arXiv subjects

Ohad Vilk

Publications and source records attributed to Ohad Vilk.

18 recordsLinked to original sources

arXiv2026

Lattice Brownian bees with cooperative reproduction: steady states, collapse, and spreading

We extend the ``Brownian bees'' model of Berestycki et al. (2021, 2022) to cooperative reproduction, $kA\to(k{+}1)A$, of a population of $N$ symmetric random walkers with removal, at each birth event, of the particle farthest from the origin. Working in the limit $N\to\infty$, we formulate a hydrodynamic free-boundary problem for this model. Using this formalism, we determine steady state population densities for all~$k$ and prove their linear stability for $k\le 2$ and instability for $k\ge 4$. In the marginal case $k=3$, there is a whole continuous family of steady states at a single, critical ratio of the reproduction and diffusion rates. Above criticality the population undergoes an asymptotically self-similar finite-time collapse to the origin. Below the criticality the population spreads diffusively, but the reproduction remains quantitatively relevant. For $k\ge 4$, the unstable steady state separates regimes of a finite-time collapse and a diffusive spreading. Here the collapse dynamics is asymptotically self-similar, and the population density exhibits a scale separation requiring a matched-asymptotic description. Our analytical predictions are confirmed by numerical solutions of the hydrodynamic free-boundary problem and by Monte Carlo simulations of the original microscopic model.

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arXiv2026

Emergent population dynamics of random walkers with cooperative reproduction and spatial selection

We extend the $N$ branching Brownian motions model of population invasion to higher-order asexual reproduction. Increasing reproduction order leads to qualitative changes: invasion fronts generically cease to exist beyond binary reproduction; and in the binary case itself, their speed becomes diffusion-independent. Ternary reproduction shows critical behavior, with collapse into a strongly localized `invasion bullet' in the supercritical regime, diffusive spreading in the subcritical regime, and a continuous family of fronts at criticality. These results suggest that the dominance of division and binary reproduction in nature reflects fundamental constraints on invasion dynamics.

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arXiv2026

Macroscopic localization and collective memory in Poisson renewal resetting

Stochastic renewal processes are ubiquitous across physics, biology, and the social sciences. Here, we show that continuous-time renewal dynamics can naturally produce a mixed discrete-continuous structure, with a macroscopic fraction of particles occupying a discrete state. For ensembles of continuous-time random walkers subject to Poissonian renewal resets, we develop an age-structured framework showing this discrete component corresponds to localization at the reset configuration. We next show that collective interactions can retain memory although all reset events are memoryless. Remarkably, the transition to collective memory is discontinuous, and we identify a discontinuous dynamical phase transition from weak collective bias, where the dynamics are stationary, to strong collective bias where the dynamics are nonstationary and display aging up to finite-size effects. We explicitly discuss ecological implications of our work, illustrating how continuous-time renewal dynamics shape macroscopic structure and collective organization with long-term memory.

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arXiv2025

Collective behavior of independent scaled Brownian particles with renewal resetting

We study fluctuations of an ensemble of $N$ independent particles undergoing anomalous diffusion with random renewal resetting. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. The particles independently reset to the origin, and each particle's clock is set to zero upon spatial resetting. Employing the known steady-state position distribution of a \emph{single} particle undergoing the sBm with renewal resetting [Bodrova et al., Phys. Rev. E \textbf{100}, 012120 (2019)], we study the statistics of the system radius $\ell$ and of the center of mass (COM) of $N\gg 1$ particles. Typical fluctuations of $\ell$ fall under the Gumbel universality class for all $H>0$, and we use extreme value statistics to calculate the moments of $\ell$. We show that, for $H>1/2$, large deviations of the COM exhibit an anomalous scaling behavior. We also uncover a singularity in the corresponding rate function at $N\to\infty$, which is caused by a ``big jump" effect.

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arXiv2025

Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

We develop an age-structured hydrodynamic (HD) theory which describes the collective behavior of $N\gg 1$ anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle -- the time since its last reset -- as an explicit dynamical variable and allows for resetting rules which introduce global inter-particle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, $D(t)\sim t^{2H-1}$, where $H>0$. We apply this theory to three different resetting protocols: independent resetting to the origin (model~A), resetting to the origin of the particle farthest from it (model~B), and a scaled-diffusion extension of the ``Brownian bees" model of Berestycki et al, Ann. Probab. \textbf{50}, 2133 (2022). In all these models non-equilibrium steady states are reached at long times, and we determine the steady-state densities. For model A the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: in particular, they have compact supports for all $H>0$. The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols which introduce global inter-particle correlations.

