A seven-mode truncation of the plane incompressible Navier-Stokes equations

A model obtained by a seven-mode truncation of the Navier-Stokes equations for a two-dimensional incompressible fluid on a torus is studied. This model, extending a previously studied five-mode one, exhibits a very rich and varied phenomenology including some remarkable properties of hysteresis (i.e., coexistence of attractors). A stochastic behavior is found for high values of the Reynolds number, when no stable fixed points, closed orbits, or tori are present.     

Noise-induced extinction in Bazykin-Berezovskaya population model

A nonlinear Bazykin-Berezovskaya prey-predator model under the influence of parametric stochastic forcing is considered. Due to Allee effect, this conceptual population model even in the deterministic case demonstrates both local and global bifurcations with the change of predator mortality. It is shown that random noise can transform system dynamics from the regime of coexistence, in equilibrium or periodic modes, to the extinction of both species. Geometry of attractors and separatrices, dividing basins of attraction, plays an important role in understanding the probabilistic mechanisms of these stochastic phenomena. Parametric analysis of noise-induced extinction is carried out on the base of the direct numerical simulation and new analytical stochastic sensitivity functions technique taking into account the arrangement of attractors and separatrices.     

Individual-based model for coevolving competing populations

Classical models for competition between two species usually predict exclusion or divergent evolution of resource exploitation. However, recent experimental data show that coexistence is possible for very similar species competing for the same resources without niche partition. Motivated by this experimental challenge to classical competition theory, we propose an individual-based stochastic competition model, which is essentially a modification of a deterministic Lotka-Volterra type model. The proposed model of competition dynamics incorporates the effects of a discrete genotype, which determines the individual's adaptation to the environment, as well as its interaction with the other species.     

Neutral Community Theory: How Stochasticity and Dispersal-Limitation Can Explain Species Coexistence

Neutral community theory explains biodiversity, i.e. the coexistence of several species, as the result of a stochastic balance between immigration and extinction on a local level, and between speciation and extinction on a regional level. The most popular model, presented by Hubbell in 2001, has seen many analytical developments in recent years, which can be used in model analysis, model testing and model comparison. We review these developments here, and present alternative derivations and shine previously unnoticed lights on them.     

Basins of coexistence and extinction in spatially extended ecosystems of cyclically competing species

Microscopic models based on evolutionary games on spatially extended scales have recently been developed to address the fundamental issue of species coexistence. In this pursuit almost all existing works focus on the relevant dynamical behaviors originated from a single but physically reasonable initial condition. To gain comprehensive and global insights into the dynamics of coexistence, here we explore the basins of coexistence and extinction and investigate how they evolve as a basic parameter of the system is varied. Our model is cyclic competitions among three species as described by the classical rock-paper-scissors game, and we consider both discrete lattice and continuous space, incorporating species mobility and intraspecific competitions. Our results reveal that, for all cases considered, a basin of coexistence always emerges and persists in a substantial part of the parameter space, indicating that coexistence is a robust phenomenon. Factors such as intraspecific competition can, in fact, promote coexistence by facilitating the emergence of the coexistence basin. In addition, we find that the extinction basins can exhibit quite complex structures in terms of the convergence time toward the final state for different initial conditions. We have also developed models based on partial differential equations, which yield basin structures that are in good agreement with those from microscopic stochastic simulations. To understand the origin and emergence of the observed complicated basin structures is challenging at the present due to the extremely high dimensional nature of the underlying dynamical system.     

Stochastic Bistable Systems: Competing Hysteresis and Phase Coexistence

In this paper we describe the solution of a stochastic bistable system from a dynamical perspective. We show how a single framework with variable noise can explain hysteresis at zero temperature and two-state coexistence in the presence of noise. This feature is similar to the phase transition of thermodynamics. Our mathematical model for bistable systems also explains how the width of a hysteresis loop shrinks in the presence of noise, and how variation in initial conditions can take such systems to different final states.     

输入方波信号的过阻尼谐振子的随机共振

计算了在输入方波信号情况下过阻尼谐振子的输出功率谱和信噪比,详细讨论了相应的随机共振现象,并与输入余弦信号情况进行对比.研究发现:它们均出现随机共振现象,且均存在共振和抑制并存的现象.但这一并存现象,在两模型中分别出现于不同的共振曲线. 关键词： 过阻尼谐振子 随机共振 方波信号     

Method of confidence domains in the analysis of noise-induced extinction for tritrophic population system

A problem of the analysis of the noise-induced extinction in multidimensional population systems is considered. For the investigation of conditions of the extinction caused by random disturbances, a new approach based on the stochastic sensitivity function technique and confidence domains is suggested, and applied to tritrophic population model of interacting prey, predator and top predator. This approach allows us to analyze constructively the probabilistic mechanisms of the transition to the noise-induced extinction from both equilibrium and oscillatory regimes of coexistence. In this analysis, a method of principal directions for the reducing of the dimension of confidence domains is suggested. In the dispersion of random states, the principal subspace is defined by the ratio of eigenvalues of the stochastic sensitivity matrix. A detailed analysis of two scenarios of the noise-induced extinction in dependence on parameters of considered tritrophic system is carried out.     

