Exact spatiotemporal dynamics of lattice random walks in hexagonal and honeycomb domains.

A variety of transport processes in natural and man-made systems are intrinsically random. To model their stochasticity, lattice random walks have been employed for a long time, mainly by considering Cartesian lattices. However, in many applications in bounded space the geometry of the domain may have profound effects on the dynamics and ought to be accounted for. We consider here the cases of the six-neighbor (hexagonal) and three-neighbor (honeycomb) lattices, which are utilized in models ranging from adatoms diffusing in metals and excitations diffusing on single-walled carbon nanotubes to animal foraging strategy and the formation of territories in scent-marking organisms. In these and other examples, the main theoretical tool to study the dynamics of lattice random walks in hexagonal geometries has been via simulations. Analytic representations have in most cases been inaccessible, in particular in bounded hexagons, given the complicated "zigzag" boundary conditions that a walker is subject to. Here we generalize the method of images to hexagonal geometries and obtain closed-form expressions for the occupation probability, the so-called propagator, for lattice random walks both on hexagonal and honeycomb lattices with periodic, reflective, and absorbing boundary conditions. In the periodic case, we identify two possible choices of image placement and their corresponding propagators. Using them, we construct the exact propagators for the other boundary conditions, and we derive transport-related statistical quantities such as first-passage probabilities to one or multiple targets and their means, elucidating the effect of the boundary condition on transport properties.

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach.

We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

Sociogenesis in unbounded space: modelling self-organised cohesive collective motion.

Maintaining cohesion between randomly moving agents in unbounded space is an essential functionality for many real-world applications requiring distributed multi-agent systems. We develop a bio-inspired collective movement model in 1D unbounded space to ensure such functionality. Using an internal agent belief to estimate the mesoscopic state of the system, agent motion is coupled to a dynamically self-generated social ranking variable. This coupling between social information and individual movement is exploited to induce spatial self-sorting and produces an adaptive, group-relative coordinate system that stabilises random motion in unbounded space. We investigate the state-space of the model in terms of its key control parameters and find two separate regimes for the system to attain dynamical cohesive states, including a Partial Sensing regime in which the system self-selects nearest-neighbour distances so as to ensure a near-constant mean number of sensed neighbours. Overall, our approach constitutes a novel theoretical development in models of collective movement, as it considers agents who make decisions based on internal representations of their social environment that explicitly take into account spatial variation in a dynamic internal variable.

Toward Engineering Biosystems With Emergent Collective Functions.

Many complex behaviors in biological systems emerge from large populations of interacting molecules or cells, generating functions that go beyond the capabilities of the individual parts. Such collective phenomena are of great interest to bioengineers due to their robustness and scalability. However, engineering emergent collective functions is difficult because they arise as a consequence of complex multi-level feedback, which often spans many length-scales. Here, we present a perspective on how some of these challenges could be overcome by using multi-agent modeling as a design framework within synthetic biology. Using case studies covering the construction of synthetic ecologies to biological computation and synthetic cellularity, we show how multi-agent modeling can capture the core features of complex multi-scale systems and provide novel insights into the underlying mechanisms which guide emergent functionalities across scales. The ability to unravel design rules underpinning these behaviors offers a means to take synthetic biology beyond single molecules or cells and toward the creation of systems with functions that can only emerge from collectives at multiple scales.

Closed-form solutions to the dynamics of confined biased lattice random walks in arbitrary dimensions.

Biased lattice random walks (BLRW) are used to model random motion with drift in a variety of empirical situations in engineering and natural systems such as phototaxis, chemotaxis, or gravitaxis. When motion is also affected by the presence of external borders resulting from natural barriers or experimental apparatuses, modelling biased random movement in confinement becomes necessary. To study these scenarios, confined BLRW models have been employed but so far only through computational techniques due to the lack of an analytic framework. Here, we lay the groundwork for such an analytical approach by deriving the Green's functions, or propagators, for the confined BLRW in arbitrary dimensions and arbitrary boundary conditions. By using these propagators we construct explicitly the time-dependent first-passage probability in one dimension for reflecting and periodic domains, while in higher dimensions we are able to find its generating function. The latter is used to find the mean first-passage passage time for a d-dimensional box, d-dimensional torus or a combination of both. We show the appearance of surprising characteristics such as the presence of saddles in the spatiotemporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimization of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multitarget environment with the presence of a bias shorter mean first-passage times can be achieved by placing fewer targets close to boundaries in contrast to many targets away from them.