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arXiv2024

Strong anomalous diffusion for free-ranging birds

Diffusion and anomalous diffusion are widely observed and used to study movement across organisms, resulting in extensive use of the mean and mean-squared displacement (MSD). However, these measures - corresponding to specific displacement moments - do not capture the full complexity of movement behavior. Using high-resolution data from over 70 million localizations of young and adult free-ranging Barn Owls (\textit{Tyto alba}), we reveal strong anomalous diffusion as nonlinear growth of displacement moments. The moment spectrum function $\lambda_t(q)$ -- defined by $\left<|\bm{x}(t)|^q\right> \sim t^{\lambda_t(q)}$ -- displays piecewise linearity in $q$, with a critical moment marking the crossover between scaling regimes. This highlights the need of a broad spectrum of displacement moments to characterize movement, which we link to age-specific ecological drivers. Furthermore, a characteristic timescale of five minutes marks an unexpected transition from a convex to a concave $\lambda_t(q)$. Using two stochastic models - a bounded L\'evy walk and a multi-mode behavioral model - we account for the observed phenomena, showing good agreement with data, relating age-specific behavioral states to environmentally confined movement, and demonstrating how L\'evy walk-like patterns can arise from underlying behavioral structure. Finally, we discuss the ecological significance of our results, arguing that strong anomalous diffusion may be widespread in animal movement.

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arXiv2024

Non-Markovian Rock-Paper-Scissors games

There is mounting evidence that species interactions often involve long-term memory, with highly-varying waiting times between successive events and long-range temporal correlations. Accounting for memory undermines the common Markovian assumption, and dramatically impacts key ingredients of population dynamics including birth, foraging, predation, and competition processes. Here, we study a critical aspect of population dynamics, namely non-Markovian multi-species competition. This is done in the realm of the zero-sum rock-paper-scissors (zRPS) model that is broadly used in the life sciences to metaphorically describe cyclic competition between three interacting species. We develop a general non-Markovian formalism for multi-species dynamics, allowing us to determine the regions of the parameter space where each species dominates. In particular, when the dynamics are Markovian, the waiting times are exponentially distributed and the fate of the zRPS model in large well-mixed populations is encoded in a remarkably simple condition, often referred to as the ``law of the weakest'' (LOW), stating that the species with the lowest growth rate is the most likely to prevail. We show that the survival behavior and LOW of the zRPS model are critically affected by non-exponential waiting times, and especially, by their coefficient of variation. Our findings provide key insight into the influence of long waiting times on non-Markovian evolutionary processes.

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arXiv2024

Efficient weighted-ensemble network simulations of the SIS model of epidemics

The presence of erratic or unstable paths in standard kinetic Monte Carlo simulations significantly undermines the accurate simulation and sampling of transition pathways. While typically reliable methods, such as the Gillespie algorithm, are employed to simulate such paths, they encounter challenges in efficiently identifying rare events due to their sequential nature and reliance on exact Monte Carlo sampling. In contrast, the weighted ensemble method effectively samples rare events and accelerates the exploration of complex reaction pathways by distributing computational resources among multiple replicas, where each replica is assigned a weight reflecting its importance, and evolves independently from the others. Here, we implement the highly efficient and robust weighted ensemble method to model susceptible-infected-susceptible (SIS) dynamics on large heterogeneous population networks, and explore the interplay between stochasticity and contact heterogeneity which ultimately gives rise to disease clearance. Studying a wide variety of networks characterized by fat-tailed asymmetric degree distributions, we are able to compute the mean time to extinction and quasi-stationary distribution around it in previously-inaccessible parameter regimes.

