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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.

Interactions between moving entities and spatial disorder have a fundamental impact on the transport properties of natural and artificial systems. The unprecedented resolution brought by modern technologies in particle or individual tracking and in recording of the spatial environment demand more advanced quantitative techniques to interpret how movement is affected by spatial heterogeneities. Here we tackle this challenge by developing a mathematical framework to model inert particle-environment interactions in domains of arbitrary shape and dimensions. By constructing a fully analytic formalism we predict how the spatio-temporal dynamics of randomly moving entities change in the presence of reflecting or partially reflecting obstacles, regions of increased or decreased diffusivity, one-way gates, open partitions, reversible traps as well as long range connections to far away areas. To demonstrate the vast applicability of our techniques we employ them to study three examples spanning different spatial and temporal scales in physiology, ecology and molecular biology.

Random transmission events between individuals occurring at short scales control patterns emerging at much larger scales in natural and artificial systems. Examples range from the spatial propagation of an infectious pathogen in an animal population to the spread of misinformation in online social networks or the sharing of target locations between robot units in a swarm. Despite the ubiquity of information transfer events, a general methodology to quantify spatio-temporal transmission processes has remained elusive. The challenge in predicting when and where information is passed from one individual to another stems from the limited number of analytic approaches and from the large fluctuations and inherent computational cost of stochastic simulation outputs, the main theoretical tool available to study such processes so far. Here we overcome these limitations by developing an analytic theory of transmission dynamics between randomly moving agents in arbitrary spatial domains and with arbitrary information transfer efficiency. We move beyond well-known approximations employed to study reaction diffusion phenomena, such as the motion and reaction limited regimes, by quantifying exactly the mean reaction time in presence of multiple heterogeneous reactive locations. To demonstrate the wide applicability of our theory we employ it in different scenarios. We show how the type of spatial confinement may change by many orders of magnitude the time scale at which transmission occurs. When acquiring information represents the ability to capture, we use our formalism to uncover counterintuitive evasive strategies in a predator–prey contest between territorial animals. When information transmission represents the transfer of an infectious pathogen, we consider a population with susceptible, infected and recovered individuals that move and pass infection upon meeting and predict analytically the basic reproduction number. Finally we show how to apply the transmission theory semi-analytically when the topology of where individuals move is that of a network.

This foreword is to introduce the reason for this special issue of the journal. Professor Vasudev Mangesh (Nitant) Kenkre has had a distinguished career as a statistical physicist, teacher, educator and a magnet for a great many statistical physicists from around the world across nearly five decades. This volume is to celebrate Nitant’s work. The contributors are mostly his former students, friends and collaborators. This foreword captures a fraction of what Nitant means to us.

A particle roaming in a potential in the presence of spatial heterogeneities such as traps or physical barriers is a mathematical representation of a vast number of processes across multiple disciplines, ranging from the dynamics of chemical reactions and the movement of foraging animals to the performance of a financial asset. Among the approaches to tackle such problems, a powerful one is the so-called defect technique. Within a continuous space–time formulation, the technique has been successfully used to analyze diffusion problems with partially absorbing boundaries, but has not been employed to study other types of boundaries. Here we exploit the continuity equation to extend the defect technique to when partially reflecting barriers are present. For each boundary type, we apply the defect formalism to the Ornstein–Uhlenbeck process, recovering known analytical results and presenting new ones.

We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (1) Type A dynamics whereby the interface is placed between two lattice points, and (2) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green's function or propagator for the composite system in unbounded domain as well as domains confined with reflecting, absorbing, and mixed boundaries. For the case with reflecting confinement in the absence of bias, the steady-state probability shows a step-like behavior for the Type A dynamics, while it is uniform for the Type B dynamics. We also derive explicit expressions for the first-passage probability and the mean first-passage time, and compare the hitting time dependence to a single target. Finally, considering the continuous-space continuous-time limit of the propagator, we obtain the boundary conditions at the interface. At the interface, while the flux is the same, the probability density is discontinuous for Type A and is continuous for Type B. For the latter we derive a generalized version of the so-called leather boundary condition in the appropriate limit., Comment: Submitted version: 30 pages, 8 figures. Accepted for publication in J. Stat. Mech.: Theory Exp

