effects. We find that the effects of delaying reproduction is not stabilizing, but that the reproductive delay is a mechanism that acts to simplify the system dynamics. Density dependence in the reproduction schedule tends to lead to oscillations of large period and towards more unstable dynamics. The methods allow us to establish a conjecture of Levin and Goodyear about the form of the stability in discrete Leslie matrix models.This research was supported in part by the US Environmental Protection Agency under cooperation agreement CR-816081     

Stability of the human respiratory control system I. Analysis of a two-dimensional delay state-space model

A number of mathematical models of the human respiratory control system have been developed since 1940 to study a wide range of features of this complex system. Among them, periodic breathing (including Cheyne-Stokes respiration and apneustic breathing) is a collection of regular but involuntary breathing patterns that have important medical implications. The hypothesis that periodic breathing is the result of delay in the feedback signals to the respiratory control system has been studied since the work of Grodins et al. in the early 1950's [12]. The purpose of this paper is to study the stability characteristics of a feedback control system of five differential equations with delays in both the state and control variables presented by Khoo et al. [17] in 1991 for modeling human respiration. The paper is divided in two parts. Part I studies a simplified mathematical model of two nonlinear state equations modeling arterial partial pressures of O2 and CO2 and a peripheral controller. Analysis was done on this model to illuminate the effect of delay on the stability. It shows that delay dependent stability is affected by the controller gain, compartmental volumes and the manner in which changes in the ventilation rate is produced (i.e., by deeper breathing or faster breathing). In addition, numerical simulations were performed to validate analytical results. Part II extends the model in Part I to include both peripheral and central controllers. This, however, necessitates the introduction of a third state equation modeling CO2 levels in the brain. In addition to analytical studies on delay dependent stability, it shows that the decreased cardiac output (and hence increased delay) resulting from the congestive heart condition can induce instability at certain control gain levels. These analytical results were also confirmed by numerical simulations.     

The Effect of Time Distribution Shape on a Complex Epidemic Model

In elaborating a model of the progress of an epidemic, it is necessary to make assumptions about the distributions of latency times and infectious times. In many models, the often implicit assumption is that these times are independent and exponentially distributed. We explore the effects of altering the distribution of latency and infectious times in a complex epidemic model with regional divisions connected by a travel intensity matrix. We show a delay in spread with more realistic latency times. More realistic infectiousness times lead to faster epidemics. The effects are similar but accentuated when compared to a purely homogeneous mixing model.