Presence Of Stochastic Perturbations Research Articles

Optimal dispersal and diffusion-enhanced robustness in two-patch metapopulations: origin’s saddle-source nature matters

A two-patch logistic metapopulation model is investigated both analytically and numerically focusing on the impact of dispersal on population dynamics. First, the dependence of the global dynamics on the stability type of the full extinction equilibrium point is tackled. Then, the behaviour of the total population with respect to the dispersal is studied analytically. Our findings demonstrate that diffusion plays a crucial role in the preservation of both subpopulations and the full metapopulation under the presence of stochastic perturbations. At low diffusion, the origin is a repulsor, causing the orbits to flow nearly parallel to the axes, risking stochastic extinctions. Higher diffusion turns the repeller into a saddle point. Orbits then quickly converge to the saddle’s unstable manifold, reducing extinction chances. This change in the vector field enhances metapopulation robustness. On the other hand, the well-known fact that asymmetric conditions on the patches is beneficial for the total population is further investigated. This phenomenon has been studied in previous works for large enough or small enough values of the dispersal. In this work, we complete the theory for all values of the dispersal. In particular, we derive analytically a formula for the optimal value of the dispersal that maximizes the total population.

A stochastic variant of replicator dynamics in zero-sum games and its invariant measures

We study the behavior of a stochastic variant of replicator dynamics in two-agent zero-sum games. Inspired by the well studied deterministic replicator equation, this model gives arguably a more realistic description of real world settings and, as we demonstrate here, exhibits totally distinct behavior when compared to its deterministic counterpart which is known to be recurrent. In more detail, we characterize the statistics of such systems by their invariant measures which can be shown to be entirely supported on the boundary of the space of mixed strategies. Depending on the noise strength we can furthermore characterize these invariant measures by finding accumulation of mass at specific parts of the boundary. In particular, regardless of the magnitude of noise, we show that any invariant probability measure is a convex combination of Dirac measures on pure strategy profiles, which correspond to vertices/corners of the agents’ simplices. Thus, in the presence of stochastic perturbations, even in the most classic zero-sum settings, such as Matching Pennies, we observe a stark disagreement between the axiomatic prediction of Nash equilibrium and the evolutionary emergent behavior derived by an assumption of stochastically adaptive, learning agents.

Aperiodic Event-Triggered Synchronization Control for Neural Networks With Stochastic Perturbations and Time Delay

In the presence of stochastic perturbations and time delay, the aperiodic event-triggered synchronization control for neural networks is studied in this paper. First, a suitable linear event-triggered control is designed to ensure the synchronization of the delayed stochastic neural networks in mean square. Then, by using Itô-differential formula, the synchronization criteria are derived under the event-triggered control. Moreover, a new method is proposed to prove that Zeno behavior does not occur, which ensures that the event-triggered control is feasible. Finally, an example and its simulation are provided to substantiate the effectiveness of the theory.

Perspectives on the formation of peakons in the stochastic Camassa–Holm equation

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

Optical Manipulation of Biological Cells With a Robot-Tweezers System: A Stochastic Control Approach

Cell manipulation is a key technique involved in many cell surgery applications. Various automatic frameworks have been developed for optical manipulation of biological cells without consideration of stochastic perturbations, such as Brownian motion. In this brief, a stochastic control scheme for optical manipulation of biological cells is developed. On the basis of the stochastic dynamic model of an optically trapped biological cell, a vectorial backstepping control strategy and its adaptive case is formulated for cell trajectory tracking. The global noise-to-state stability of a closed-loop system in the presence of stochastic perturbations is analyzed. Simulation results demonstrate the performance of the proposed approach.

Asymptotic behaviors of a two prey one predator model with cooperation among the prey species in a stochastic environment

In this paper, a two prey and one predator model with logistic growth in prey species have been considered in presence of stochastic perturbation. The cooperation among the preys against predation is assumed. Necessary conditions for asymptotic behavior around the equilibrium points of its deterministic system with and without stochasticity have been discussed. Finally, to illustrate the analytical results, numerical simulations have been performed.