非理想MHD流的奇异吸引子与混沌现象

本文提出并研究了在剪切磁场中非理想MHD流的Rayleigh-Bnard问题的一个模型,得到了关于这个模型的一个新的非线性微分方程组。理论和数值分析表明:这组方程蕴含一个奇异吸引子,它具有不同于Lorenz吸引子的一些新特性;更重要的是,已知的三条通往混沌的道路并存于这个模型之中。应当指出,在迄今所有已知的向混沌态过渡的三条道路共存的模型中，我们的方程组是唯一没有外部周期驱动项的，更直接地体现了非线性确定论系统的“内在”随机件、另外，对这个简单模型进行数值模拟．我们观察到磁力线的随机运动、磁力线重联和磁岛 关键词：     

A model problem with the coexistence of stochastic and integrable behaviour

A one parameter family of piecewise linear measure preserving transformations of a torus which can be viewed as a perturbation of the twist mapping is introduced. Theorems on their ergodic properties for an infinite set of parameters are proved. For some parameters coexistence of stochastic and integrable behaviour is obtained.     

Moving average rules as a source of market instability

Despite the pervasiveness of the efficient markets paradigm in the academic finance literature, the use of various moving average (MA) trading rules remains popular with financial market practitioners. This paper proposes a stochastic dynamic financial market model in which demand for traded assets has both a fundamentalist and a chartist component. The chartist demand is governed by the difference between current price and a (long-run) MA. Our simulations show that the MA is a source of market instability, and the interaction of the MA and market noises can lead to the tendency for the market price to take long excursions away from the fundamental. The model reveals various market price phenomena, the coexistence of apparent market efficiency and a large chartist component, price resistance levels, long memory and skewness and kurtosis of returns.     

Cyclic dominance and biodiversity in well-mixed populations

Coevolutionary dynamics is investigated in chemical catalysis, biological evolution, social and economic systems. The dynamics of these systems can be analyzed within the unifying framework of evolutionary game theory. In this Letter, we show that even in well-mixed finite populations, where the dynamics is inherently stochastic, biodiversity is possible with three cyclic-dominant strategies. We show how the interplay of evolutionary dynamics, discreteness of the population, and the nature of the interactions influences the coexistence of strategies. We calculate a critical population size above which coexistence is likely.     

Spatio-temporal dynamics in simple asymmetric hypercycles under weak parasitic coupling

Hypercycles represent an example of cooperation at the molecular level suggested to be involved in the evolution of the first autonomous, self-reproducing molecular living-like systems. However, they are vulnerable to parasites: an established hypercycle may decay if a selfish replicator appears i.e. one that receives catalytic help but does not reciprocate catalysis. In this work we study the dynamics of a two-member hypercycle with an attached, weak parasite. We are using a mean field model and a stochastic cellular automaton (CA) focusing on the role of the kinetic properties of both hypercycle replicators, exploited by the parasite. Both approaches show three possible, parameter-dependent outcomes: (i) hypercycle stability and parasite extinction; (ii) extinction of the entire system; and (iii) coexistence between the hypercycle and the parasite. Scenario (iii) is shown to be structurally unstable in the mean field model but the addition of space allows a wider coexistence phase. Moreover the CA model also indicates that the degree of hypercycle’s asymmetry may be relevant in the survival of the catalytic replicators, being also responsible for qualitatively different dynamics e.g. if the parasite is attached to the fittest hypercycle unit, low-dimensional chaos arises. The coexistence phase is characterized by the emergence of hypercyclic aggregates with a broad distribution of cluster sizes and mixed patterns of replicators able to provide the hypercycle with resistance against the parasites. Nevertheless the resistance to the parasite is shown to reduce with increase of diffusion. The role of the kinetic properties of the hypercycle and of diffusion in the survival of the hypercycle is discussed in the context of prebiotic evolution.     

Mixed Ising system designed with integer and half-integer spins: dynamic behaviors under oscillating magnetic field

By using the simplified effective-field theory based on Glauber-type stochastic dynamics, namely the dynamic simplified effective-field theory, the dynamic phase diagrams of a two-dimensional mixed spin-1 and spin-3/2 Blume–Capel model are studied in an oscillating external magnetic field. The dynamic equations are derived for two interpenetrating square lattices. The time variations of average magnetizations and the temperature dependence of the dynamic magnetizations are examined, and the dynamic phase diagrams are presented in the two different planes. The dynamic phase diagrams illustrate several ordered phases, the coexistence phase region and critical points as well as a re-entrant behavior depending on the interaction parameters. Finally, the discussion and comparison of the dynamic phase diagrams are given briefly.     