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arXiv2024

Escape from a metastable state in non-Markovian population dynamics

We study the long-time dynamics in non-Markovian single-population stochastic models, where one or more reactions are modelled as a stochastic process with a fat-tailed non-exponential distribution of waiting times, mimicking long-term memory. We focus on three prototypical examples: genetic switching, population establishment and population extinction, all with non-exponential production rates. The system is studied in two regimes. In the first, the distribution of waiting times has a finite mean. Here, the system approaches a (quasi)stationary steady state at long times, and we develop a general WKB approach for these non-Markovian systems. We derive explicit results for the mean population size and mean escape time from the metastable state of the stochastic dynamics. In this realm, we reveal that for sufficiently strong memory, a memory-induced (meta)stable state can emerge in the system. In the second regime, the waiting time distribution is assumed to have an infinite mean. Here, for bistable systems we find two distinct scaling regimes, separated by an exponentially long time which may strongly depend on the initial conditions of the system.

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arXiv2023

Non-Markovian gene expression

We study two non-Markovian gene-expression models in which protein production is a stochastic process with a fat-tailed non-exponential waiting time distribution (WTD). For both models, we find two distinct scaling regimes separated by an exponentially long time, proportional to the mean first passage time (MFPT) to a ground state (with zero proteins) of the dynamics, from which the system can only exit via a non-exponential reaction. At times shorter than the MFPT the dynamics are stationary and ergodic, entailing similarity across different realizations of the same process, with an increased Fano factor of the protein distribution, even when the WTD has a finite cutoff. Notably, at times longer than the MFPT the dynamics are nonstationary and nonergodic, entailing significant variability across different realizations. The MFPT to the ground state is shown to directly affect the average population sizes and we postulate that the transition to nonergodicity is universal in such non-Markovian models.

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arXiv2022

Competition in a system of Brownian particles: Encouraging achievers

We introduce and study analytically and numerically a simple model of inter-agent competition, where underachievement is strongly discouraged. We consider $N\gg 1$ particles performing independent Brownian motions on the line. Two particles are selected at random and at random times, and the particle closest to the origin is reset to it. We show that, in the limit of $N\to \infty$, the dynamics of the coarse-grained particle density field can be described by a nonlocal hydrodynamic theory which was encountered in a study of the spatial extent of epidemics in a critical regime. The hydrodynamic theory predicts relaxation of the system toward a stationary density profile of the "swarm" of particles, which exhibits a power-law decay at large distances. An interesting feature of this relaxation is a non-stationary "halo" around the stationary solution, which continues to expand in a self-similar manner. The expansion is ultimately arrested by finite-$N$ effects at a distance of order $\sqrt{N}$ from the origin, which gives an estimate of the average radius of the swarm. The hydrodynamic theory does not capture the behavior of the particle farthest from the origin -- the current leader. We suggest a simple scenario for typical fluctuations of the leader's distance from the origin and show that the mean distance continues to grow indefinitely as $\sqrt{t}$. Finally, we extend the inter-agent competition from $n=2$ to an arbitrary number $n$ of competing Brownian particles ($n\ll N$). Our analytical predictions are supported by Monte-Carlo simulations.

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arXiv2022

Fluctuations and first-passage properties of systems of Brownian particles with reset

We study stationary fluctuations in two models involving $N$ Brownian particles undergoing stochastic resetting to the origin in 1d. We start with the basic reset model where the particles reset independently (model A). Then we introduce nonlocal interparticle correlations by postulating that only the particle farthest from the origin is reset (model B). At long times both models approach nonequilibrium steady states. In the limit of $N\to \infty$, the steady-state particle density in model A has an infinite support, whereas in model B it has a compact support. A finite system radius, which scales at large $N$ as $\ln N$, appears in model A when $N$ is finite. In both models we study stationary fluctuations of the center of mass of the system and of the system's radius due to the random character of the Brownian motion and of the resetting events. In model A we determine exact distributions of these two quantities. The variance of the center of mass for both models scales as $1/N$. The variance of the radius is independent of $N$ in model A and exhibits an unusual scaling $(\ln N)/N$ in model B. The latter scaling is intimately related to the $1/f$ noise in the radius autocorrelations. Finally, we evaluate the mean first-passage time (MFPT) to a distant target in model A, model B, and the BBM. For model A we obtain an exact asymptotic expression for the MFPT which scales as $1/N$. For model B, and for the "Brownian bees" model, we propose a sharp upper bound for the MFPT. The bound assumes an ``evaporation" scenario, where the first passage requires multiple attempts of a single particle, which breaks away from the rest of the particles, to reach the target. The resulting MFPT for model B and the Brownian bees model scales exponentially with $\sqrt{N}$. We verify this bound by performing highly efficient weighted-ensemble simulations of the first passage in model B.

