Stochastic Lattice Model Research Articles

Localization in Boundary-Driven Lattice Models

Several systems display an equilibrium condensation transition, where a finite fraction of a conserved quantity is spatially localized. The presence of two conservation laws may induce the emergence of such transition in an out-of-equilibrium setup, where boundaries are attached to different and subcritical heat baths. We study this phenomenon in a class of stochastic lattice models, where the local energy is a general convex function of the local mass, mass and energy being both globally conserved in the isolated system. We obtain exact results for the nonequilibrium steady state (spatial profiles, mass and energy currents, Onsager coefficients) and we highlight important differences between equilibrium and out-of-equilibrium condensation.

Walk this way: modeling foraging ant dynamics in multiple food source environments

Foraging for resources is an essential process for the daily life of an ant colony. What makes this process so fascinating is the self-organization of ants into trails using chemical pheromone in the absence of direct communication. Here we present a stochastic lattice model that captures essential features of foraging ant dynamics inspired by recent agent-based models while forgoing more detailed interactions that may not be essential to trail formation. Nevertheless, our model’s results coincide with those presented in more sophisticated theoretical models and experiments. Furthermore, it captures the phenomenon of multiple trail formation in environments with multiple food sources. This latter phenomenon is not described well by other more detailed models. We complement the stochastic lattice model by describing a macroscopic PDE which captures the basic structure of lattice model. The PDE provides a continuum framework for the first-principle interactions described in the stochastic lattice model and is amenable to analysis. Linear stability analysis of this PDE facilitates a computational study of the impact various parameters impart on trail formation. We also highlight universal features of the modeling framework that may allow this simple formation to be used to study complex systems beyond ants.

Dynamical behaviors of an impulsive stochastic neural field lattice model

This paper is concerned with the asymptotic behaviors of the solutions of an impulsive stochastic neural field lattice model driven by nonlinear noise. We first show the existence and uniqueness of weak pullback mean random attractors for the impulsive stochastic systems. Then by the properties of Markov processes, the existence of evolution system of measures for the impulsive stochastic systems is established. To this end, we employ the idea of uniform estimates on the tails of the solutions to show the tightness of a family of distributions of the solutions of the lattice systems.

Nonequilibrium Thermodynamics of the Majority Vote Model.

The majority vote model is one of the simplest opinion systems yielding distinct phase transitions and has garnered significant interest in recent years. This model, as well as many other stochastic lattice models, are formulated in terms of stochastic rules with no connection to thermodynamics, precluding the achievement of quantities such as power and heat, as well as their behaviors at phase transition regimes. Here, we circumvent this limitation by introducing the idea of a distinct and well-defined thermal reservoir associated to each local configuration. Thermodynamic properties are derived for a generic majority vote model, irrespective of its neighborhood and lattice topology. The behavior of energy/heat fluxes at phase transitions, whether continuous or discontinuous, in regular and complex topologies, is investigated in detail. Unraveling the contribution of each local configuration explains the nature of the phase diagram and reveals how dissipation arises from the dynamics.

Lotka-Volterra predator-prey model with periodically varying carrying capacity.

We study the stochastic spatial Lotka-Volterra model for predator-prey interaction subject to a periodically varying carrying capacity. The Lotka-Volterra model with on-site lattice occupation restrictions (i.e., finite local carrying capacity) that represent finite food resources for the prey population exhibits a continuous active-to-absorbing phase transition. The active phase is sustained by the existence of spatiotemporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis in order to specifically delineate the impact of stochastic fluctuations and spatial correlations. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The (quasi-)stationary regime of our periodically varying Lotka-Volterra predator-prey system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity-switching environments, the mean-field rate equationspredict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast and slow periodic switching. The analysis of the mean-field equationsin the fast-switching regime enables a semiquantitative description of the (quasi-)stationary state.

Steady-state thermodynamics: Description equivalence and violation of reservoir independence.