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

We study the general properties of stochastic two-species models for predator-prey competition and coexistence with Lotka–Volterra type interactions defined on a d-dimensional lattice. Introducing spatial degrees of freedom and allowing for stochastic fluctuations generically invalidates the classical, deterministic mean-field picture. Already within mean-field theory, however, spatial constraints, modeling locally limited resources, lead to the emergence of a continuous active-to-absorbing state phase transition. Field-theoretic arguments, supported by Monte Carlo simulation results, indicate that this transition, which represents an extinction threshold for the predator population, is governed by the directed percolation universality class. In the active state, where predators and prey coexist, the classical center singularities with associated population cycles are replaced by either nodes or foci. In the vicinity of the stable nodes, the system is characterized by essentially stationary localized clusters of predators in a sea of prey. Near the stable foci, however, the stochastic lattice Lotka–Volterra system displays complex, correlated spatio-temporal patterns of competing activity fronts. Correspondingly, the population densities in our numerical simulations turn out to oscillate irregularly in time, with amplitudes that tend to zero in the thermodynamic limit. Yet in finite systems these oscillatory fluctuations are quite persistent, and their features are determined by the intrinsic interaction rates rather than the initial conditions. We emphasize the robustness of this scenario with respect to various model perturbations.     

Analyzing the coexistence of DSRC and Wi-Fi networks using the Poisson line Cox process

Vehicular networks can aid in traffic monitoring, autonomous driving, and car accidents prevention. Yet, the deployment of these networks has been delayed due to the limited spectrum, especially for the case of unlicensed operations. To handle this issue, the Federal Communications Commission (FCC) proposed to permit Wi-Fi devices to operate in the 5.9 GHz band allocated to the intelligent transportation system (ITS). In a recent work, we analyzed the impact of the coexistence of dedicated short range communications (DSRC) and Wi-Fi on future DSRC network deployments by developing a stochastic geometry analytical model that considers a dynamic medium access probability (MAP) of DSRC nodes which uses carrier sense multiple access with collision avoidance (CSMA/CA). This previous work was based on the standard 2D homogeneous Poisson Point Process (PPP) model. In this work, we model the roads using the more applicable but more complex Poisson line process (PLP) Cox point process. We generate performance metrics represented through coverage probability and area system throughput, and we compare these results to our earlier work. The importance of this work is two-fold. First, it allows a further understanding of the impact of DSRC-Wi-Fi coexistence on future DSRC network deployments, and second, it highlights the effectiveness of the PLP in modeling the distribution of vehicles in an area by producing more accurate performance results.     

Equilibration and relaxation times at the chiral phase transition including reheating

We investigate the relaxational dynamics of the order parameter of chiral symmetry breaking, the sigma mean-field, with a heat bath consisting of quarks and antiquarks. A semiclassical stochastic Langevin equation of motion is obtained from the linear sigma model with constituent quarks. The equilibration of the system is studied for a first order phase transition and a critical point, where a different behavior is found. At the first order phase transition we observe the phase coexistence and at a critical point the phenomenon of critical slowing down with large relaxation times. We go beyond existing Langevin studies and include reheating of the heat bath by determining the energy dissipation during the relaxational process. The energy of the entire system is conserved. In a critical point scenario we again observe critical slowing down.     

A maximum entropy mean field method for driven diffusive systems

We introduce a method of generating systematic mean field (MF) approximations for the nonequilibrium steady state of ferromagnetic Ising driven diffusive systems (DDS), based on the maximum entropy principle due to Jaynes. In the phase coexistence region, MF approximations to the master equation do not provide a closed system of equations in the MF variables. This can be traced to the conservation of the order parameter by the stochastic dynamics. Our maximum entropy mean field (MEMF) approximation method is applicable to high temperatures as well to the low-temperature phase coexistence region. It is based on a derivation of a generalized variational free energy from the maximum entropy principle, with the MF evolution equations playing the role of constraints. In the phase coexistence region this free energy is nonconvex and is interpreted by means of a Maxwell construction. We use a pair-level variant of the MEMF approximation to calculate quantities of interest for the ferro-magnetic Ising DDS on a square lattice. Results of calculations with several different choices of transition rates satisfying local detailed balance are discussed and compared with those obtained by other methods.     

Effects of migration on the evolutionary game dynamics in finite populations with community structures

We investigate the impacts of migration on the evolutionary game dynamics in finite populations with community structures in the framework of evolutionary game theory. In contrast to deterministic dynamics, our model incorporates stochastic factors induced by the finite population size. Based on the analysis of the stationary distribution of the evolutionary process in the limit of rare mutations, we prove that it is most likely to find the population in the community where all individuals have the lower migration rate. Furthermore, we show that reducing the difference between the migration rates of distinct communities can increase the first hitting time to the homogeneous absorbing state and can prolong the coexistence time of different species, promoting the conservation of biodiversity.     

周期外力对频率涨落的过阻尼谐振子所作的功和能量随机共振

研究了周期外力对频率涨落的过阻尼谐振子系统作功的特点. 结果揭示了瞬时功率随时间的周期变化出现不对称性. 研究结果还揭示周期外力一个周期对系统所做的功随乘法噪声强度的变化出现非单调行为, 系统是否出现能量随机共振与抑制并存, 由乘法噪声与加法噪声之间互关联的符号决定. 关键词： 过阻尼谐振子 频率涨落 周期外力作功 能量随机共振与抑制