Random Walks on Comb-like Structures under Stochastic Resetting.

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

Phase Transition in a 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. Exact results for the probability of visiting a given number of sites are derived and a practical WKB approximation to treat the nonstationary problem is developed. A mean-field version of this model, first suggested by Song et al., [Modelling the scaling properties of human mobility, Nat. Phys. 6, 818 (2010)NPAHAX1745-247310.1038/nphys1760] was shown to well describe human movement data. We show that our generalized model adequately describes empirical movement data of Egyptian fruit bats (Rousettus aegyptiacus) when accounting for interindividual variation in the population. We also study the probability of visiting any site a given number of times and derive a mean-field equation. Our analysis yields 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.

Informational Entropy Threshold as a Physical Mechanism for Explaining Tree-like Decision Making in Humans.

While approaches based on physical grounds (such as the drift-diffusion model-DDM) have been exhaustively used in psychology and neuroscience to describe perceptual decision making in humans, similar approaches to complex situations, such as sequential (tree-like) decisions, are still scarce. For such scenarios that involve a reflective prospection of future options, we offer a plausible mechanism based on the idea that subjects can carry out an internal computation of the uncertainty about the different options available, which is computed through the corresponding Shannon entropy. When the amount of information gathered through sensory evidence is enough to reach a given threshold in the entropy, this will trigger the decision. Experimental evidence in favor of this entropy-based mechanism was provided by exploring human performance during navigation through a maze on a computer screen monitored with the help of eye trackers. In particular, our analysis allows us to prove that (i) prospection is effectively used by humans during such navigation tasks, and an indirect quantification of the level of prospection used is attainable; in addition, (ii) the distribution of decision times during the task exhibits power-law tails, a feature that our entropy-based mechanism is able to explain, unlike traditional (DDM-like) frameworks.

Foraging success under uncertainty: search tradeoffs and optimal space use.

Understanding the structural complexity and the main drivers of animal search behaviour is pivotal to foraging ecology. Yet, the role of uncertainty as a generative mechanism of movement patterns is poorly understood. Novel insights from search theory suggest that organisms should collect and assess new information from the environment by producing complex exploratory strategies. Based on an extension of the first passage time theory, and using simple equations and simulations, we unveil the elementary heuristics behind search behaviour. In particular, we show that normal diffusion is not enough for determining optimal exploratory behaviour but anomalous diffusion is required. Searching organisms go through two critical sequential phases (approach and detection) and experience fundamental search tradeoffs that may limit their encounter rates. Using experimental data, we show that biological search includes elements not fully considered in contemporary physical search theory. In particular, the need to consider search movement as a non-stationary process that brings the organism from one informational state to another. For example, the transition from remaining in an area to departing from it may occur through an exploratory state where cognitive search is challenged. Therefore, a more comprehensive view of foraging ecology requires including current perspectives about movement under uncertainty.

Dynamic redundancy as a mechanism to optimize collective random searches.

We explore the case of a group of random walkers looking for a target randomly located in space, such that the number of walkers is not constant but new ones can join the search, or those that are active can abandon it, with constant rates r_{b} and r_{d}, respectively. Exact analytical solutions are provided both for the fastest-first-passage time and for the collective time cost required to reach the target, for the exemplifying case of Brownian walkers with r_{d}=0. We prove that even for such a simple situation there exists an optimal rate r_{b} at which walkers should join the search to minimize the collective search costs. We discuss how these results open a new line to understand the optimal regulation in searches conducted through multiparticle random walks, e.g., in chemical or biological processes.

Intermittent random walks under stochastic resetting.

We analyze a one-dimensional intermittent random walk on an unbounded domain in the presence of stochastic resetting. In this process, the walker alternates between local intensive search, diffusion, and rapid ballistic relocations in which it does not react to the target. We demonstrate that Poissonian resetting leads to the existence of a non-equilibrium steady state. We calculate the distribution of the first arrival time to a target along with its mean and show the existence of an optimal reset rate. In particular, we prove that the initial condition of the walker, i.e., either starting diffusely or relocating, can significantly affect the long-time properties of the search process. Moreover, we demonstrate the presence of distinct parameter regimes for the global optimization of the mean first arrival time when ballistic and diffusive movements are in direct competition.

Spatiotemporal organization of ant foraging from a complex systems perspective.