Gillespie-Lindenmayer systems for stochastic simulation of morphogenesis

Abstract Lindenmayer systems (L-systems) provide a useful framework for modelling the development of multicellular structures and organisms. The parametric extension of L-systems allows for incorporating molecular-level processes into the models. Until now, the dynamics of these processes has been expressed using differential equations, implying continuously valued concentrations of the substances involved. This assumption is not satisfied, however, when the numbers of molecules are small. A further extension that accounts for the stochastic effects arising in this case is thus needed. We integrate L-systems and the Gillespie’s Stochastic Simulation Algorithm to simulate stochastic processes in fixed and developing linear structures. We illustrate the resulting formalism with stochastic implementations of diffusion-decay, reaction-diffusion and auxin-transport-driven morphogenetic processes. Our method and software can be used to simulate molecular and higher-level spatially explicit stochastic processes in static and developing structures, and study their behaviour in the presence of stochastic perturbations.

Stochastic Optical Trapping and Manipulation of a Micro Object With Neural-Network Adaptation

Optical tweezers are capable of manipulating micro/nano-objects without any physical contact and, therefore, are widely used in biomedical engineering and bio-logical science. While much progress has been achieved in automated optical manipulation of micro objects, the Brownian motion is commonly ignored in the stability analysis in order to simplify the control problem. However, random Brownian perturbations exist in the micromanipulation problem and, therefore, may result in failure of optical trapping due to the escape of the micro object from the trap. In addition, it is usually assumed in the development of a controller that the model of trapping stiffness is known, but the model is difficult to obtain because of its spatially varying feature around the center of laser beam, and variations with laser power and dimensions of objects. In this paper, a neural-network control method is proposed for optical trapping and manipulation of the micro object in the presence of stochastic perturbations and unknown trapping stiffness. The unknown trapping stiffness and dynamic parameters of micro objects, which vary with different laser power settings and sizes of the objects, are approximated by using adaptive neural networks. The stability analysis is carried out from stochastic perspectives by considering the effect of the Brownian motion in the dynamic model. Both experimental results and simulation results are presented.

Application of Wavelet-Based Methods for Accelerating Multi-Time-Scale Simulation of Bistable Heterogeneous Catalysis

We report results from a numerical study of multi-time-scale bistable dynamics for CO oxidation on a catalytic surface in a flowing, well-mixed gas stream. The problem is posed in terms of surface and gas-phase submodels that dynamically interact in the presence of stochastic perturbations, reflecting the impact of molecular-scale fluctuations on the surface and turbulence in the gas. Wavelet-based methods are used to encode and characterize the temporal dynamics produced by each submodel and detect the onset of sudden state shifts (bifurcations) caused by nonlinear kinetics. When impending state shifts are detected, a more accurate but computationally expensive integration scheme can be used. This appears to make it possible, at least in some cases, to decrease the net computational burden associated with simulating multi-time-scale, nonlinear reacting systems by limiting the amount of time in which the more expensive integration schemes are required. Critical to achieving this is being able to detect un...

Penalty-Regulated Dynamics and Robust Learning Procedures in Games

Starting from a heuristic learning scheme for strategic N-person games, we derive a new class of continuous-time learning dynamics consisting of a replicator-like drift adjusted by a penalty term that renders the boundary of the game’s strategy space repelling. These penalty-regulated dynamics are equivalent to players keeping an exponentially discounted aggregate of their ongoing payoffs and then using a smooth best response to pick an action based on these performance scores. Owing to this inherent duality, the proposed dynamics satisfy a variant of the folk theorem of evolutionary game theory and they converge to (arbitrarily precise) approximations of Nash equilibria in potential games. Motivated by applications to traffic engineering, we exploit this duality further to design a discrete-time, payoff-based learning algorithm that retains these convergence properties and only requires players to observe their in-game payoffs. Moreover, the algorithm remains robust in the presence of stochastic perturbations and observation errors, and it does not require any synchronization between players.