Fokker-Planck representations of non-Markov Langevin equations: application to delayed systems.

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Making ecological models adequate.

Critical evaluation of the adequacy of ecological models is urgently needed to enhance their utility in developing theory and enabling environmental managers and policymakers to make informed decisions. Poorly supported management can have detrimental, costly or irreversible impacts on the environment and society. Here, we examine common issues in ecological modelling and suggest criteria for improving modelling frameworks. An appropriate level of process description is crucial to constructing the best possible model, given the available data and understanding of ecological structures. Model details unsupported by data typically lead to over parameterisation and poor model performance. Conversely, a lack of mechanistic details may limit a model's ability to predict ecological systems' responses to management. Ecological studies that employ models should follow a set of model adequacy assessment protocols that include: asking a series of critical questions regarding state and control variable selection, the determinacy of data, and the sensitivity and validity of analyses. We also need to improve model elaboration, refinement and coarse graining procedures to better understand the relevancy and adequacy of our models and the role they play in advancing theory, improving hind and forecasting, and enabling problem solving and management.

Identifying influential neighbors in animal flocking.

Schools of fish and flocks of birds can move together in synchrony and decide on new directions of movement in a seamless way. This is possible because group members constantly share directional information with their neighbors. Although detecting the directionality of other group members is known to be important to maintain cohesion, it is not clear how many neighbors each individual can simultaneously track and pay attention to, and what the spatial distribution of these influential neighbors is. Here, we address these questions on shoals of Hemigrammus rhodostomus, a species of fish exhibiting strong schooling behavior. We adopt a data-driven analysis technique based on the study of short-term directional correlations to identify which neighbors have the strongest influence over the participation of an individual in a collective U-turn event. We find that fish mainly react to one or two neighbors at a time. Moreover, we find no correlation between the distance rank of a neighbor and its likelihood to be influential. We interpret our results in terms of fish allocating sequential and selective attention to their neighbors.

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.

Flying foxes create extensive seed shadows and enhance germination success of pioneer plant species in deforested Madagascan landscapes.

Seed dispersal plays a significant role in forest regeneration and maintenance. Flying foxes are often posited as effective long-distance seed dispersers due to their large home ranges and ability to disperse seeds when flying. We evaluate the importance of the Madagascan flying fox Pteropus rufus in the maintenance and regeneration of forests in one of the world's priority conservation areas. We tested germination success of over 20,000 seeds from the figs Ficus polita, F. grevei and F. lutea extracted from bat faeces and ripe fruits under progressively more natural conditions, ranging from petri-dishes to outdoor environments. Seeds from all fig species showed increased germination success after passing through the bats' digestive tracts. Outside, germination success in F. polita was highest in faecal seeds grown under semi-shaded conditions, and seeds that passed through bats showed increased seedling establishment success. We used data from feeding trials and GPS tracking to construct seed shadow maps to visualize seed dispersal patterns. The models use Gaussian probability density functions to predict the likelihood of defecation events occurring after feeding. In captivity, bats had short gut retention times (often < 30 mins), but were sometimes able to retain seeds for over 24h. In the wild, bats travelled 3-5 km within 24-280 min after feeding, when defecation of ingested seeds is very likely. They produced extensive seed shadows (11 bats potentially dispersing seeds over 58,000 ha over 45 total days of tracking) when feeding on figs within their large foraging areas and dispersed the seeds in habitats that were often partially shaded and hence would facilitate germination up to 20 km from the feeding tree. Because figs are important pioneer species, P. rufus is an important dispersal vector that makes a vital contribution to the regeneration and maintenance of highly fragmented forest patches in Madagascar.

Measuring site fidelity and spatial segregation within animal societies.

Animals often display a marked tendency to return to previously visited locations that contain important resources, such as water, food, or developing brood that must be provisioned. A considerable body of work has demonstrated that this tendency is strongly expressed in ants, which exhibit fidelity to particular sites both inside and outside the nest. However, thus far many studies of this phenomena have taken the approach of reducing an animal's trajectory to a summary statistic, such as the area it covers.Using both simulations of biased random walks, and empirical trajectories from individual rock ants, Temnothorax albipennis, we demonstrate that this reductive approach suffers from an unacceptably high rate of false negatives.To overcome this, we describe a site-centric approach which, in combination with a spatially-explicit null model, allows the identification of the important sites towards which individuals exhibit statistically significant biases.Using the ant trajectories, we illustrate how the site-centric approach can be combined with social network analysis tools to detect groups of individuals whose members display similar space-use patterns.We also address the mechanistic origin of individual site fidelity; by examining the sequence of visits to each site, we detect a statistical signature associated with a self-attracting walk - a non-Markovian movement model that has been suggested as a possible mechanism for generating individual site fidelity.