We use complex systems science to explore the emergent behavioral patterns that typify eusocial species, using collective ant foraging as a paradigmatic example. Our particular aim is to provide a methodology to quantify how the collective orchestration of foraging provides functional advantages to ant colonies. For this, we combine (i) a purpose-built experimental arena replicating ant foraging across realistic spatial and temporal scales, and (ii) a set of analytical tools, grounded in information theory and spin-glass approaches, to explore the resulting data. This combined approach yields computational replicas of the colonies; these are high-dimensional models that store the experimental foraging patterns through a training process, and are then able to generate statistically similar patterns, in an analogous way to machine learning tools. These in silico models are then used to explore the colony performance under different resource availability scenarios. Our findings highlight how replicas of the colonies trained under constant and predictable experimental food conditions exhibit heightened foraging efficiencies, manifested in reduced times for food discovery and gathering, and accelerated transmission of information under similar conditions. However, these same replicas demonstrate a lack of resilience when faced with new foraging conditions. Conversely, replicas of colonies trained under fluctuating and uncertain food conditions reveal lower efficiencies at specific environments but increased resilience to shifts in food location.

Optimal intermittence in search strategies under speed-selective target detection.

Random search theory has been previously explored for both continuous and intermittent scanning modes with full target detection capacity. Here we present a new class of random search problems in which a single searcher performs flights of random velocities, the detection probability when it passes over a target location being conditioned to the searcher speed. As a result, target detection involves an N-passage process for which the mean search time is here analytically obtained through a renewal approximation. We apply the idea of speed-selective detection to random animal foraging since a fast movement is known to significantly degrade perception abilities in many animals. We show that speed-selective detection naturally introduces an optimal level of behavioral intermittence in order to solve the compromise between fast relocations and target detection capability.

Density-dependent dispersal and population aggregation patterns.

We have derived reaction-dispersal-aggregation equations from Markovian reaction-random walks with density-dependent jump rate or density-dependent dispersal kernels. From the corresponding diffusion limit we recover well-known reaction-diffusion-aggregation and reaction-diffusion-advection-aggregation equations. It is found that the ratio between the reaction and jump rates controls the onset of spatial patterns. We have analyzed the qualitative properties of the emerging spatial patterns. We have compared the conditions for the possibility of spatial instabilities for reaction-dispersal and reaction-diffusion processes with aggregation and have found that dispersal process is more stabilizing than diffusion. We have obtained a general threshold value for dispersal stability and have analyzed specific examples of biological interest.

Two-point approximation to the Kramers problem with coloured noise.

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.

Nonstandard diffusion under Markovian resetting in bounded domains.

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results, we find that when the waiting time probability has first and second finite moments, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset point to the boundary, or (iii) always beneficial. Instead, when the waiting time probability has the first or the two first moments diverging we find that resetting is always beneficial. Finally, we have also found that the optimal strategy to exit the domain depends on the reset rate. For low reset rates, walkers with exponential waiting times are found to be optimal, while for high reset rate, anomalous waiting times optimize the search process.

Conditioned backward and forward times of diffusion with stochastic resetting: A renewal theory approach.

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward B and forward F times being the times since the last and until the next reset, respectively, given that the current state of the system X(t) is known. These magnitudes are introduced with the paradigmatic case of diffusion under resetting, for which the backward and forward times are conditioned to the position of the walker. We find analytical expressions for the conditioned backward and forward time probability density functions (PDFs), and we compare them with numerical simulations. The general expressions allow us to study particular scenarios. For instance, for power-law reset time PDFs such that φ(t)∼t^{-1-α}, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of α due to the composition between the long-time scaling of diffusion P(x,t)∼1/sqrt[t] and the reset time PDF.

Effect of environmental fluctuations on invasion fronts.

We determine the density profile and velocity of invasion fronts in one-dimensional infinite habitats in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction-diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain analytical expressions for the front profile and velocity by constructing a variational principle. The effect of the noise differs, depending on whether it affects the density-independent growth rate, the intraspecific competition term or the Allee threshold. Fluctuations in the density-independent growth rate increase the invasion velocity and the population density of the invaded area. Fluctuations in the competition term also change the population density of the invaded area, but modify the invasion velocity only for certain initial conditions. Fluctuations in the Allee threshold can induce pulled or pushed invasion fronts as well as invasion failure. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction-diffusion equations with multiplicative noise.

Optimal escape-and-feeding dynamics of random walkers: Rethinking the convenience of ballistic strategies.

Excited random walks represent a convenient model to study food intake in a media which is progressively depleted by the walker. Trajectories in the model alternate between (i) feeding and (ii) escape (when food is missed and so it must be found again) periods, each governed by different movement rules. Here, we explore the case where the escape dynamics is adaptive, so at short times an area-restricted search is carried out, and a switch to extensive or ballistic motion occurs later if necessary. We derive for this case explicit analytical expressions of the mean escape time and the asymptotic growth of the depleted region in one dimension. These, together with numerical results in two dimensions, provide surprising evidence that ballistic searches are detrimental in such scenarios, a result which could explain why ballistic movement is barely observed in animal searches at microscopic and millimetric scales, therefore providing significant implications for biological foraging.

Continuous time random walks under Markovian resetting.

