Bounding the quality of stochastic oscillations in population models

We analyze general two-species stochastic models, of the kind generally used for the study of population dynamics. Although usually defined a priori, the deterministic version of these models can be obtained as the infinite volume limit of many stochastic models (which are necessarily defined by more parameters than the deterministic one). It is known that damped oscillations in a deterministic model usually correspond to oscillatory-like fluctuations in their deterministic counterparts. The quality of these “oscillations" depends on details of each stochastic model. We show, however, that the parameters of the deterministic system are generally enough to obtain very good bounds for the quality of “oscillations" in any of its stochastic counterparts. These bounds are shown to depend on only one dimensionless parameter.     

Oscillations vs. chaotic waves: Attractor selection in bistable stochastic reaction–diffusion systems

In bistable systems, the long-term behavior of solutions depends on the location of the initial conditions. In a deterministic setting, where the initial condition is kept fixed in one particular basin of attraction, repeated numerical simulations will always lead to the same long-term behavior. The other possible asymptotic solution type will never be observed. This clear distinction does not hold anymore if the system is forced by random fluctuations. In this case, both asymptotic solutions can be attained, and the relative frequency of different long-term behaviors observed in many repeated simulation runs will follow a certain probability distribution. We present a simple reaction–diffusion model of spatial predator–prey interaction, where depending on the initial spatial distribution of the two populations either spatially homogeneous or spatiotemporal irregular oscillations may be observed. We show by repeated stochastic simulations that, when starting in the basin of attraction of the spatiotemporal irregular solution, in the randomly forced system the probability to observe spatially homogeneous oscillations instead of spatiotemporally irregular oscillations follows a non-trivial bimodal distribution.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noisy identical case

We theoretically investigate the collective phase synchronization between interacting groups of globally coupled noisy identical phase oscillators exhibiting macroscopic rhythms. Using the phase reduction method, we derive coupled collective phase equations describing the macroscopic rhythms of the groups from microscopic Langevin phase equations of the individual oscillators via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we determine the type of the collective phase coupling function, i.e., whether the groups exhibit in-phase or antiphase synchronization. We show that the macroscopic rhythms can exhibit effective antiphase synchronization even if the microscopic phase coupling between the groups is in-phase, and vice versa. Moreover, near the onset of collective oscillations, we analytically obtain the collective phase coupling function using center-manifold and phase reductions of the nonlinear Fokker-Planck equations.     

Relationships between a microscopic parameter and the stochastic equations for interface's evolution of two growth models

The relationship between a microscopic parameter p, that is related to the probability of choosing a mechanism of deposition, and the stochastic equation for the interface's evolution is studied for two different models. It is found that in one model, that is similar to ballistic deposition, the corresponding stochastic equation can be represented by a Kardar-Parisi-Zhang (KPZ) equation where both λ and ν depend on p in the following way: ν(p) = νp and λ(p) = λp ^3/2. Furthermore, in the other studied model, which is similar to random deposition with relaxation, the stochastic equation can be represented by an Edwards-Wilkinson (EW) equation where ν depends on p according to ν(p) = νp ². It is expected that these results will help to find a framework for the development of stochastic equations starting from microscopic details of growth models. Received 26 August 2002 / Received in final form 20 November 2002 Published online 6 March 2003 RID="a" ID="a"e-mail: ealbano@inifta.unlp.edu.ar     

Quantum microscopic vs. classical macroscopic calculations on the phenomenon of electrostatic influence

In order to compare microscopic and macroscopic approaches to the phenomenon of electrostatic influence, we have studied the atomic charges of an electric conductor, obtained either from macroscopic classical electrostatics, or microscopic quantum ab initio calculations. A torus was chosen as conducting material, built from valence monoelectronic atoms and influenced by an external point charge. The classical electric charges are obtained by integrating the macroscopic density over “atomic" sectors. This density is determined from a numerical integration of linearized electrostatic equations. The quantum charges are defined from Natural Orbitals in MP2/6-31G* calculations on clusters of different sizes. The overall agreement is good, with reasonable discrepancies due (i) to the continuity of the macroscopic model, which ignores the oscillations on atomic distances; and (ii) to the linearity constraint in the macroscopic equations.     

Numerical strategies for filtering partially observed stiff stochastic differential equations

In this paper, we present a fast numerical strategy for filtering stochastic differential equations with multiscale features. This method is designed such that it does not violate the practical linear observability condition and, more importantly, it does not require the computationally expensive cross correlation statistics between multiscale variables that are typically needed in standard filtering approach. The proposed filtering algorithm comprises of a “macro-filter” that borrows ideas from the Heterogeneous Multiscale Methods and a “micro-filter” that reinitializes the fast microscopic variables to statistically reflect the unbiased slow macroscopic estimate obtained from the macro-filter and macroscopic observations at asynchronous times. We will show that the proposed micro-filter is equivalent to solving an inverse problem for parameterizing differential equations. Numerically, we will show that this microscopic reinitialization is an important novel feature for accurate filtered solutions, especially when the microscopic dynamics is not mixing at all.     

Homogenized Dynamics of Stochastic Partial Differential Equations with Dynamical Boundary Conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with a static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes’ boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday.     

Phase synchronization between collective rhythms of globally coupled oscillator groups: noiseless nonidentical case

We theoretically study the synchronization between collective oscillations exhibited by two weakly interacting groups of nonidentical phase oscillators with internal and external global sinusoidal couplings of the groups. Coupled amplitude equations describing the collective oscillations of the oscillator groups are obtained by using the Ott-Antonsen ansatz, and then coupled phase equations for the collective oscillations are derived by phase reduction of the amplitude equations. The collective phase coupling function, which determines the dynamics of macroscopic phase differences between the groups, is calculated analytically. We demonstrate that the groups can exhibit effective antiphase collective synchronization even if the microscopic external coupling between individual oscillator pairs belonging to different groups is in-phase, and similarly effective in-phase collective synchronization in spite of microscopic antiphase external coupling between the groups.     