For stochastic lattice models in spatially uniform nonequilibrium steady states, an effective thermodynamic temperature T and chemical potential μ can be defined via coexistence with heat and particle reservoirs. We verify that the probability distribution P_{N} for the number of particles in the driven lattice gas with nearest-neighbor exclusion in contact with a particle reservoir with dimensionless chemical potential µ^{*} possesses a large-deviation form in the thermodynamic limit. This implies that the thermodynamic properties determined in isolation (fixed particle number representation) and in contact with a particle reservoir (fixed dimensionless chemical potential representation) are equal. We refer to this as description equivalence. This finding motivates investigation of whether the effective intensive parameters so obtained depend on the nature of the exchange between system and reservoir. For example, a stochastic particle reservoir is usually taken to insert or remove a single particle in each exchange, but one may also consider a reservoir that inserts or removes a pair of particles in each event. In equilibrium, equivalence of pair and single-particle reservoirs is guaranteed by the canonical form of the probability distribution on configuration space. Remarkably, this equivalence is violated in nonequilibrium steady states, limiting the generality of steady-state thermodynamics based on intensive variables.

A Dimensionless Parameter for Predicting Convective Self‐Aggregation Onset in a Stochastic Reaction‐Diffusion Model of Tropical Radiative‐Convective Equilibrium

AbstractWe introduce a minimal stochastic lattice model for the column relative humidity (R) in the tropics, which incorporates convective moistening, horizontal transport and subsidence drying. The probability of convection occurring in a location increases with R, based on Tropical Rainfall Measuring Mission observations, providing a positive feedback that could lead to aggregation. We show that the simple model reproduces many aspects of full‐physics cloud resolving model experiments. Depending on model parameter settings and domain size and resolution choices, it can produce both random and aggregated equilibrium states. Clustering occurs more readily with larger domains and coarser resolutions, in agreement with full‐physics models. Using dimensional arguments and fits from empirical data, we derive a dimensionless parameter which we call the aggregation number, Nag, that predicts whether a specific model and experiment setup will result in an aggregated or random state. The parameter includes the moistening feedback strength, the horizontal moisture transport efficiency, the subsidence timescale, the domain size and spatial resolution. Using large ensembles of experiments, we show that the transition between random and aggregated states occurs at a critical value of Nag. We argue that Nag could help to understand the differences in aggregation states between full‐physics, cloud resolving models, which show little consensus about the robustness of self‐organized patterns, whose emergence appears to be sensitive to the model setup, physics and parameterizations.

The onset of whole-cell modeling using the martini force field.

Molecular dynamics (MD) is a well established simulation method, which has successfully been applied to study a wide range of biomolecular processes. Continuous improvements in both computational infrastructure and modelling methods have enabled scientists to study mesoscopic, multi-component systems using MD simulations. Understanding how biomolecular functions emerge from millions of interacting molecules is viewed as the next research objective. Since biomolecular processes function on a hierarchy of interconnected scales, decoding the complexity of cellular environments requires us to study them as a whole. We present our ongoing effort to construct a whole-cell model at a molecular resolution, using the Martini coarse-grained force field. We focus our efforts on modeling a genetically minimal cell: the JCVI-syn3A. This cell is engineered to have minimal complexity, making it an ideal starting point for this project. The cell's structural organization and composition is gathered in an integrative modeling approach, combining experimentally resolved data and stochastic lattice model simulations of the whole cell. Incorporating the necessary data into our MD model requires specialized tools, which will be part of the Martini3 ecosystem. The first version of our whole-cell model is presented, showing that performing MD simulations at this scale is feasible. Despite simplifying the cell composition in some key components, this model will provide a starting point for further improvements and pushing the boundaries of realistic whole-cell modeling. Computational microscopy of entire cells and cell organelles will provide valuable insights in a wide range of problems, ranging from drug design to understanding the internal organization of the cellular environments.

