Lévy movements and a slowly decaying memory allow efficient collective learning in groups of interacting foragers.

Many animal species benefit from spatial learning to adapt their foraging movements to the distribution of resources. Learning involves the collection, storage and retrieval of information, and depends on both the random search strategies employed and the memory capacities of the individual. For animals living in social groups, spatial learning can be further enhanced by information transfer among group members. However, how individual behavior affects the emergence of collective states of learning is still poorly understood. Here, with the help of a spatially explicit agent-based model where individuals transfer information to their peers, we analyze the effects on the use of resources of varying memory capacities in combination with different exploration strategies, such as ordinary random walks and Lévy flights. We find that individual Lévy displacements associated with a slow memory decay lead to a very rapid collective response, a high group cohesion and to an optimal exploitation of the best resource patches in static but complex environments, even when the interaction rate among individuals is low.

Channel-facilitated transport under resetting dynamics.

The transport of particles through channels holds immense significance in physics, chemistry, and biological sciences. For instance, the motion of solutes through biological membranes is facilitated by specialized proteins that create water-filled channels. Valuable insights can be obtained by studying the transition paths of particles through a channel and gathering information on their lifetimes inside the channel as well as their exit probabilities. In a similar vein, we consider a one-dimensional model of channel-facilitated transport where a diffusive particle is subject to attractive interactions with the walls of the channel. We study the statistics of conditional and unconditional escape times in the presence of resetting-an intermittent dynamics that brings the particle back to its initial coordinate stochastically. We determine analytically the physical conditions under which such a resetting mechanism becomes beneficial for the faster escape of the particles from the channel, thus enhancing transport. Our theory has been verified with the aid of Brownian dynamics simulations for various interaction strengths and extents. The overall results presented herein highlight the scope of resetting-based strategies to be universally promising for complex transport processes of single or long molecules through biological membranes.

Fick-Jacobs description and first passage dynamics for diffusion in a channel under stochastic resetting.

The transport of particles through channels is of paramount importance in physics, chemistry, and surface science due to its broad real world applications. Much insight can be gained by observing the transition paths of a particle through a channel and collecting statistics on the lifetimes in the channel or the escape probabilities from the channel. In this paper, we consider the diffusive transport through a narrow conical channel of a Brownian particle subject to intermittent dynamics, namely, stochastic resetting. As such, resetting brings the particle back to a desired location from where it resumes its diffusive phase. To this end, we extend the Fick-Jacobs theory of channel-facilitated diffusive transport to resetting-induced transport. Exact expressions for the conditional mean first passage times, escape probabilities, and the total average lifetime in the channel are obtained, and their behavior as a function of the resetting rate is highlighted. It is shown that resetting can expedite the transport through the channel-rigorous constraints for such conditions are then illustrated. Furthermore, we observe that a carefully chosen resetting rate can render the average lifetime of the particle inside the channel minimal. Interestingly, the optimal rate undergoes continuous and discontinuous transitions as some relevant system parameters are varied. The validity of our one-dimensional analysis and the corresponding theoretical predictions is supported by three-dimensional Brownian dynamics simulations. We thus believe that resetting can be useful to facilitate particle transport across biological membranes-a phenomenon that can spearhead further theoretical and experimental studies.

Lévy flight movements prevent extinctions and maximize population abundances in fragile Lotka-Volterra systems.

Multiple-scale mobility is ubiquitous in nature and has become instrumental for understanding and modeling animal foraging behavior. However, the impact of individual movements on the long-term stability of populations remains largely unexplored. We analyze deterministic and stochastic Lotka-Volterra systems, where mobile predators consume scarce resources (prey) confined in patches. In fragile systems (that is, those unfavorable to species coexistence), the predator species has a maximized abundance and is resilient to degraded prey conditions when individual mobility is multiple scaled. Within the Lévy flight model, highly superdiffusive foragers rarely encounter prey patches and go extinct, whereas normally diffusing foragers tend to proliferate within patches, causing extinctions by overexploitation. Lévy flights of intermediate index allow a sustainable balance between patch exploitation and regeneration over wide ranges of demographic rates. Our analytical and simulated results can explain field observations and suggest that scale-free random movements are an important mechanism by which entire populations adapt to scarcity in fragmented ecosystems.