A carbon cycle coupled climate model of Neoproterozoic glaciation: Explicit carbon cycle with stochastic perturbations

[1] It has been suggested that a negative climate feedback may have operated during the Neoproterozoic Era as a consequence of the existence of a massive oceanic pool of dissolved organic carbon (DOC). As climate cooled so as to induce intense glaciation, the drawdown of oxygen into the Neoproterozoic ocean would have been enhanced because of the temperature dependence of the solubility of oxygen in seawater. Such increasing ventilation would have enhanced DOC remineralization, thus increasing the content of dissolved inorganic carbon (DIC) in the ocean. CO2 concentration in the atmosphere increases rapidly with DIC, thereby inhibiting further climate cooling. The model employed to illustrate the resulting climate dynamical behavior was an idealized one in which stochastic influence was assumed to be absent. However, such influence is expected to exist due to the action of processes that are not explicitly included in the model. Furthermore, the paleogeography assumed for the purpose of the published analyses was more appropriate to the Marinoan glaciation than to the earlier Sturtian event. In this paper, we fully investigate the stability of the system represented by the carbon cycle coupled climate model for both the Marinoan and Sturtian continental configurations and in the presence of stochastic perturbations. It is found that the hysteresis predicted by the ice sheet coupled model is sensitive to both continental configuration and to the strength of the negative feedback which arises due to carbon cycle coupling. Nevertheless, the very low frequency cyclic glaciation process predicted by the initial version of the model is found to persist in the presence of noise of significant amplitude. However, this cyclic behavior may be arrested entirely if the glaciation process were to result in insufficient alkalinity being delivered to the ocean basins. In this case the system would be expected to execute only a single excursion into and escape from the glacial state.

The emergence of rational behavior in the presence of stochastic perturbations

We study repeated games where players use an exponential learning scheme in order to adapt to an ever-changing environment. If the game's payoffs are subject to random perturbations, this scheme leads to a new stochastic version of the replicator dynamics that is quite different from the "aggregate shocks" approach of evolutionary game theory. Irrespective of the perturbations' magnitude, we find that strategies which are dominated (even iteratively) eventually become extinct and that the game's strict Nash equilibria are stochastically asymptotically stable. We complement our analysis by illustrating these results in the case of congestion games.

GENERAL ANALYSIS OF TYPE I PLANETARY MIGRATION WITH STOCHASTIC PERTURBATIONS

This paper presents a generalized treatment of Type I planetary migration in the presence of stochastic perturbations. In many planet-forming disks, the Type I migration mechanism, driven by asymmetric torques, acts on a short time scale and compromises planet formation. If the disk also supports MHD instabilities, however, the corresponding turbulent fluctuations produce additional stochastic torques that modify the steady inward migration scenario. This work studies the migration of planetary cores in the presence of stochastic fluctuations using complementary methods, including a Fokker-Planck approach and iterative maps. Stochastic torques have two main effects: [1] Through outward diffusion, a small fraction of the planetary cores can survive in the face of Type I inward migration. [2] For a given starting condition, the result of any particular realization of migration is uncertain, so that results must be described in terms of the distributions of outcomes. In addition to exploring different regimes of parameter space, this paper considers the effects of the outer disk boundary condition, varying initial conditions, and time-dependence of the torque parameters. For disks with finite radii, the fraction of surviving planets decreases exponentially with time. We find the survival fractions and decay rates for a range of disk models, and find the expected distribution of locations for surviving planets. For expected disk properties, the survival fraction lies in the range $0.01 < p_S < 0.1$.