A two-dimensional stochastic fractional non-local diffusion lattice model with delays

The well-posedness, regularity and general stability of solutions to a two-dimensional stochastic non-local delay diffusion lattice system with a time Caputo fractional operator of order [Formula: see text] are investigated in [Formula: see text] spaces for [Formula: see text]. First, the global existence and uniqueness of solutions are established by using a temporally weighted norm, the Burkholder–Davis–Gundy inequality and the Banach fixed point theorem. Then the continuous dependence of solutions on initial values is established in the sense of [Formula: see text]th moment. In particular, the [Formula: see text]th moment Hölder regularities in time and [Formula: see text]th moment general stability, including polynomial and logarithmic stability of solutions, are obtained.

Phase Growth with Heat Diffusion in a Stochastic Lattice Model

When a stable phase is adjacent to a metastable phase with a planar interface, the stable phase grows. We propose a stochastic lattice model describing the phase growth accompanying heat diffusion. The model is based on an energy-conserving Potts model with a kinetic energy term defined on a two-dimensional lattice, where each site is sparse-randomly connected in one direction and local in the other direction. For this model, we calculate the stable and metastable phases exactly using statistical mechanics. Performing numerical simulations, we measure the displacement of the interface R(t). We observe the scaling relation \(R(t)=L_x \bar{\mathcal {R}} (Dt/L_x^2)\), where D is the thermal diffusion constant and \(L_x\) is the system size between the two heat baths. The scaling function \(\bar{\mathcal {R}}(z)\) shows \(\bar{\mathcal {R}}(z) \simeq z^{0.5}\) for \(z \ll z_c\) and \(\bar{\mathcal {R}}(z) \simeq z^{\alpha }\) for \(z \gg z_c\), where the cross-over value \(z_c\) and exponent \(\alpha \) depend on the temperatures of the baths, and \(0.5\le \alpha \le 1\). We then confirm that a deterministic phase-field model exhibits the same scaling relation. Moreover, numerical simulations of the phase-field model show that the cross-over value \(\bar{\mathcal {R}}(z_c)\) approaches zero when the stable phase becomes neutral.

Interaction network structure and spatial patterns influence invasiveness and invasibility in a stochastic model of plant communities

The factors governing invasiveness and invasibility are important for understanding invasions in plant communities, but most previous work does not incorporate network‐level interaction structure. We addressed this gap by simulating perennial plant communities at multiple life stages that varied in species richness and interaction network structures (transitive and intransitive), characteristics associated with species coexistence, using a stochastic lattice model that incorporated local competition, facilitation, and dispersal. Resident communities were invaded with alien species that varied in their fecundity, dispersal, and competitive abilities. We found that community structures that are known to promote coexistence also increased invasibility; communities with intransitive interaction networks on the whole were more invasible than transitive communities. However, the interaction between species richness and network structure was complex, as intransitive communities with few species were relatively resistant and transitive communities with many species were relatively invasible. Thus, simple rules linking intransitivity and diversity may not reflect the complex relationships between community structure and dynamics that occur in more realistic spatial systems. Spatial patterns of clustering and segregation were associated with higher invasibility and successful invaders intensified these patterns in communities, particularly at large spatial scales. Invasiveness and alien species abundance were associated with species‐level characteristics (fecundity and dispersal), but community‐level characteristics were also important, particularly competitive tolerance to the resident community. These results emphasize the importance of a community‐level approach, involving network‐level structure and spatial patterns, in adding nuance to our understanding of invasions in plant communities.

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia.

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain’s division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains—but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Stochastic self-assembly of reaction-diffusion patterns in synaptic membranes.

