Effects of time-lagged niche construction on metapopulation dynamics and environmental heterogeneity

Time-delayed responses to environmental changes and disturbance can beget profound effects on the spatiotemporal dynamics of metapopulations. Here, we first examined the effect of three forms of time-lag (that is, equal-weight, recency and primacy effects) on population dynamics, using a spatially structured lattice model. The time-lag was incorporated in the niche construction process of the system (an organism–environment feedback). Using bifurcations diagrams and numerical simulations, we found that the time-lag can form a phase-locked oscillation. Three typical spatial patterns emerged: spiral wave, spiral-broken wave and circular wave. These spatial patterns gradually become immobile as a result of the self-organized ecological imprinting due to niche construction. Therefore, the phase-locked oscillation and the ecological imprinting process together determine the spatial structure of metapopulations and the environmental heterogeneity.     

On nonlinear oscillations in a suspension bridge system

In this paper, we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations. By applying the Leray-Schauder degree theory, it is proved that the suspension bridge system has at least two solutions, one is a near-equilibrium oscillation, and the other is a large amplitude oscillation.

     

Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis

We consider hysteresis perturbations of a system of ODEs which possesses an asymptotically stable periodic solution z_∗. Where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity, it is shown that the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of z_∗. The main result is a theorem on the stability of this continuum.     

The inhomogeneous Dirichlet problem for the Stokes system in Lipschitz domains with unit normal close to VMO

The goal of this work is to study the inhomogeneous Dirichlet problem for the Stokes system in a Lipschitz domain Ω ⊆ ℝ ⁿ , n⩾2. Our main result is that this problem is well posed in Besov-Triebel-Lizorkin spaces, provided that the unit normal ν to Ω has small mean oscillation.     

Numerical approach for solving fractional relaxation–oscillation equation

In this study, we will obtain the approximate solutions of relaxation–oscillation equation by developing the Taylor matrix method. A relaxation oscillator is a kind of oscillator based on a behavior of physical system’s return to equilibrium after being disturbed. The relaxation–oscillation equation is the primary equation of relaxation and oscillation processes. The relaxation–oscillation equation is a fractional differential equation with initial conditions. For this propose, generalized Taylor matrix method is introduced. This method is based on first taking the truncated fractional Taylor expansions of the functions in the relaxation–oscillation equation and then substituting their matrix forms into the equation. Hence, the result matrix equation can be solved and the unknown fractional Taylor coefficients can be found approximately. The reliability and efficiency of the proposed approach are demonstrated in the numerical examples with aid of symbolic algebra program, Maple.     

Center manifold approach to the control of a tethered satellite system

It is well known that in a linearized analysis the in-plane oscillation of a tethered satellite system about the radial earth pointing position decouples from the out-of-plane oscillation. By tension control, therefore, only the in-plane but not the out-of-plane oscillation can be affected. Hence, using tension control linearization of the equations of motion cannot be used and a nonlinear problem must be treated. For a simple mechanical model of a tethered satellite system we show by means of center manifold theory that for the nonlinear system the out-of-plane oscillations can be stabilized by tension control.     

具有无界时滞的微分方程解的振动性

具有无界时滞的微分方程解的振动性刘正荣（云南大学数学系和云南省应用数学研究所，昆明６５００９１）俞元洪（中国科学院应用数学研究所，北京１０００８０）基金项目：国家自然科学基金资助项目１９９２年３月２５日收到．一、引言考虑泛函微分方程ｉ＝１其中，．关于...     

Nonlinear features in protein circuitry

The noise involved in protein circuit can result in fluctuations in protein concentrations. Then we have explored the effect of such noise on the feedback loop between p53 and its repressor Mdm2, the negative feedback dynamics and oscillatory activities are presented. Recent experimental results show that under certain conditions, the activity of the average protein level of p53 behaves with dampened oscillation in response to DNA damage, and it has non-decaying oscillatory behavior in individual cells, and we show that the dampening is induced by intrinsic noise, namely the uncertainty associated with chemical kinetics in dealing with when and in what order reactions take place in the p53 system. Furthermore, the experimental results are reproduced in this paper.     

A note on the theorems of paley and stein

We show that, under the conditions of the theorems of Paley and Stein, a system with bounded norms in L∞ can be replaced by a system with bounded norms in the space of functions of bounded mean oscillation.     

Super-high-frequency oscillations in a discontinuous dynamic system with time delay

We study oscillations in the discontinuous dynamic system with time delay . This is a typical model of relay feedback with delay. It is known that stable modes in this system have a bounded oscillation frequency. Here we consider transient processes and obtain the following result: under some restrictions ofF, the average oscillation frequency of any solution becomes finite after a period of time, i.e. super-high-frequency oscillations (with infinite frequency) exist only in a finite time interval. Moreover, we give an effective upper bound on the length of this interval.     

