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.     

A stage structured predator-prey model and its dependence on maturation delay and death rate

Many of the existing models on stage structured populations are single species models or models which assume a constant resource supply. In reality, growth is a combined result of birth and death processes, both of which are closely linked to the resource supply which is dynamic in nature. From this basic standpoint, we formulate a general and robust predator-prey model with stage structure with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and computational study. Our work indicates that if the juvenile death rate (through-stage death rate) is nonzero, then for small and large values of maturation time delays, the population dynamics takes the simple form of a globally attractive steady state. Our linear stability work shows that if the resource is dynamic, as in nature, there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics.Work is partially supported by NSF grant DMS-0077790.Mathamatics Subject Classification (2000):92D25, 35R10Revised version: 26 February 2004     

Delay-decomposing approach to robust stability for switched interval networks with state-dependent switching

This paper is concerned with a class of nonlinear uncertain switched networks with discrete time-varying delays . Based on the strictly complete property of the matrices system and the delay-decomposing approach, exploiting a new Lyapunov–Krasovskii functional decomposing the delays in integral terms, the switching rule depending on the state of the network is designed. Moreover, by piecewise delay method, discussing the Lyapunov functional in every different subintervals, some new delay-dependent robust stability criteria are derived in terms of linear matrix inequalities, which lead to much less conservative results than those in the existing references and improve previous results. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.     

Mean square stability of uncertain stochastic BAM neural networks with interval time-varying delays

The robust asymptotic stability analysis for uncertain BAM neural networks with both interval time-varying delays and stochastic disturbances is considered. By using the stochastic analysis approach, employing some free-weighting matrices and introducing an appropriate type of Lyapunov functional which takes into account the ranges for delays, some new stability criteria are established to guarantee the delayed BAM neural networks to be robustly asymptotically stable in the mean square. Unlike the most existing mean square stability conditions for BAM neural networks, the supplementary requirements that the time derivatives of time-varying delays must be smaller than 1 are released and the lower bounds of time varying delays are not restricted to be 0. Furthermore, in the proposed scheme, the stability conditions are delay-range-dependent and rate-dependent/independent. As a result, the new criteria are applicable to both fast and slow time-varying delays. Three numerical examples are given to illustrate the effectiveness of the proposed criteria.     

A novel global exponential stability result for discrete-time cellular neural networks with variable delays

Global exponential stability is considered for a class of discrete-time cellular neural networks with variable delays. By employing a discrete Halanay inequality, a new result is presented ensuring global exponential stability of the unique equilibrium point of the networks. The result extends and improves the earlier publications due to the fact that it removes some restrictions on the delay. An example is given to illustrate the effectiveness of the global exponential stability condition provided here.     

A new criterion for the global stability of simultaneous cell replication and maturation processes

 We analyze a population model of cells that are capable of simultaneous and independent proliferation and maturation. This model is described by a first order partial differential equation with a time delay and a retardation of the maturation variable, both due to cell replication. We provide a general criterion for global stability in such equations. Received: 26 August 1996 / Revised version: 22 March 1997     

Global and robust stability of interval Hopfield neural networks with time-varying delays

In this paper, we investigate the problem of global and robust stability of a class of interval Hopfield neural networks that have time-varying delays. Some criteria for the global and robust stability of such networks are derived, by means of constructing suitable Lyapunov functionals for the networks. As a by-product, for the conventional Hopfield neural networks with time-varying delays, we also obtain some new criteria for their global and asymptotic stability.     

Delay-dependent robust exponential stability for uncertain recurrent neural networks with time-varying delays

This paper considers the problem of robust exponential stability for a class of recurrent neural networks with time-varying delays and parameter uncertainties. The time delays are not necessarily differentiable and the uncertainties are assumed to be time-varying but norm-bounded. Sufficient conditions, which guarantee that the concerned uncertain delayed neural network is robustly, globally, exponentially stable for all admissible parameter uncertainties, are obtained under a weak assumption on the neuron activation functions. These conditions are dependent on the size of the time delay and expressed in terms of linear matrix inequalities. Numerical examples are provided to demonstrate the effectiveness and less conservatism of the proposed stability results.     

A schistosomiasis model with mating structure and time delay

A system of homogeneous equations with a time delay is used to model the population dynamics of schistosomes. The model includes the parasite’s mating structure, multiple resistant schistosome strains, and biological complexity associated with the parasite’s life cycle. Invasion criteria of resistant strains and coexistence threshold conditions are derived. These results are used to explore the impact of drug treatment on resistant strain survival. Numerical simulations indicate that the dynamical behaviors of the current model are not qualitatively different from those derived from an earlier model that ignores the impact of time delays associated with the multiple stages in parasite’s life cycle. However, quantitatively the time delays make it more likely for drug-resistant strains to invade in a parasite population.     

Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model

The present work employs a nonlocal delay reaction-diffusion model to study the impacts of the density dependent birth function, maturation time delay and population dispersal on single species dynamics (i.e., extinction, survival, extinction-survival). It is shown that the maturation time and the birth function are two major factors determining the fate of single species. Whereas the dispersal acts as a subsidiary factor that only affects the spatial patterns of population densities. When the birth function has a compensating density dependence, maturation time delay cannot destabilize the population survival at the positive equilibrium. Nevertheless, when the birth function has an over-compensating density dependence, the population densities of single species fluctuate in the spatial domain due to the increased maturation time delay. With the Allee effect and over-compensating density dependence, the increases in the maturation time may cause extinction of the single species in the entire spatial domain. The numerical simulations suggest that the solutions of the general model may temporarily remain nearby a stationary wave pulse or a stationary wavefront of the reduced model. The former indicates the survival of single species in a narrow region of the spatial domain. Whereas the latter represents the survival in the entire left-half or right-half of the spatial domain.     

Global robust power-rate stability of delayed genetic regulatory networks with noise perturbations

In this paper, by using the Lyapunov method, Itô’s differential formula and linear matrix inequality (LMI) approach, the global robust power-rate stability in mean square is discussed for genetic regulatory networks with unbounded time-varying delay, noise perturbations and parameter uncertainties. Sufficient conditions are given to ensure the robust power-rate stability (in mean square) of the genetic regulatory networks. Meanwhile, the criteria ensuring global power-rate stability in mean square are a byproduct of the criteria guaranteeing global robust power-rate stability in mean square. The obtained conditions are derived in terms of linear matrix inequalities (LMIs) which are easy to be verified via the LMI toolbox. An illustrative example is given to show the effectiveness of the obtained result.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Mean square exponential and robust stability of stochastic discrete-time genetic regulatory networks with uncertainties

This paper aims to analyze global robust exponential stability in the mean square sense of stochastic discrete-time genetic regulatory networks with stochastic delays and parameter uncertainties. Comparing to the previous research works, time-varying delays are assumed to be stochastic whose variation ranges and probability distributions of the time-varying delays are explored. Based on the stochastic analysis approach and some analysis techniques, several sufficient criteria for the global robust exponential stability in the mean square sense of the networks are derived. Moreover, two numerical examples are presented to show the effectiveness of the obtained results.     

变时延神经网络的全局指数稳定性及收敛性

利用Lyapunov泛函方法和线性矩阵不等式（LMI）技术,通过引入一系列参数,给出全局指数稳定的平衡点的判别条件和时延的最大上界和神经网络的收敛速度,所得结果较之一些文献中的结果简单、实用并且对于具体设计带时延神经网络有重要的指导意义.最后,通过实例表明给出的判定条件是有效、可行的.     

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.     

一个时滞微分系统的稳定性与Hopf分支

给出了一个三维时滞微分系统的平衡点的全时滞稳定的代数判据。也讨论并给出了这个系统存在Hopf分支的条件,两个例子说明了本文定理的应用。     

含有两个时滞项的退化时滞微分方程周期解

讨论了含有两个时滞项退化时滞微分方程的周期解的问题,特别的,给出了此类方程存在非常数周期解的充要条件,并对二维退化微分方程给出了非常数周期解存在性的代数判据,并在最后给出一个例子验证了判据的有效性.     

On the estimation of robustness and filtering ability of dynamic biochemical networks under process delays, internal parametric perturbations and external disturbances

Inherently, biochemical regulatory networks suffer from process delays, internal parametrical perturbations as well as external disturbances. Robustness is the property to maintain the functions of intracellular biochemical regulatory networks despite these perturbations. In this study, system and signal processing theories are employed for measurement of robust stability and filtering ability of linear and nonlinear time-delay biochemical regulatory networks. First, based on Lyapunov stability theory, the robust stability of biochemical network is measured for the tolerance of additional process delays and additive internal parameter fluctuations. Then the filtering ability of attenuating additive external disturbances is estimated for time-delay biochemical regulatory networks. In order to overcome the difficulty of solving the Hamilton Jacobi inequality (HJI), the global linearization technique is employed to simplify the measurement procedure by a simple linear matrix inequality (LMI) method. Finally, an example is given in silico to illustrate how to measure the robust stability and filtering ability of a nonlinear time-delay perturbative biochemical network. This robust stability and filtering ability measurement for biochemical network has potential application to synthetic biology, gene therapy and drug design.     

Design of state estimator for genetic regulatory networks with time-varying delays and randomly occurring uncertainties

In this paper, the design problem of state estimator for genetic regulatory networks with time delays and randomly occurring uncertainties has been addressed by a delay decomposition approach. The norm-bounded uncertainties enter into the genetic regulatory networks (GRNs) in random ways, and such randomly occurring uncertainties (ROUs) obey certain mutually uncorrelated Bernoulli distributed white noise sequences. Under these circumstances, the state estimator is designed to estimate the true concentration of the mRNA and the protein of the uncertain GRNs. Delay-dependent stability criteria are obtained in terms of linear matrix inequalities by constructing a Lyapunov–Krasovskii functional and using some inequality techniques (LMIs). Then, the desired state estimator, which can ensure the estimation error dynamics to be globally asymptotically robustly stochastically stable, is designed from the solutions of LMIs. Finally, a numerical example is provided to demonstrate the feasibility of the proposed estimation schemes.