Synaptic receptor and scaffold molecules self-assemble into membrane protein domains, which play an important role in signal transmission across chemical synapses. Experiment and theory have shown that the formation of receptor-scaffold domains of the characteristic size observed in nerve cells can be understood from the receptor and scaffold reaction and diffusion processes suggested by experiments. We employ here kinetic Monte Carlo (KMC) simulations to explore the self-assembly of synaptic receptor-scaffold domains in a stochastic lattice model of receptor and scaffold reaction-diffusion dynamics. For reaction and diffusion rates within the ranges of values suggested by experiments we find, in agreement with previous mean-field calculations, self-assembly of receptor-scaffold domains of a size similar to that observed in experiments. Comparisons between the results of our KMC simulations and mean-field solutions suggest that the intrinsic noise associated with receptor and scaffold reaction and diffusion processes accelerates the self-assembly of receptor-scaffold domains, and confers increased robustness to domain formation. In agreement with experimental observations, our KMC simulations yield a prevalence of scaffolds over receptors in receptor-scaffold domains. Our KMC simulations show that receptor and scaffold reaction-diffusion dynamics can inherently give rise to plasticity in the overall properties of receptor-scaffold domains, which may be utilized by nerve cells to regulate the receptor number at chemical synapses.

Stochastic analysis of lattice, nonlocal continuous beams in vibration

In this paper, we study the stochastic behavior of some lattice beams, called Hencky bar-chain model and their non-local continuous beam approximations. Hencky bar-chain model is a beam lattice composed of rigid segments, connected by some homogeneous rotational elastic links. In the present stochastic analysis, the stiffness of these elastic links is treated as a continuous random variable, with given probability density function. The fundamental eigenfrequency of the linear difference eigenvalue problem is also a random variable in this context. The reliability is defined as the probability that this fundamental frequency is less than an excitation frequency. This reliability function is exactly calculated for the lattice beam in conjunction with various boundary conditions. An exponential distribution is considered for the random stiffness of the elastic links. The stochastic lattice model is then compared to a stochastic nonlocal beam model, based on the continualization of the difference equation of the lattice model. The efficiency of the nonlocal beam model to approximate the lattice beam model is shown in presence of rotational elastic link randomness. We also compare such stochastic function with the one of a continuous local Euler-Bernoulli beam, with a special emphasis on scale effect in presence of randomness. Scale effect is captured both in deterministic and non-deterministic frameworks.

Long-term effects of abrupt environmental perturbations in model of group chase and escape with the presence of non-conservative processes

This paper examines the influence of environmental perturbations on dynamical regimes of model ecosystems. We study a stochastic lattice model describing the dynamics of a group chasing and escaping between predators and prey. The model includes smart pursuit (predators to prey) and evasion (prey from predators). Both species can affect their movement by visual perception within their finite sighting range. Non-conservative processes that change the number of individuals within the population, such as breeding and physiological dying, are implemented in the model. The model contains five parameters that control the breeding and physiological dying of predators and prey: the birth and two death rates of predators and two parameters characterizing the birth and death of prey. We study the response of our model of group chase and escape to sudden perturbations in values of parameters that characterize the non-conservative processes. Temporal dependencies of the number of predators and prey are compared for various perturbation events with different abrupt changes of probabilities affecting the non-conservative processes.

Dynamic spatiotemporal coordination of neural stem cell fate decisions occurs through local feedback in the adult vertebrate brain.

SummaryNeural stem cell (NSC) populations persist in the adult vertebrate brain over a lifetime, and their homeostasis is controlled at the population level through unknown mechanisms. Here, we combine dynamic imaging of entire NSC populations in their in vivo niche over several weeks with pharmacological manipulations, mathematical modeling, and spatial statistics and demonstrate that NSCs use spatiotemporally resolved local feedback signals to coordinate their decision to divide in adult zebrafish brains. These involve Notch-mediated short-range inhibition from transient neural progenitors and a dispersion effect from the dividing NSCs themselves exerted with a delay of 9–12 days. Simulations from a stochastic NSC lattice model capturing these interactions demonstrate that these signals are linked by lineage progression and control the spatiotemporal distribution of output neurons. These results highlight how local and temporally delayed interactions occurring between brain germinal cells generate self-propagating dynamics that maintain NSC population homeostasis and coordinate specific spatiotemporal correlations.

