Mixed analytical-stochastic simulation method for the recovery of a Brownian gradient source from probability fluxes to small windows.

Is it possible to recover the position of a source from the steady-state fluxes of Brownian particles to small absorbing windows located on the boundary of a domain? To address this question, we develop a numerical procedure to avoid tracking Brownian trajectories in the entire infinite space. Instead, we generate particles near the absorbing windows, computed from the analytical expression of the exit probability. When the Brownian particles are generated by a steady-state gradient at a single point, we compute asymptotically the fluxes to small absorbing holes distributed on the boundary of half-space and on a disk in two dimensions, which agree with stochastic simulations. We also derive an expression for the splitting probability between small windows using the matched asymptotic method. Finally, when there are more than two small absorbing windows, we show how to reconstruct the position of the source from the diffusion fluxes. The present approach provides a computational first principle for the mechanism of sensing a gradient of diffusing particles, a ubiquitous problem in cell biology

Critical scaling and aging near the flux-line-depinning transition

We utilize Langevin molecular dynamics simulations to study dynamical critical behavior of magnetic flux lines near the depinning transition in type-II superconductors subject to randomly distributed attractive point defects. We employ a coarse-grained elastic line Hamiltonian for the mutually repulsive vortices and purely relaxational kinetics. In order to infer the stationary-state critical exponents for the continuous non-equilibrium depinning transition at zero temperature T = 0 and at the critical driving current density j_c, we explore two-parameter scaling laws for the flux lines' gyration radius and mean velocity as functions of the two relevant scaling fields T and j - j_c. We also investigate critical aging scaling for the two-time height auto-correlation function in the early-time non-equilibrium relaxation regime to independently measure critical exponents. We provide numerical exponent values for the distinct universality classes of non-interacting and repulsive vortices

Evolutionary dynamics and competition stabilize three-species predator–prey communities

We perform individual-based Monte Carlo simulations in a community consisting of two predator species competing for a single prey species, with the purpose of studying biodiversity stabilization in this simple model system. Predators are characterized with predation efficiency and death rates, to which Darwinian evolutionary adaptation is introduced. Competition for limited prey abundance drives the populations' optimization with respect to predation efficiency and death rates. We study the influence of various ecological elements on the final state, finding that both indirect competition and evolutionary adaptation are insufficient to yield a stable ecosystem. However, stable three-species coexistence is observed when direct interaction between the two predator species is implemented

Particle-based multiscale modeling of calcium puff dynamics

© 2016 SIAM. Intracellular calcium is regulated in part by the release of Ca2+ ions from the endo-plasmic reticulum via inositol-4,5-triphosphate receptor (IP3R) channels (among other possibilities such as RyR and L-type calcium channels). The resulting dynamics are highly diverse and lead to local calcium "puffs" as well as global waves propagating through cells, as observed in Xenopus oocytes, neurons, and other cell types. Local fluctuations in the number of calcium ions play a crucial role in the onset of these features. Previous modeling studies of calcium puff dynamics stemming from IP3R channels have predominantly focused on stochastic channel models coupled to deterministic diffusion of ions, thereby neglecting local fluctuations of the ion number. Tracking of individual ions is computationally difficult due to the scale separation in the Ca2+ concentration when channels are in the open or closed states. In this paper, a spatial multiscale model for investigating of the dynamics of puffs is presented. It couples Brownian motion (diffusion) of ions with a stochastic channel gating model. The model is used to analyze calcium puff statistics. Concentration time traces as well as channel state information are studied. We identify the regime in which puffs can be found and develop a mean-field theory to extract the boundary of this regime. Puffs are possible only when the time scale of channel inhibition is sufficiently large. Implications for the understanding of puff generation and termination are discussed.This work was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement 239870

Stochastic population dynamics in spatially extended predator-prey systems

Spatially extended population dynamics models that incorporate intrinsic noise serve as case studies for the role of fluctuations and correlations in biological systems. Including spatial structure and stochastic noise in predator-prey competition invalidates the deterministic Lotka-Volterra picture of neutral population cycles. Stochastic models yield long-lived erratic population oscillations stemming from a resonant amplification mechanism. In spatially extended predator-prey systems, one observes noise-stabilized activity and persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively. The critical dynamics and the non-equilibrium relaxation kinetics at the predator extinction threshold are characterized by the directed percolation universality class. Spatial or environmental variability results in more localized patches which enhances both species densities. Affixing variable rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of cyclic competition with rock-paper-scissors interactions illustrate connections between population dynamics and evolutionary game theory, and demonstrate how space can help maintain diversity. In two dimensions, three-species cyclic competition models of the May-Leonard type are characterized by the emergence of spiral patterns whose properties are elucidated by a mapping onto a complex Ginzburg-Landau equation. Extensions to general food networks can be classified on the mean-field level, which provides both a fundamental understanding of ensuing cooperativity and emergence of alliances. Novel space-time patterns emerge as a result of the formation of competing alliances, such as coarsening domains that each incorporate rock-paper-scissors competition games