We investigate the effects of Markovian resetting events on continuous time random walks where the waiting times and the jump lengths are random variables distributed according to power-law probability density functions. We prove the existence of a nonequilibrium stationary state and finite mean first arrival time. However, the existence of an optimum reset rate is conditioned to a specific relationship between the exponents of both power-law tails. We also investigate the search efficiency by finding the optimal random walk which minimizes the mean first arrival time in terms of the reset rate, the distance of the initial position to the target, and the characteristic transport exponents.

Extinction conditions for isolated populations affected by environmental stochasticity.

We determine the critical patch size below which extinction occurs for populations living in one-dimensional habitats surrounded by completely hostile environments in the presence of environmental fluctuations. The population dynamics is reformulated in terms of a stochastic reaction-diffusion equation and is reduced to a deterministic equation that incorporates the systematic contributions of the noise. We obtain bifurcation diagrams and relations for the mean population density at the stationary state, the critical patch size, and the mean number of individuals in the habitat. The effect of the noise differs, depending on whether it affects the net growth rate or the intraspecific competition term. Fluctuations in the net growth rate decrease the critical patch size, whereas fluctuations in the competition term do not change the critical patch size. We compare our analytical results with numerical solutions of the stochastic partial differential equations and show that our procedure proves useful in dealing with reaction-diffusion equations with multiplicative noise.

Persistent random motion: uncovering cell migration dynamics.

In this paper we study analytically the stick-slip models recently introduced to explain the stochastic migration of free cells. We show that persistent motion of cells of many different types is compatible with stochastic reorientation models which admit an analytical mesoscopic treatment. This is proved by examining and discussing experimental data compiled from different sources in the literature, and by fitting some of these results too. We are able to explain many of the 'apparently complex' migration patterns obtained recently from cell tracking data, like power-law dependences in the mean square displacement or non-Gaussian behavior for the kurtosis and the velocity distributions, which depart from the predictions of the classical Ornstein-Uhlenbeck process.

Extinction and chaotic patterns in map lattices under hostile conditions.

Population dynamics in spatially extended systems can be modeled by Coupled Map Lattices (CML). We employ such equations to study the behavior of populations confined to a finite patch surrounded by a completely hostile environment. By means of the Galerkin projection and the normal solution ansatz, we are able to find analytical expressions for the critical patch size and show the existence of chaotic patterns. The analytical solutions provided are shown to fit, under the appropriate approximations, the dynamics of a logistic map. This interesting result, together with our discussion, suggests the existence of a universal class of spatially extended systems directly linked to the well-known characteristics of the logistic map.

Occupancy patterns in superorganisms: a spin-glass approach to ant exploration.

Emergence of collective, as well as superorganism-like, behaviour in biological populations requires the existence of rules of communication, either direct or indirect, between organisms. Because reaching an understanding of such rules at the individual level can be often difficult, approaches carried out at higher, or effective, levels of description can represent a useful alternative. In the present work, we show how a spin-glass approach characteristic of statistical physics can be used as a tool to characterize the properties of the spatial occupancy patterns of a biological population. We exploit the presence of pairwise interactions in spin-glass models for detecting correlations between occupancies at different sites in the media. Such correlations, we claim, represent a proxy to the existence of planned and/or social strategies in the spatial organization of the population. Our spin-glass approach does not only identify those correlations but produces a statistical replica of the system (at the level of occupancy patterns) that can be subsequently used for testing alternative conditions/hypothesis. Here, this methodology is presented and illustrated for a particular case of study: we analyse occupancy patterns of Aphaenogaster senilis ants during foraging through a simplified environment consisting of a discrete (tree-like) artificial lattice. Our spin-glass approach consistently reproduces the experimental occupancy patterns across time, and besides, an intuitive biological interpretation of the parameters is attainable. Likewise, we prove that pairwise correlations are important for reproducing these dynamics by showing how a null model, where such correlations are neglected, would perform much worse; this provides a solid evidence to the existence of superorganism-like strategies in the colony.

A model for plant invasions: the role of distributed generation times.

An analytical model consisting of adult plants and two types of seeds (unripe and mature) is considered and successfully tested using experimental data available for some invasive weeds (Echium plantagineum, Cytisus scoparius, Carduus nutans andCarduus acanthoides) from their native and exotic ranges. The model accounts for probability distribution functions (pdfs) for times of germination, growth, death and dispersal on two dimensions, so the general life-cycle of individuals is considered with high level of description. Our work provides for the first time, for a model containing all that life-cycle information, explicit relationship conditions for the invasive success and expressions for the speed of invasive fronts, which can be useful tools for invasions assessment. The expressions derived allow us to prove that the different phenotypes showed by the weeds in their native (exotic) ranges can explain their corresponding non-invasive (invasive) behavior.