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arXiv2022

Unravelling the origins of anomalous diffusion: from molecules to migrating storks

Anomalous diffusion or, more generally, anomalous transport, with nonlinear dependence of the mean-squared displacement on the measurement time, is ubiquitous in nature. It has been observed in processes ranging from microscopic movement of molecules to macroscopic, large-scale paths of migrating birds. Using data from multiple empirical systems, spanning 12 orders of magnitude in length and 8 orders of magnitude in time, we employ a method to detect the individual underlying origins of anomalous diffusion and transport in the data. This method decomposes anomalous transport into three primary effects: long-range correlations ("Joseph effect"), fat-tailed probability density of increments ("Noah effect"), and non-stationarity ("Moses effect"). We show that such a decomposition of real-life data allows to infer nontrivial behavioral predictions, and to resolve open questions in the fields of single particle tracking in living cells and movement ecology.

physics.data-an↗

arXiv2022

Ergodicity breaking in area-restricted search of avian predators

Quantifying and comparing patterns of dynamical ecological systems require averaging over measurable quantities. For example, to infer variation in movement and behavior, metrics such as step length and velocity are averaged over large ensembles. Yet, in nonergodic systems such averaging is inconsistent; thus, identifying ergodicity breaking is essential in ecology. Using rich high-resolution movement datasets ($>\! 7 \times 10^7$ localizations) from 70 individuals and continuous-time random walk modeling, we find subdiffusive behavior and ergodicity breaking in the localized movement of three species of avian predators. Small-scale, within-patch movement was found to be qualitatively different, not inferrable and separated from large-scale inter-patch movement. Local search is characterized by long power-law-distributed waiting times with diverging mean, giving rise to ergodicity breaking in the form of considerable variability uniquely observed at this scale. This implies that wild animal movement is scale specific with no typical waiting time at the local scale.

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arXiv2022

Classification of anomalous diffusion in animal movement data using power spectral analysis

The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find $1/f$ noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations.

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arXiv2021

Phase transition in non-Markovian animal exploration model with preferential returns

We study a non-Markovian and nonstationary model of animal mobility incorporating both exploration and memory in the form of preferential returns. We derive exact results for the probability of visiting a given number of sites and develop a practical WKB approximation to treat the nonstationary problem. We further show that this model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for inter-individual variation in the population. Finally, we study the probability of visiting any site a given number of times and derive the corresponding mean-field equation. Here, we find a remarkable phase transition occurring at preferential returns which scale linearly with past visits. Following empirical evidence, we suggest that this phase transition reflects a trade-off between extensive and intensive foraging modes.

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arXiv2019

Extinction risk of a Metapopulation under the Allee Effect

We study the extinction risk of a fragmented population residing on a network of patches coupled by migration, where the local patch dynamics include the Allee effect. We show that mixing between patches dramatically influences the population's viability. Slow migration is shown to always increase the population's global extinction risk compared to the isolated case. At fast migration, we demonstrate that synchrony between patches minimizes the population's extinction risk. Moreover, we discover a critical migration rate that maximizes the extinction risk of the population, and identify an early-warning signal when approaching this state. Our theoretical results are confirmed via the highly-efficient weighted ensemble method. Notably, our analysis can also be applied to studying switching in gene regulatory networks with multiple transcriptional states.

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arXiv2018

Population Extinction under Bursty Reproduction in a Time Modulated Environment

In recent years non-demographic variability has been shown to greatly affect dynamics of stochastic populations. For example, non-demographic noise in the form of a bursty reproduction process with an a-priori unknown burst size, or environmental variability in the form of time-varying reaction rates, have been separately found to dramatically impact the extinction risk of isolated populations. In this work we investigate the extinction risk of an isolated population under the combined influence of these two types of non-demographic variation. Using the so-called momentum-space WKB approach we arrive at a set of time-dependent Hamilton equations. In order to account for the explicit time dependence, we find the instanton of the time-perturbed Hamiltonian numerically, where analytical expressions are presented in particular limits using various perturbation techniques. We focus on two classes of time-varying environments: periodically-varying rates corresponding to seasonal effects, and a sudden decrease in the birth rate corresponding to a catastrophe. All our theoretical results are tested against numerical Monte Carlo simulations with time-dependent rates and also against a numerical solution of the corresponding time-dependent Hamilton equations.

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