Disordered vortex matter out of equilibrium: a Langevin molecular dynamics study

We discuss the use of Langevin molecular dynamics in the investigation of the non-equilibrium properties of disordered vortex matter. Our special focus is set on values of system parameters that are realistic for disordered high-$T_c$ superconductors such as YBCO. Using a discretized elastic line model, we study different aspects of vortices far from thermal equilibrium. On the one hand we investigate steady-state properties of driven magnetic flux lines in a disordered environment, namely the current-voltage characteristics, the gyration radius, and the pinning time statistics. On the other hand we study the complex relaxation processes and glassy-like dynamics that emerge in type-II superconductors due to the intricate competition between the long-range vortex-vortex repulsion and flux pinning due to randomly placed point defects. To this end we consider different types of sudden perturbations: temperature, magnetic field, and external current quenches

Flux line relaxation kinetics following current quenches in disordered type-II superconductors

We investigate the relaxation dynamics of magnetic vortex lines in type-II superconductors following rapid changes of the external driving current by means of an elastic line model simulated with Langevin molecular dynamics. A system of flux vortices in a sample with randomly distributed point-like defects is subjected to an external current of appropriate strength for a sufficient period of time so as to be in a moving non-equilibrium steady state. The current is then instantaneously lowered to a value that pertains to either the moving or pinned regime. The ensuing relaxation of the flux lines is studied via one-time observables such as their mean velocity and radius of gyration. We have in addition measured the two-time flux line height autocorrelation function to investigate dynamical scaling and aging behavior in the system, which in particular emerge after quenches into the glassy pinned state

Co-existence in the two-dimensional May-Leonard model with random rates

We employ Monte Carlo simulations to numerically study the temporal evolution and transient oscillations of the population densities, the associated frequency power spectra, and the spatial correlation functions in the (quasi-)steady state in two-dimensional stochastic May--Leonard models of mobile individuals, allowing for particle exchanges with nearest-neighbors and hopping onto empty sites. We therefore consider a class of four-state three-species cyclic predator-prey models whose total particle number is not conserved. We demonstrate that quenched disorder in either the reaction or in the mobility rates hardly impacts the dynamical evolution, the emergence and structure of spiral patterns, or the mean extinction time in this system. We also show that direct particle pair exchange processes promote the formation of regular spiral structures. Moreover, upon increasing the rates of mobility, we observe a remarkable change in the extinction properties in the May--Leonard system (for small system sizes): (1) As the mobility rate exceeds a threshold that separates a species coexistence (quasi-)steady state from an absorbing state, the mean extinction time as function of system size N crosses over from a functional form ~ e^{cN} / N (where c is a constant) to a linear dependence; (2) the measured histogram of extinction times displays a corresponding crossover from an (approximately) exponential to a Gaussian distribution. The latter results are found to hold true also when the mobility rates are randomly distributed.Comment: 9 pages, 4 figures; to appear in Eur. Phys. J. B (2011

On the expected number of internal equilibria in random evolutionary games with correlated payoff matrix

The analysis of equilibrium points in random games has been of great interest in evolutionary game theory, with important implications for understanding of complexity in a dynamical system, such as its behavioural, cultural or biological diversity. The analysis so far has focused on random games of independent payoff entries. In this paper, we overcome this restrictive assumption by considering multi-player two-strategy evolutionary games where the payoff matrix entries are correlated random variables. Using techniques from the random polynomial theory we establish a closed formula for the mean numbers of internal (stable) equilibria. We then characterise the asymptotic behaviour of this important quantity for large group sizes and study the effect of the correlation. Our results show that decreasing the correlation among payoffs (namely, of a strategist for different group compositions) leads to larger mean numbers of (stable) equilibrium points, suggesting that the system or population behavioural diversity can be promoted by increasing independence of the payoff entries. Numerical results are provided to support the obtained analytical results.Comment: Revision from the previous version; 27 page