Macroscopic-microscopic calculations of ground state properties of superheavy nuclei

We systematically calculate the ground state properties of superheavy even-even nuclei with proton number Z=94–118. The calculations are based on the liquid drop macroscopic model and the microscopic model with the modified single-particle oscillator potential. The calculated binding energies and α-decay energies agree well with the experimental data. The reliability of the macroscopic-microscopic(MM)model for superheavy nuclei is confirmed by the good agreement between calculated results and experimental ones. Detailed comparisons between our calculations and M?ller’s are made. It is found that the calculated results also agree with M?ller’s results and that the MM model is insensitive to the microscopic single-particle potential. Calculated results are also compared with results from relativistic mean-field (RMF) model and from Skyrme-Hatree-Fock(SHF) model. In addition, half-lives, deformations and shape coexistence are also investigated. The properties of some unknown nuclei are predicted and they will be useful for future experimental researches of superheavy nuclei.     

Nonlinear voter models: the transition from invasion to coexistence

In nonlinear voter models the transitions between two states depend in a nonlinear manner on the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to m = 4 neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicit expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion; (ii) random coexistence; and – most interestingly – (iii) correlated coexistence. These findings are contrasted with predictions from the mean-field phase diagram and are confirmed by extensive computer simulations of the microscopic dynamics.     

On stochastic parameter estimation using data assimilation

Data assimilation-based parameter estimation can be used to deterministically tune forecast models. This work demonstrates that it can also be used to provide parameter distributions for use by stochastic parameterization schemes. While parameter estimation is (theoretically) straightforward to perform, it is not clear how one should physically interpret the parameter values obtained. Structural model inadequacy implies that one should not search for a deterministic “best” set of parameter values, but rather allow the parameter values to change as a function of state; different parameter values will be needed to compensate for the state-dependent variations of realistic model inadequacy. Over time, a distribution of parameter values will be generated and this distribution can be sampled during forecasts. The current work addresses the ability of ensemble-based parameter estimation techniques utilizing a deterministic model to estimate the moments of stochastic parameters. It is shown that when the system of interest is stochastic the expected variability of a stochastic parameter is biased when a deterministic model is employed for parameter estimation. However, this bias is ameliorated through application of the Central Limit Theorem, and good estimates of both the first and second moments of the stochastic parameter can be obtained. It is also shown that the biased variability information can be utilized to construct a hybrid stochastic/deterministic integration scheme that is able to accurately approximate the evolution of the true stochastic system.     

On the controversy around Daganzo’s requiem for and Aw-Rascle’s resurrection of second-order traffic flow models

Daganzo’s criticisms of second-order fluid approximations of traffic flow [C. Daganzo, Transpn. Res. B. 29, 277 (1995)] and Aw and Rascle’s proposal how to overcome them [A. Aw, M. Rascle, SIAM J. Appl. Math. 60, 916 (2000)] have stimulated an intensive scientific activity in the field of traffic modeling. Here, we will revisit their arguments and the interpretations behind them. We will start by analyzing the linear stability of traffic models, which is a widely established approach to study the ability of traffic models to describe emergent traffic jams. Besides deriving a collection of useful formulas for stability analyses, the main attention is put on the characteristic speeds, which are related to the group velocities of the linearized model equations. Most macroscopic traffic models with a dynamic velocity equation appear to predict two characteristic speeds, one of which is faster than the average velocity. This has been claimed to constitute a theoretical inconsistency. We will carefully discuss arguments for and against this view. In particular, we will shed some new light on the problem by comparing Payne’s macroscopic traffic model with the Aw-Rascle model and macroscopic with microscopic traffic models.     

Density-Profile Processes Describing Biological Signaling Networks: Almost Sure Convergence to Deterministic Trajectories

We introduce jump processes in ℝ ^k , called density-profile processes, to model biological signaling networks. Our modeling setup describes the macroscopic evolution of a finite-size spin-flip model with k types of spins with arbitrary number of internal states interacting through a non-reversible stochastic dynamics. We are mostly interested on the multi-dimensional empirical-magnetization vector in the thermodynamic limit, and prove that, within arbitrary finite time-intervals, its path converges almost surely to a deterministic trajectory determined by a first-order (non-linear) differential equation with explicit bounds on the distance between the stochastic and deterministic trajectories. As parameters of the spin-flip dynamics change, the associated dynamical system may go through bifurcations, associated to phase transitions in the statistical mechanical setting. We present a simple example of spin-flip stochastic model, associated to a synthetic biology model known as repressilator, which leads to a dynamical system with Hopf and pitchfork bifurcations. Depending on the parameter values, the magnetization random path can either converge to a unique stable fixed point, converge to one of a pair of stable fixed points, or asymptotically evolve close to a deterministic orbit in ℝ ^k . We also discuss a simple signaling pathway related to cancer research, called p53 module.     

Kinetic Limit for a System of Coagulating Planar Brownian Particles

We study a model of mass-bearing coagulating planar Brownian particles. The coagulation occurs when two particles are within a distance of order ε. We assume that the initial number of particles N is of order |logε|. Under suitable assumptions of the initial distribution of particles and the microscopic coagulation propensities, we show that the macroscopic particle densities satisfy a Smoluchowski-type equation.