Diffusive epidemic process in 3D: a two-species reaction–diffusion phase transition

By the use of Monte Carlo simulation we study the critical behavior of a three-dimensional stochastic lattice model describing a diffusive epidemic propagation process. In this model, healthy (A) and sick (B) individuals diffuse on the lattice with diffusion constants d A and d B , respectively, and undergo reactions B → A and A + B → 2B. We determine the absorbing phase transition between a steady reactive state and a vacuum state. We obtained the order parameter, order parameter fluctuations, correlation length and their critical exponents by the use of steady state and short-time dynamics simulations. We studied three different diffusion regimes: the case of species A diffusing much slower than species B (d A ≪ d B ), the case of species with equal diffusion constants (d A = d B ) and the case of species A diffusing much faster than species B (d A ≫ d B ). We found only second order transition for all three cases. We did not identify any signal of first order transition for the case d A > d B as predicted by field theory in first order approximation.

A Novel Method for Interpolating Daily Station Rainfall Data Using a Stochastic Lattice Model

AbstractRain gauge data are routinely recorded and used around the world. However, their sparsity and inhomogeneity make them inadequate for climate model calibration and many other climate change studies. Various algorithms and interpolation techniques have been developed over the years to obtain adequately distributed datasets. Objective interpolation methods such as inverse distance weighting (IDW) are the most widely used and have been employed to produce some of the most popular gridded daily rainfall datasets (e.g., India Meteorological Department gridded daily rainfall). Unfortunately, the skill of these techniques becomes very limited to nonexistent in areas located far away from existing recording stations. This is problematic as many areas of the world lack adequate rain gauge coverage throughout the recording history. Here, we introduce a new probabilistic interpolation method in an attempt to address this issue. The new algorithm employs a multitype particle interacting stochastic lattice model that assigns a binned rainfall value, from a given number of bins to each lattice site or grid cell, with a certain probability according to the rainfall amounts observed in neighboring sites and a background climatological rain rate distribution, drawn from the available data. Grid cells containing recording stations are not affected and are being used as “boundary” input conditions by the stochastic model. The new stochastic model is successfully tested and compared against two widely used gridded daily rainfall datasets over the Indian landmass for data from the summer monsoon seasons (June–September) for 1951–70.

Throughput reduction on an air-ground transport system by the simultaneous effect of multiple traveling routes equipped with parking sites

This paper examines the traffic flows on a two-dimensional stochastic lattice model that comprises a junction of two traveling routes: the domestic route and the international route each of which has parking sites. In our model, the system distributes the arrived particles to either of the two routes and selects one of the parking sites in the route for each particle, which stops at the parking site once during its travel. Because each particle has antennas in the back and front directions to detect other approaching particles, the effect of the volume exclusion of each particle extends in the moving direction. The system displays interesting behavior; remarkably, the dependence of the throughput on the distribution ratio of particles to the domestic route reduces after reaching the maximum parking capacity of the domestic route. Our simulations and analysis with the queueing model describe this phenomenon and suggest the following fact: as the distribution ratio of particles to the international route decreases, the throughput of the international route reduces, and simultaneously, that of the domestic route saturates. The simultaneous effect of the decrease and saturation causes a reduction in the throughput of the entire system.

Asymptotic autonomous attractors for a stochastic lattice model with random viscosity

We study forward asymptotic autonomy of a pullback random attractor for a non-autonomous random lattice system and establish the criteria in terms of convergence, recurrence, forward-pullback absorption and asymptotic smallness of the discrete random dynamical system. By applying the abstract result to both non-autonomous and autonomous stochastic lattice equations with random viscosity, we show the existence of both pullback and global random attractors such that the time-component of the pullback attractor semi-converges to the global attractor as the time-parameter tends to infinity.