Bumblebees can discriminate between scent-marks deposited by conspecifics.

Bumblebees secrete a substance from their tarsi wherever they land, which can be detected by conspecifics. These secretions are referred to as scent-marks, which bumblebees are able to use as social cues. Although it has been found that bumblebees can detect and associate scent-marks with rewarding or unrewarding flowers, their ability at discriminating between scent-marks from bumblebees of differing relatedness is unknown. We performed three separate experiments with bumblebees (Bombus terrestris), where they were repeatedly exposed to rewarding and unrewarding artificial flowers simultaneously. Each flower type carried scent-marks from conspecifics of differing relatedness or were unmarked. We found that bumblebees are able to distinguish between 1. Unmarked flowers and flowers that they themselves had scent-marked, 2. Flowers scent-marked by themselves and flowers scent-marked by others in their nest (nestmates), and 3. Flowers scent-marked by their nestmates and flowers scent-marked by non-nestmates. The bumblebees found it more difficult to discriminate between each of the flower types when both flower types were scent-marked. Our findings show that bumblebees have the ability to discriminate between scent-marks of conspecifics, which are potentially very similar in their chemical composition, and they can use this ability to improve their foraging success.

Delayed response and biosonar perception explain movement coordination in trawling bats.

Animal coordinated movement interactions are commonly explained by assuming unspecified social forces of attraction, repulsion and alignment with parameters drawn from observed movement data. Here we propose and test a biologically realistic and quantifiable biosonar movement interaction mechanism for echolocating bats based on spatial perceptual bias, i.e. actual sound field, a reaction delay, and observed motor constraints in speed and acceleration. We found that foraging pairs of bats flying over a water surface swapped leader-follower roles and performed chases or coordinated manoeuvres by copying the heading a nearby individual has had up to 500 ms earlier. Our proposed mechanism based on the interplay between sensory-motor constraints and delayed alignment was able to recreate the observed spatial actor-reactor patterns. Remarkably, when we varied model parameters (response delay, hearing threshold and echolocation directionality) beyond those observed in nature, the spatio-temporal interaction patterns created by the model only recreated the observed interactions, i.e. chases, and best matched the observed spatial patterns for just those response delays, hearing thresholds and echolocation directionalities found to be used by bats. This supports the validity of our sensory ecology approach of movement coordination, where interacting bats localise each other by active echolocation rather than eavesdropping.

Exact dynamics of stochastic linear delayed systems: application to spatiotemporal coordination of comoving agents.

Delayed dynamics result from finite transmission speeds of a signal in the form of energy, mass, or information. In stochastic systems the resulting lagged dynamics challenge our understanding due to the rich behavioral repertoire encompassing monotonic, oscillatory, and unstable evolution. Despite the vast literature, quantifying this rich behavior is limited by a lack of explicit analytic studies of high-dimensional stochastic delay systems. Here we fill this gap for systems governed by a linear Langevin equation of any number of delays and spatial dimensions with additive Gaussian noise. By exploiting Laplace transforms we are able to derive an exact time-dependent analytic solution of the Langevin equation. By using characteristic functionals we are able to construct the full time dependence of the multivariate probability distribution of the stochastic process as a function of the delayed and nondelayed random variables. As an application we consider interactions in animal collective movement that go beyond the traditional assumption of instantaneous alignment. We propose models for coordinated maneuvers of comoving agents applicable to recent empirical findings in pigeons and bats whereby individuals copy the heading of their neighbors with some delay. We highlight possible strategies that individual pairs may adopt to reduce the variance in their velocity difference and/or in their spatial separation. We also show that a minimum in the variance of the spatial separation at long times can be achieved with certain ratios of measurement to reaction delay.

Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality.

