Theory of home range estimation from displacement measurements of animal populations

The flight of the hornbill: drift and diffusion in arboreal avian movement

Capturing movement of animals in mathematical models has long been a keenly pursued direction of research¹. Any good model of animal movement is built upon information about the animal’s environment and the available resources including whether prey is in abundance or scarce, densely distributed or sparse². Such an approach could enable the identification of certain quantities or measures from the model that are species-specific characteristics. We propose here a mechanistic model to describe the movement of two species of Asian hornbills in a resource-abundant heterogenous landscape which includes degraded forests and human settlements. Hornbill telemetry data was used to this end. The birds show a bias both towards features of attraction such as nesting and roosting sites as well as possible bias away from points of repulsion such as human presence. These biases are accounted for with suitable potentials. The spatial patterns of movement are analyzed using the Fokker–Planck equation, which helps explain the variation in movement of different individuals. Search times to target locations were calculated using first passage time equations dual to the Fokker–Planck equations. We also find that the diffusion coefficients are larger for breeding birds than for non-breeding ones—a manifestation of repeated switching of directions to move back to the nest from foraging sites. The degree of directedness towards nests and roosts is captured by the drift coefficients. Non-breeding hornbills show similar values of the ratio of the two coefficients irrespective of the fact that their movement data is available from different seasons. Therefore, the ratio of drift to diffusion coefficients is indicative of an individual’s breeding status, as seen from available data. It could possibly also characterize different species. For all individuals, first passage times increase with proximity to human settlements, in agreement with the premise that anthropogenic activities close to nesting/roosting sites are not desirable.

Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting

We compute exactly the mean perimeter and the mean area of the convex hull of a two-dimensional isotropic Brownian motion of duration t and diffusion constant D, in the presence of resetting to the origin at a constant rate r. We show that for any t, the mean perimeter is given by 〈L(t)〉=2πsqrt[D/r]f_{1}(rt) and the mean area is given by 〈A(t)〉=2πD/rf_{2}(rt) where the scaling functions f_{1}(z) and f_{2}(z) are computed explicitly. For large t≫1/r, the mean perimeter grows extremely slowly as 〈L(t)〉∝ln(rt) with time. Likewise, the mean area also grows slowly as 〈A(t)〉∝ln^{2}(rt) for t≫1/r. Our exact results indicate that the convex hull, in the presence of resetting, approaches a circular shape at late times due to the isotropy of the Brownian motion. Numerical simulations are in perfect agreement with our analytical predictions.

Ecology of an ocelot population at the northern edge of the species' distribution in northern Sonora, Mexico

We used data from eight years of camera trapping at Rancho El Aribabi, a cattle ranch and conservation property in northern Sonora, Mexico, to examine the ecology of the northern-most known breeding population of ocelots (Leopardus pardalis). Ocelots were found mostly in two discrete and disjunct areas: a riverine riparian canyon at just less than 1,000 masl elevation and along arroyos in an oak-mesquite savanna in the Sierra Azul at 1,266–1,406 masl. An ocelot was also detected at a site between those two areas, in an area of a Sonoran desertscrub-foothills thornscrub ecotone at 1,300 masl. At least 18 ocelots, both males and females, were detected during the 2007–2011 and 2014–2018 sampling periods. A female with a kitten was documented in 2011. No individual ocelots were photographed in both areas, which are separated by a minimum of 11.29 km, and no individuals were photographed in both time periods. In a binary logistic regression, key environmental variables predicting ocelot presence were, in order of importance, distance to a paved road, distance to human habitation, proximity to water, and an anthropogenic influences index that was dominated by cattle. Another analysis corroborated the finding regarding ocelot presence and cattle. Contrary to previous studies, ocelot presence was not tied to vegetation cover close to the ground. We present information about the types of habitats and sites ocelots used, short-term movements, daily and seasonal activity patterns, and behavior, including occurrence of different individuals at or near the same site over short periods of time. We discuss ocelot home range, density, and movements, but small sample sizes and study design problems limit the value of estimates derived from our work. Rancho El Aribabi is a private, conservation ranch for which the owners have made voluntary conservation commitments that provide habitat and protection for ocelots and other animals and plants. This northern-most known breeding population is a likely source of ocelots that are periodically detected in southeastern Arizona. Our results should help facilitate conservation of the ocelot in other semi-arid areas of northwestern Mexico and adjacent USA.

Transport properties of random walks under stochastic noninstantaneous resetting

Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce noninstantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean-square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport subdiffusive and even reaching a stationary MSD, i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.

The time frame of home-range studies: from function to utilization

As technological and statistical innovations open new avenues in movement ecology, I review the fundamental implications of the time frame of home-range studies, with the aim of associating terminologies consistently with research objectives and methodologies. There is a fundamental distinction between (a) extrapolations of stationary distributions, associated with long time scales and aiming at asymptotic consistency, and (b) period-specific techniques, aiming at specificity but typically sensitive to the sampling design. I then review the difference between function and utilization in home-range studies. Most home-range studies are based on phenomenological descriptions of the time budgets of the study animals, not the function of the visited areas. I highlight emerging trends in automated pattern-recognition techniques for inference about function rather than utilization.