We consider the dynamics of lattice random walks with resetting. The walker moving randomly on a lattice of arbitrary dimensions resets at every time step to a given site with a constant probability $r$. We construct a discrete renewal equation and present closed-form expressions for different quantities of the resetting dynamics in terms of the underlying reset-free propagator or Green's function. We apply our formalism to the biased random walk dynamics in one-dimensional unbounded space and show how one recovers in the continuous limits results for diffusion with resetting. The resetting dynamics of biased random walker in one-dimensional domain bounded with periodic and reflecting boundaries is also analyzed. Depending on the bias the first-passage probability in periodic domain shows multi-fold non-monotonicity as $r$ is varied. Finally, we apply our formalism to study the transmission dynamics of two lattice walkers with resetting in one-dimensional domain bounded by periodic and reflecting boundaries. The probability of a definite transmission between the walkers shows non-monotonic behavior as the resetting probabilities are varied., Comment: v1: 32 pages, 10 figures

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.

Exploiting powerful techniques from physics and mathematics, this book studies animal movement in ecology, with a focus on epidemic spread. Pulmonary syndrome is not only feared in epidemics of recent times, such as COVID-19, but is also characteristic of epidemics studied earlier such as Hantavirus. The Hantavirus is one of the book's central topics. Correlations between epidemic outbreaks and precipitation events like El Niño are analyzed and spatial reservoirs of infection in off-period of the epidemic, known as refugia, are studied. Predicted traveling waves of infection are successfully compared to field observations. Territoriality in scent-marking animals is presented, with parallels drawn with the theory of melting. The flocking and herding of birds and mammals are described in terms of collective excitations. For scientists interested in movement ecology and epidemic spread, this book provides effective solutions to long-standing problems.

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 spatio-temporal dynamics of the propagator with reflecting boundaries, bimodal features in the first-passage probability in periodic domains and the minimisation of the mean first-return time for a bias of intermediate strength in rectangular domains. Furthermore, we quantify how in a multi-target 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.

Many complex behaviours 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 multiple length-scales. Here, we present a perspective on how some of these challenges could be overcome by using multi-agent modelling 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 modelling 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 behaviours offers a means to take synthetic biology beyond single molecules or cells and towards the creation of systems with functions that can only emerge from collectives at multiple scales.

We consider a random walker whose motion is tethered around a focal point. We use two models that exhibit the same spatial dependence in the steady state but widely different dynamics. In one case, the walker is subject to a deterministic bias towards the focal point, while in the other case, it resets its position to the focal point at random times. The deterministic tendency of the biased walker makes the forays away from the focal point more unlikely when compared to the random nature of the returns of the resetting walker. This difference has consequences on the spatio-temporal dynamics at intermediate times. To show the differences in the two models, we analyze their probability distribution and their dynamics in presence and absence of partially or fully absorbing traps. We derive analytically various quantities: (i) mean first-passage times to one target, where we recover results obtained earlier by a different technique, (ii) splitting probabilities to either of two targets as well as survival probabilities when one or either target is partially absorbing. The interplay between confinement, diffusion and absorbing traps produces interesting non-monotonic effects in various quantities, all potentially accessible in experiments. The formalism developed here may have a diverse range of applications, from study of animals roaming within home ranges and of electronic excitations moving in organic crystals to developing efficient search algorithms for locating targets in a crowded environment., Comment: 26 pages, 7 figures

A mechanism to generate spatially segregated territories in two dimensions consists of individuals depositing their own marks and avoiding foreign ones. This form of environment-mediated interaction, called stigmergy, makes the emerging spatiotemporal territory dynamics quite rich. Short-lived marks produce rapidly morphing and highly mobile territories. Long-lived marks yield slow territories with a narrowly defined shape distribution. The change in territory mobility depending on the time for which individual marks remain active is accompanied by a liquid-hexatic-solid transition akin to the Kosterlitz-Thouless melting scenario.