Using posterior predictive distributions to analyse epidemic models: COVID-19 in Mexico City.

Epidemiological models usually contain a set of parameters that must be adjusted based on available observations. Once a model has been calibrated, it can be used as a forecasting tool to make predictions and to evaluate contingency plans. It is customary to employ only point estimators of model parameters for such predictions. However, some models may fit the same data reasonably well for a broad range of parameter values, and this flexibility means that predictions stemming from them will vary widely, depending on the particular values employed within the range that gives a good fit. When data are poor or incomplete, model uncertainty widens further. A way to circumvent this problem is to use Bayesian statistics to incorporate observations and use the full range of parameter estimates contained in the posterior distribution to adjust for uncertainties in model predictions. Specifically, given an epidemiological model and a probability distribution for observations, we use the posterior distribution of model parameters to generate all possible epidemic curves, whose information is encapsulated in posterior predictive distributions. From these, one can extract the worst-case scenario and study the impact of implementing contingency plans according to this assessment. We apply this approach to the evolution of COVID-19 in Mexico City and assess whether contingency plans are being successful and whether the epidemiological curve has flattened.

First Hitting Times to Intermittent Targets.

In noisy environments such as the cell, many processes involve target sites that are often hidden or inactive, and thus not always available for reaction with diffusing entities. To understand reaction kinetics in these situations, we study the first hitting time statistics of a one-dimensional Brownian particle searching for a target site that switches stochastically between visible and hidden phases. At high crypticity, an unexpected rate limited power-law regime emerges for the first hitting time density, which markedly differs from the classic t^{-3/2} scaling for steady targets. Our problem admits an asymptotic mapping onto a mixed, or Robin, boundary condition. Similar results are obtained with non-Markov targets and particles diffusing anomalously.

Quantifying uncertainty due to fission-fusion dynamics as a component of social complexity.

Groups of animals (including humans) may show flexible grouping patterns, in which temporary aggregations or subgroups come together and split, changing composition over short temporal scales, (i.e. fission and fusion). A high degree of fission-fusion dynamics may constrain the regulation of social relationships, introducing uncertainty in interactions between group members. Here we use Shannon's entropy to quantify the predictability of subgroup composition for three species known to differ in the way their subgroups come together and split over time: spider monkeys (Ateles geoffroyi), chimpanzees (Pan troglodytes) and geladas (Theropithecus gelada). We formulate a random expectation of entropy that considers subgroup size variation and sample size, against which the observed entropy in subgroup composition can be compared. Using the theory of set partitioning, we also develop a method to estimate the number of subgroups that the group is likely to be divided into, based on the composition and size of single focal subgroups. Our results indicate that Shannon's entropy and the estimated number of subgroups present at a given time provide quantitative metrics of uncertainty in the social environment (within which social relationships must be regulated) for groups with different degrees of fission-fusion dynamics. These metrics also represent an indirect quantification of the cognitive challenges posed by socially dynamic environments. Overall, our novel methodological approach provides new insight for understanding the evolution of social complexity and the mechanisms to cope with the uncertainty that results from fission-fusion dynamics.

Localization Transition Induced by Learning in Random Searches.

We solve an adaptive search model where a random walker or Lévy flight stochastically resets to previously visited sites on a d-dimensional lattice containing one trapping site. Because of reinforcement, a phase transition occurs when the resetting rate crosses a threshold above which nondiffusive stationary states emerge, localized around the inhomogeneity. The threshold depends on the trapping strength and on the walker's return probability in the memoryless case. The transition belongs to the same class as the self-consistent theory of Anderson localization. These results show that similarly to many living organisms and unlike the well-studied Markovian walks, non-Markov movement processes can allow agents to learn about their environment and promise to bring adaptive solutions in search tasks.

Random walks with preferential relocations to places visited in the past and their application to biology.

Strongly non-Markovian random walks offer a promising modeling framework for understanding animal and human mobility, yet, few analytical results are available for these processes. Here we solve exactly a model with long range memory where a random walker intermittently revisits previously visited sites according to a reinforced rule. The emergence of frequently visited locations generates very slow diffusion, logarithmic in time, whereas the walker probability density tends to a Gaussian. This scaling form does not emerge from the central limit theorem but from an unusual balance between random and long-range memory steps. In single trajectories, occupation patterns are heterogeneous and have a scale-free structure. The model exhibits good agreement with data of free-ranging capuchin monkeys.