Time scales of adaptive behavior and motor learning in the presence of stochastic perturbations

In this paper, the major assumptions of influential approaches to the structure of variability in practice conditions are discussed from the perspective of a generalized evolving attractor landscape model of motor learning. The efficacy of the practice condition effects is considered in relation to the theoretical influence of stochastic perturbations in models of gradient descent learning of multiple dimension landscapes. A model for motor learning is presented combining simulated annealing and stochastic resonance phenomena against the background of different time scales for adaptation and learning processes. The practical consequences of the model’s assumptions for the structure of practice conditions are discussed, together with their implications for teaching and coaching.

Best performance analysis for stochastic perturbation systems with spectral factorization

We investigate the best performance for linear feedback control systems in the case that plant uncertainty is to be considered. First, we define an average integral square criterion of tracking error over a class of stochastic model errors. By utilizing spectral factorization to minimize the performance index, we derive an optimal controller design method and further study best performance in the presence of stochastic perturbation. The results can be used to evaluate optimal performance in practical control system designs.

Species competition: uncertainty on a double invariant loop

The Tribolium (flour beetle) competition experiments conducted by Park have been highly influential in ecology. We have previously shown that the dynamics of single-species Tribolium populations can be well-described by the discrete-time, 3-dimensional larva–pupa–adult (LPA) model. Motivated by Park's experiments, we explore the dynamics of a 6-dimensional “competition LPA model” consisting of two LPA models coupled through cannibalism. The model predicts a double-loop coexistence attractor, as well as an unstable exclusion equilibrium with a 5-dimensional stable manifold that plays an important role in causing one of the species to go extinct in the presence of stochastic perturbations. We also present a stochastic version of the model, using binomial and Poisson distributions to describe the aggregation of demographic events within life stages. A novel “stochastic outcome diagram,” the stochastic counterpart to a bifurcation diagram, summarizes the model-predicted dynamics of uncertainty on the double-loop. We hypothesize that the model predictions provide an explanation for Park's data. This “stochastic double-loop hypothesis” is accessible to experimental verification.

Analysis and reconstruction of stochastic coupled map lattice models

The Letter introduces a general stochastic coupled lattice map model together with an algorithm to estimate the nodal equations involved based only on a small set of observable variables and in the presence of stochastic perturbations. More general forms of the Frobenius–Perron and the transfer operators, which describe the evolution of densities under the action of the CML transformation, are derived.

On Stochastic Bifurcation Behavior of the Atom Movement at the Tip of a Crack in Fatigue Damage System

By virtue of the stochastic bifurcation theory the transition of the atom moveme nt at a crack tip in fatigue damage system is investigated.Using the singular po int theory of one－dimensional diffusion process and the stochastic averaging ap proach of energy envelope a micro－model to describe the atom movement at the cr ack tip in homoclinic bifurcation fatigue damage system which is in the presence of stochastic perturbation is established.After the study on the characteristic of the diffusion exponent the drift exponent and the character exponent of the fatigue damage diffusion process on singular boundary the bifurcation behavior o f a homoclinic bifurcation fatigue damage system which is in the presence of par ametric white noise is examined.

Cosmological models expressible as gradient vector fields

Classes of cosmological models, for which Einstein’s equations reduce to two-dimensional dynamical systems, are studied in the presence of stochastic perturbations using the steady state of the associated Fokker–Planck equations and Zeeman’s notion of ε-stability. In all cases, a set of variables is found for which the dynamical systems are expressible as gradient vector flows, showing that the associated cosmologies are stochastically stable. Such models are also important in connection with application of catastrophy theory to cosmology.

Path integrals in Hamiltonian systems: Breakup of the last Kolmogorov-Arnold-Moser torus due to random forces.

Transport in three-dimensional nonintegrable Hamiltonian flows is studied and the destruction of Kolmogorov-Arnold-Moser barriers in the presence of stochastic perturbations described. We extend the action principle to Hamiltonians with small noise, which provides a framework to determine universal scaling of characteristic times as a function of the noise.