Online coverage path planning is a canonical multi-robot task, where the objective is to minimize the time it takes for robots to visit every point in an unknown area. Two general major approaches have been explored in the literature: a stigmergic approach, inspired by ant behavior, relies on active marking of the environment. In contrast, the collaborative approach relies instead on localization, memory of positions, and global communications. In this paper, we report on a new approach, inspired by territorial bird chirping, which borrows from both previous approaches: it relies on localization and memory, but not on global communications. We provide a detailed analytic and empirical evaluation of this model.

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.

We solve an adaptive search model where a random walker or L\'evy flight stochastically resets to previously visited sites on a $d$-dimensional lattice containing one trapping site. Due to reinforcement, a phase transition occurs when the resetting rate crosses a threshold above which non-diffusive 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., Comment: 5 pages, 5 figures + 4 pages of Supplemental Information. Accepted in Physical Review Letters

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.

Maximilian Konzack, Tim Ophelders, Michel A. Westenberg and Kevin Buchin are supported by the Netherlands Organisation for Scientific Research (NWO) under grant no. 612.001.207 (Maximilian Konzack, Michel A. Westenberg and Kevin Buchin) and grant no. 639.023.208 (Tim Ophelders). The analysis of interaction between movement trajectories is of interest for various domains when movement of multiple objects is concerned. Interaction often includes a delayed response, making it difficult to detect interaction with current methods that compare movement at specific time intervals. We propose analyses and visualizations, on a local and global scale, of delayed movement responses, where an action is followed by a reaction over time, on trajectories recorded simultaneously. We developed a novel approach to compute the global delay in subquadratic time using a fast Fourier transform (FFT). Central to our local analysis of delays is the computation of a matching between the trajectories in a so-called delay space. It encodes the similarities between all pairs of points of the trajectories. In the visualization, the edges of the matching are bundled into patches, such that shape and color of a patch help to encode changes in an interaction pattern. To evaluate our approach experimentally, we have implemented it as a prototype visual analytics tool and have applied the tool on three bidimensional data sets. For this we used various measures to compute the delay space, including the directional distance, a new similarity measure, which captures more complex interactions by combining directional and spatial characteristics. We compare matchings of various methods computing similarity between trajectories. We also compare various procedures to compute the matching in the delay space, specifically the Fréchet distance, dynamic time warping (DTW), and edit distance (ED). Finally, we demonstrate how to validate the consistency of pairwise matchings by computing matchings between more than two trajectories. Publisher PDF

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.

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’.

We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can be also useful for understanding spatial learning by living organisms., Comment: 26 pages, 10 figures

Summary 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.

Starting from a Langevin equation with memory describing the attraction of a particle to a center, we investigate its transport and response properties corresponding to two special forms of the memory: one is algebraic, i.e., power-law, and the other involves a delay. We examine the properties of the Green function of the Langevin equation and encounter Mittag-Leffler and Lambert W-functions well-known in the literature. In the presence of white noise, we study two experimental situations, one involving the motional narrowing of spectral lines and the other the steady-state size of the particle under consideration. By comparing the results to counterparts for a simple exponential memory, we uncover instructive similarities and differences. Perhaps surprisingly, we find that the Balescu-Swenson theorem that states that non-Markoffian equations do not add anything new to the description of steady-state or equilibrium observables is violated for our system in that the saturation size of the particle in the steady-state depends on the memory function utilized. A natural generalization of the Smoluchowski equation for the time-local case is examined and found to satisfy the Balescu-Swenson theorem and describe accurately the first moment but not the second and higher moments. We also calculate two-time correlation functions for all three cases of the memory, and show how they differ from (tend to) their Markoffian counterparts at small (large) values of the difference between the two times.

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.