Active particle in one dimension subjected to resetting with memory.

The study of diffusion with preferential returns to places visited in the past has attracted increased attention in recent years. In these highly non-Markov processes, a standard diffusive particle intermittently resets at a given rate to previously visited positions. At each reset, a position to be revisited is randomly chosen with a probability proportional to the accumulated amount of time spent by the particle at that position. These preferential revisits typically generate a very slow diffusion, logarithmic in time, but still with a Gaussian position distribution at late times. Here we consider an active version of this model, where between resets the particle is self-propelled with constant speed and switches direction in one dimension according to a telegraphic noise. Hence there are two sources of non-Markovianity in the problem. We exactly derive the position distribution in Fourier space, as well as the variance of the position at all times. The crossover from the short-time ballistic regime, dominated by activity, to the long-time anomalous logarithmic growth induced by memory is studied. We also analytically derive a large deviation principle for the position, which exhibits a logarithmic time scaling instead of the usual algebraic form. Interestingly, at large distances, the large deviations become independent of time and match the nonequilibrium steady state of a particle under resetting to its starting position only.

Diffusion with two resetting points.

We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight m of the further site is comprised in the interval [2.9028...,8.5603...]. When the initial position is a random variable that follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that m exceeds the critical value m_{c}=6.6008.... This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.

Optimizing the random search of a finite-lived target by a Lévy flight.

In many random search processes of interest in chemistry, biology, or during rescue operations, an entity must find a specific target site before the latter becomes inactive, no longer available for reaction or lost. We present exact results on a minimal model system, a one-dimensional searcher performing a discrete time random walk, or Lévy flight. In contrast with the case of a permanent target, the capture probability and the conditional mean first passage time can be optimized. The optimal Lévy index takes a nontrivial value, even in the long lifetime limit, and exhibits an abrupt transition as the initial distance to the target is varied. Depending on the target lifetime, this transition is discontinuous or continuous, separated by a nonconventional tricritical point. These results pave the way to the optimization of search processes under time constraints.

Buckling instability in ordered bacterial colonies.

Bacterial colonies often exhibit complex spatio-temporal organization. This collective behavior is affected by a multitude of factors ranging from the properties of individual cells (shape, motility, membrane structure) to chemotaxis and other means of cell-cell communication. One of the important but often overlooked mechanisms of spatio-temporal organization is direct mechanical contact among cells in dense colonies such as biofilms. While in natural habitats all these different mechanisms and factors act in concert, one can use laboratory cell cultures to study certain mechanisms in isolation. Recent work demonstrated that growth and ensuing expansion flow of rod-like bacteria Escherichia coli in confined environments leads to orientation of cells along the flow direction and thus to ordering of cells. However, the cell orientational ordering remained imperfect. In this paper we study one mechanism responsible for the persistence of disorder in growing cell populations. We demonstrate experimentally that a growing colony of nematically ordered cells is prone to the buckling instability. Our theoretical analysis and discrete-element simulations suggest that the nature of this instability is related to the anisotropy of the stress tensor in the ordered cell colony.

Thermodynamic states of nanoclusters at low pressure and low temperature: the case of 13 H(2).

A confinement model of finite-size systems that embodies an equation of state is presented. The temperature and pressure of the system are obtained from the positions and velocities of the enclosed particles after a number of molecular dynamics simulations. The pressure has static and dynamic (thermal) contributions, extending the Mie-Grüneisen equation of state to include weakly interacting anharmonic oscillators. The model is applied to a system of 13 H(2) molecules under low-pressure and low-temperature conditions in the classical regime. The confining cage in this case is a spherical hydrogen cavity. The Born-Oppenheimer molecular dynamics in conjunction with density functional theory are used for the time evolution of the particle system. The hydrogen molecules form a noncrystalline cluster structure with icosahedral symmetry that remains so in the whole temperature range investigated. The fluctuations of the interatomic distances increase with the temperature, while the orientational order of the enclosed system of molecules fades out, suggesting a gradual order-disorder transition.

First hitting times between a run-and-tumble particle and a stochastically gated target.