Animal spacing has important implications for population abundance, species demography and the environment. Mechanisms underlying spatial segregation have their roots in the characteristics of the animals, their mutual interaction and their response, collective as well as individual, to environmental variables. This review describes how the combination of these factors shapes the patterns we observe and presents a practical, usable framework for the analysis of movement data in confined spaces. The basis of the framework is the theory of interacting random walks and the mathematical description of out-of-equilibrium systems. Although our focus is on modelling and interpreting animal home ranges and territories in vertebrates, we believe further studies on invertebrates may also help to answer questions and resolve unanswered puzzles that are still inaccessible to experimental investigation in vertebrate species.

Stigmergy, collective actions, and animal social spacing.

Collective animal behavior studies have led the way in developing models that account for a large number of individuals, but mostly have considered situations in which alignment and attraction play a key role, such as in schooling and flocking. By quantifying how animals react to one another's presence, when interaction is via conspecific avoidance rather than alignment or attraction, we present a mechanistic insight that enables us to link individual behavior and space use patterns. As animals respond to both current and past positions of their neighbors, the assumption that the relative location of individuals is statistically and history independent is not tenable, underscoring the limitations of traditional space use studies. We move beyond that assumption by constructing a framework to analyze spatial segregation of mobile animals when neighbor proximity may elicit a retreat, and by linking conspecific encounter rate to history-dependent avoidance behavior. Our approach rests on the knowledge that animals communicate by modifying the environment in which they live, providing a method to analyze social cohesion as stigmergy, a form of mediated animal-animal interaction. By considering a population of animals that mark the terrain as they move, we predict how the spatiotemporal patterns that emerge depend on the degree of stigmergy of the interaction processes. We find in particular that nonlocal decision rules may generate a nonmonotonic dependence of the animal encounter rate as a function of the tendency to retreat from locations recently visited by other conspecifics, which has fundamental implications for epidemic disease spread and animal sociality.

Quantifying behavioral changes in territorial animals caused by sudden population declines.

Although territorial animals are able to maintain exclusive use of certain regions of space, movement data from neighboring individuals often suggest overlapping home ranges. To explain and unify these two aspects of animal space use, we use recently developed mechanistic models of collective animal movement. We apply our approach to a natural experiment on an urban red fox (Vulpes vulpes) population that underwent a rapid decline in population density due to a sarcoptic mange epizooty. By extracting details of movement and interaction strategies from location data, we show how foxes alter their behavior, taking advantage of sudden population-level changes by acquiring areas vacated due to neighbor mortality, while ensuring territory boundaries remain contiguous. The rate of territory border movement increased eightfold as the population declined and the foxes' response time to neighboring scent reduced by a third. By demonstrating how observed, fluctuating territorial patterns emerge from movements and interactions of individual animals, our results give the first data-validated, mechanistic explanation of the elastic disc hypothesis, proposed nearly 80 years ago.

Encounter times in overlapping domains: application to epidemic spread in a population of territorial animals.

We develop an analytical method to calculate encounter times of two random walkers in one dimension when each individual is segregated in its own spatial domain and shares with its neighbor only a fraction of the available space, finding very good agreement with numerically exact calculations. We model a population of susceptible and infected territorial individuals with this spatial arrangement, and which may transmit an epidemic when they meet. We apply the results on encounter times to determine analytically the macroscopic propagation speed of the epidemic as a function of the microscopic characteristics: the confining geometry, the animal diffusion constant, and the infection transmission probability.

Quasi-one-dimensional waves in rodent populations in heterogeneous habitats: a consequence of elevational gradients on spatio-temporal dynamics.

Wave propagation can be clearly discerned in data collected on mouse populations in the Cibola National Forest (New Mexico, USA) related to seasonal changes. During an exploration of the construction of a methodology for investigations of the spread of the Hantavirus epidemic in mice we have built a system of interacting reaction diffusion equations of the Fisher-Kolmogorov-Petrovskii-Piskunov type. Although that approach has met with clear success recently in explaining Hantavirus refugia and other spatiotemporal correlations, we have discovered that certain observed features of the wave propagation observed in the data we mention are impossible to explain unless modifications are made. However, we have found that it is possible to provide a tentative explanation/description of the observations on the basis of an assumed Allee effect proposed to exist in the dynamics. Such incorporation of the Allee effect has been found useful in several of our recent investigations both of population dynamics and pattern formation and appears to be natural to the observed system. We report on our investigation of the observations with our extended theory.