On oscillation of nonlinear second-order differential equations with damping term

The paper is concerned with oscillation of a novel class of nonlinear differential equations with a damping term. First it is demonstrated how known oscillation results for another intensively studied class of equations can be translated to the one in question, and vice versa. Advantages and drawbacks of such translation are carefully examined. Then an oscillation criterion for the new class of equations is established. The principal result of the paper is compared to those reported in the literature, and an illustrative example to which known oscillation criteria fail to apply is provided.     

The quasi-homogenized Bakhvalov–Eglit model of a thermoviscoelastic material beyond the periodic setting

The one-dimensional model of dynamics of a thermoviscoelastic Kelvin–Voigt material provided with rapidly oscillating initial distributions of specific volume, velocity, and specific internal energy is considered. It is allowed that the rapidly oscillating initial distributions do not have any ordered microstructure: periodic, quasi-periodic, random homogeneous, and so on. We rigorously justify the homogenization procedure as the frequency of rapid oscillations tends to infinity. As the result, we construct a closed limit effective model of a thermoviscoelastic material motion. This model contains an additional kinetic equation that carries complete information on the evolution of the limit oscillation regimes. We show that if the initial data are periodic, then the constructed limit model can be reduced to a system of the classical quasi-homogenized Bakhvalov–Eglit equations.     

Kinetic equation method for problems of viscous gas dynamics with rapidly oscillating density distributions

Equations describing the dynamics of a viscous gas are considered in a bounded space-time domain. It is assumed that the boundary values of density distributions oscillate rapidly. Limit regimes that arise when the oscillation frequencies tend to infinity are studied. As a result, a limit (averaged) model is constructed that contains full information on the limit oscillation regimes and includes an additional kinetic equation that has the form of the Boltzmann equation in the kinetic theory of gases.     

Stationary oscillation for high-order Hopfield neural networks with time delays and impulses

We investigate stationary oscillation for high-order Hopfield neural networks with time delays and impulses. In a recent paper [J. Zhang, Z. J. Gui, Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays, Journal of Computational and Applied Mathematics 224 (2008) 602-613], the authors claim that they obtain a criterion of existence, uniqueness, and global exponential stability of periodic solution (i.e. stationary oscillation) for high-order Hopfield neural networks with time delays and impulses. In this paper, we point out that the main result of the recent paper is unture, and present a new sufficient condition of stationary oscillation for the neural networks. A numerical example is given to illustrate the effectiveness of the obtained result.     

On perturbations of quasiperiodic Schrödinger operators

Using relative oscillation theory and the reducibility result of Eliasson, we study perturbations of quasiperiodic Schrödinger operators. In particular, we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials.     

Multiple limit cycles in an immune system

The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied by Huang has at least three limit cycles in the system. The conditions for the multiple limit cycles are useful in analyzing the nonlinear oscillation in immune response.     

A heavy traffic limit theorem for a class of open queueing networks with finite buffers

We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized d-dimensional queue length process converges in distribution to a fd-dimensional semimartingale reflecting Brownian motion (RBM) in a d-dimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks. This revised version was published online in June 2006 with corrections to the Cover Date.     

具有正负周期系数的差分方程的振动性与线性化振动性

本文得到具有正负周期系数的差分方程振动的一个充分必要条件,并利用此充要条件获得了一个线性化振动性结果。     

An oscillation criterion for discrete trigonometric systems

In this paper, we investigate oscillation properties of discrete trigonometric systems whose coefficients matrices are simultaneously symplectic and orthogonal. The main result generalizes a necessary and sufficient condition of non-oscillation of trigonometric systems proved by M. Bohner and O. Do?lý (J. Differential Equations 163 (2000), pp. 113–129) in the case when the block in the upper right corner of the coefficient matrix is symmetric and positive definite. Now, we present this oscillation criterion for an arbitrary trigonometric system. The obtained results are applied to formulate a necessary and sufficient condition for non-oscillation of even-order Sturm–Liouville difference equations.     

Complicated dynamics of a predator–prey system with Watt-type functional response and impulsive control strategy

Based on the classical predator–prey system with Watt-type functional response, an impulsive differential equations to model the process of periodic perturbations on the predator at different fixed time for pest control is proposed and investigated. It proves that there exists a globally asymptotically stable prey-eradication periodic solution when the impulse period is less than some critical value, and otherwise, the system can be permanent. Numerical results show that the system considered has more complicated dynamics involving quasi-periodic oscillation, narrow periodic window, wide periodic window, chaotic bands, period doubling bifurcation, symmetry-breaking pitchfork bifurcation, period-halving bifurcation and “crises”, etc. It will be useful for studying the dynamic complexity of ecosystems.