We study the statistics of the first hitting time between a one-dimensional run-and-tumble particle and a target site that switches intermittently between visible and invisible phases. The two-state dynamics of the target is independent of the motion of the particle, which can be absorbed by the target only in its visible phase. We obtain the mean first hitting time when the motion takes place in a finite domain with reflecting boundaries. Considering the turning rate of the particle as a tuning parameter, we find that ballistic motion represents the best strategy to minimize the mean first hitting time. However, the relative fluctuations of the first hitting time are large and exhibit nonmonotonous behaviors with respect to the turning rate or the target transition rates. Paradoxically, these fluctuations can be the largest for targets that are visible most of the time, and not for those that are mostly invisible or rapidly transiting between the two states. On the infinite line, the classical asymptotic behavior ∝t^{-3/2} of the first hitting time distribution is typically preceded, due to target intermittency, by an intermediate scaling regime varying as t^{-1/2}. The extent of this transient regime becomes very long when the target is most of the time invisible, especially at low turning rates. In both finite and infinite geometries, we draw analogies with partial absorption problems.

Diffusive transport on networks with stochastic resetting to multiple nodes.

We study the diffusive transport of Markovian random walks on arbitrary networks with stochastic resetting to multiple nodes. We deduce analytical expressions for the stationary occupation probability and for the mean and global first passage times. This general approach allows us to characterize the effect of resetting on the capacity of random walk strategies to reach a particular target or to explore the network. Our formalism holds for ergodic random walks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the efficiency of search strategies with resetting to multiple nodes. We apply the methods developed here to the dynamics with two reset nodes and derive analytical results for normal random walks and Lévy flights on rings. We also explore the effect of resetting to multiple nodes on a comb graph, Lévy flights that visit specific locations in a continuous space, and the Google random walk strategy on regular networks.

Random walks on networks with stochastic resetting.

We study random walks with stochastic resetting to the initial position on arbitrary networks. We obtain the stationary probability distribution as well as the mean and global first passage times, which allow us to characterize the effect of resetting on the capacity of a random walker to reach a particular target or to explore a finite network. We apply the results to rings, Cayley trees, and random and complex networks. Our formalism holds for undirected networks and can be implemented from the spectral properties of the random walk without resetting, providing a tool to analyze the search efficiency in different structures with the small-world property or communities. In this way, we extend the study of resetting processes to the domain of networks.

Collective learning from individual experiences and information transfer during group foraging.

Living in groups brings benefits to many animals, such as protection against predators and an improved capacity for sensing and making decisions while searching for resources in uncertain environments. A body of studies has shown how collective behaviours within animal groups on the move can be useful for pooling information about the current state of the environment. The effects of interactions on collective motion have been mostly studied in models of agents with no memory. Thus, whether coordinated behaviours can emerge from individuals with memory and different foraging experiences is still poorly understood. By means of an agent-based model, we quantify how individual memory and information fluxes can contribute to improving the foraging success of a group in complex environments. In this context, we define collective learning as a coordinated change of behaviour within a group resulting from individual experiences and information transfer. We show that an initially scattered population of foragers visiting dispersed resources can gradually achieve cohesion and become selectively localized in space around the most salient resource sites. Coordination is lost when memory or information transfer among individuals is suppressed. The present modelling framework provides predictions for empirical studies of collective learning and could also find applications in swarm robotics and motivate new search algorithms based on reinforcement.

Scale-free foraging by primates emerges from their interaction with a complex environment.

Scale-free foraging patterns are widespread among animals. These may be the outcome of an optimal searching strategy to find scarce, randomly distributed resources, but a less explored alternative is that this behaviour may result from the interaction of foraging animals with a particular distribution of resources. We introduce a simple foraging model where individual primates follow mental maps and choose their displacements according to a maximum efficiency criterion, in a spatially disordered environment containing many trees with a heterogeneous size distribution. We show that a particular tree-size frequency distribution induces non-Gaussian movement patterns with multiple spatial scales (Lévy walks). These results are consistent with field observations of tree-size variation and spider monkey (Ateles geoffroyi) foraging patterns. We discuss the consequences that our results may have for the patterns of seed dispersal by foraging primates.

Slow Lévy flights.

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy laws, as well as Gaussian distributions, can also be the limit distributions of processes with long-range memory that exhibit very slow diffusion, logarithmic in time. These processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the central limit theorem is modified in this context, keeping the usual distinction between analytic and nonanalytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.