Understanding ecological processes relies upon the knowledge of the dynamics of each individual component. In the context of animal population ecology, the way animals move and interact is of fundamental importance in explaining a variety of observed patterns. Here, we present a theoretical investigation on the movement dynamics of interacting scent-marking animals. We study how the movement statistics of territorial animals is responsible for the appearance of damped oscillations in the mean square displacement (MSD) of the animals. This non-monotonicity is shown to depend on one dimensionless parameter, given by the ratio of the correlation distance between successive steps to the size of the territory. As that parameter increases, the time dependence of the animal's MSD displays a transition from monotonic, characteristic of Brownian walks, to non-monotonic, characteristic of highly correlated walks. The results presented here represent a novel way of determining the degree of persistence in animal movement processes within confined regions.

Despite recent advances in molecular and metagenomic approaches, it is still unclear how spatiotemporal variations in microbial communities influence the biological pump, exporting organic carbon from the surface to the deep oceans. We address this important problem by constructing a simple model of a prokaryotic metacommunity in which two generalist ecotypes compete for two resources. The first and the second ecotypes have, respectively, a higher preference for particulate organic carbon (POC) and dissolved organic carbon. Sinking of POC would lead to a higher abundance of the first ecotype in the deep layer compared to the surface layer, but vertical mixing in turn generates a net upward flux of this ecotype to the surface layer. This upward movement accelerates the shifts in the community composition during the phytoplankton bloom, contributing to a higher efficiency in POC remineralization at the surface layer and reducing the carbon flux to the deep layer.

Effects of predators of juvenile mice on the spread of the Hantavirus are analyzed in the context of a recently proposed model. Two critical values of the predation probability are identified. When the smaller of them is exceeded, the hantavirus infection vanishes without extinguishing the mice population. When the larger is exceeded, the entire mice population vanishes. These results suggest the possibility of control of the spread of the epidemic by introducing predators in areas of mice colonies in a suitable way so that such control does not kill all the mice but lowers the epidemic spread.

A generalized model of the spread of the Hantavirus in mice populations is presented on the basis of recent observational findings concerning the movement characteristics of the mice that carry the infection. The factual information behind the generalization is based on mark-recapture observations reported in Giuggioli et al. [Bull. Math. Biol. 67, 1135 (2005)] that have necessitated the introduction of home ranges in the simple model of Hantavirus spread presented by Abramson and Kenkre [Phys. Rev. E 66, 11912 (2002)]. The essential feature of the model presented here is the existence of adult mice that remain largely confined to locations near their home ranges, and itinerant juvenile mice that are not so confined, and, during their search for their own homes, move and infect both other juveniles and adults that they meet during their movement. The model is presented at three levels of description: mean field, kinetic and configuration. Results of calculations are shown explicitly from the mean field equations and the simulation rules, and are found to agree in some respects and to differ in others. The origin of the differences is shown to lie in spatial correlations. It is indicated how mark-recapture observations in the field may be employed to verify the applicability of the theory.

Interactions between trajectories need to be analyzed in various domains to gain insight into movement patterns. Such interactions often take place with some delayed response. We propose an approach to analyze and visualize delayed responses on two trajectories recorded simultaneously and with the same sampling rate. Central to our approach is the computation of a matching between the trajectories in a so-called delay space. We also introduce a new similarity measure between trajectories, which combines directional and spatial characteristics. To evaluate our approach experimentally, we have implemented it as a prototype visual analytics tool and have applied the tool on two datasets.

The relaxation time approximation (RTA) is commonly employed in nonequilibrium statistical mechanics to approximate solutions to the Boltzmann equation in terms of an exponential relaxation to equilibrium. Despite its common use, the RTA suffers from the drawback that relaxation times commonly employed are independent of initial conditions. We derive a variational principle for solutions to the Boltzmann equation, which allows us to extend the standard RTA using relaxation times that depend on the initial distribution. Tests of the approach on a calculation of the mobility for a one-dimensional (1D) tight-binding band indicate that our analysis typically provides a better approximation than the standard RTA.