Analysis of Transmission of Infection in Epidemics: Confined Random Walkers in Dimensions Higher Than One

The process of transmission of infection in epidemics is analyzed by studying a pair of random walkers, the motion of each of which in two dimensions is confined spatially by the action of a quadratic potential centered at different locations for the two walks. The walkers are animals such as rodents in considerations of the Hantavirus epidemic, infected or susceptible. In this reaction-diffusion study, the reaction is the transmission of infection, and the confining potential represents the tendency of the animals to stay in the neighborhood of their home range centers. Calculations are based on a recently developed formalism (Kenkre and Sugaya in Bull Math Biol 76:3016-3027, 2014) structured around analytic solutions of a Smoluchowski equation and one of its aims is the resolution of peculiar but well-known problems of reaction-diffusion theory in two dimensions. The resolution is essential to a realistic application to field observations because the terrain over which the rodents move is best represented as a 2-d landscape. In the present analysis, reaction occurs not at points but in spatial regions of dimensions larger than 0. The analysis uncovers interesting nonintuitive phenomena one of which is similar to that encountered in the one-dimensional analysis given in the quoted article, and another specific to the fact that the reaction region is spatially extended. The analysis additionally provides a realistic description of observations on animals transmitting infection while moving on what is effectively a two-dimensional landscape. Along with the general formalism and explicit one-dimensional analysis given in Kenkre and Sugaya (2014), the present work forms a model calculational tool for the analysis for the transmission of infection in dilute systems.

Determinants of home range size and spatial overlap of Gracilinanus agilis (Mammalia: Didelphidae) in central-western Brazil

The use of space in mammals may vary according to sexual dimorphism, mating system and territorial behavior in order to ensure optimization of the reproductive success of each sex and the interactions with other species. In the present study, the determinants of home range (HR) size of males and females of Gracilinanus agilis (Burmeister 1854) were evaluated in a savanna remnant in central-western Brazil. We used live traps and capture-mark-recapture to estimate HR size. Using the method of minimum convex polygon, we estimated the HR of 24 individuals. The species showed sexual dimorphism, with males showing larger body size. The HR estimated was 0.38±0.41 ha and the highest estimated HR was for a male, with 2.08 ha. Females’ HR sizes varied according to body mass, food availability and number of captures. The more important predictor for males was the number of females found within their HRs. The overlapping areas between pairs of males were larger than those between pairs of females, suggesting that females have territorial behavior as they had mostly exclusive HRs. Considering that food availability was an important predictor for female HR size, we hypothesize that the territorial behavior in females is related to food resource.

Analysis of Confined Random Walkers with Applications to Processes Occurring in Molecular Aggregates and Immunological Systems

Explicit solutions are presented in the Laplace and time domains for a one-variable Fokker-Planck equation governing the probability density of a random walker moving in a confining potential. Illustrative applications are discussed in two unrelated physical contexts: quantum yields in a doped molecular crystal or photosynthetic system, and the motion of signal receptor clusters on the surface of a cell encountered in a problem in immunology. An interesting counterintuitive effect concerning the consequences of confinement is found in the former, and some insights into the driving force for microcluster centralization are gathered in the latter application.

Coupled effects of local movement and global interaction on contagion

By incorporating segregated spatial domain and individual-based linkage into the SIS (susceptible-infected-susceptible) model, we propose a generalized epidemic model which can change from the territorial epidemic model to the networked epidemic model. The role of the individual-based linkage between different spatial domains is investigated. As we adjust the timescale parameter τ from 0 to unity, which represents the degree of activation of the individual-based linkage, three regions are found. Within the region of 0 < τ < 0.02 , the epidemic is determined by local movement and is sensitive to the timescale τ . Within the region of 0.02 < τ < 0.5 , the epidemic is insensitive to the timescale τ . Within the region of 0.5 < τ < 1 , the outbreak of the epidemic is determined by the structure of the individual-based linkage. As we keep an eye on the first region, the role of activating the individual-based linkage in the present model is similar to the role of the shortcuts in the two-dimensional small world network. Only activating a small number of the individual-based linkage can prompt the outbreak of the epidemic globally. The role of narrowing segregated spatial domain and reducing mobility in epidemic control is checked. These two measures are found to be conducive to curbing the spread of infectious disease only when the global interaction is suppressed. A log-log relation between the change in the number of infected individuals and the timescale τ is found. By calculating the epidemic threshold and the mean first encounter time, we heuristically analyze the microscopic characteristics of the propagation of the epidemic in the present model.

Theory of the transmission of infection in the spread of epidemics: interacting random walkers with and without confinement

A theory of the spread of epidemics is formulated on the basis of pairwise interactions in a dilute system of random walkers (infected and susceptible animals) moving in [Formula: see text] dimensions. The motion of an animal pair is taken to obey a Smoluchowski equation in [Formula: see text]-dimensional space that combines diffusion with confinement of each animal to its particular home range. An additional (reaction) term that comes into play when the animals are in close proximity describes the process of infection. Analytic solutions are obtained, confirmed by numerical procedures, and shown to predict a surprising effect of confinement. The effect is that infection spread has a non-monotonic dependence on the diffusion constant and/or the extent of the attachment of the animals to the home ranges. Optimum values of these parameters exist for any given distance between the attractive centers. Any change from those values, involving faster/slower diffusion or shallower/steeper confinement, hinders the transmission of infection. A physical explanation is provided by the theory. Reduction to the simpler case of no home ranges is demonstrated. Effective infection rates are calculated, and it is shown how to use them in complex systems consisting of dense populations.

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.

Analytic solutions for some reaction-diffusion scenarios

Motivated currently by the problem of coalescence of receptor clusters in mast cells in the general subject of immune reactions, and formerly by the investigation of exciton trapping and sensitized luminescence in molecular systems and aggregates, we present analytic expressions for survival probabilities of moving entities undergoing diffusion and reaction on encounter. Results we provide cover several novel situations in simple 1-d systems as well as higher-dimensional counterparts along with a useful compendium of such expressions in chemical physics and allied fields. We also emphasize the importance of the relationship of discrete sink term analysis to continuum boundary condition studies.

Reaction-diffusion theory in the presence of an attractive harmonic potential

Problems involving the capture of a moving entity by a trap occur in a variety of physical situations, the moving entity being an electron, an excitation, an atom, a molecule, a biological object such as a receptor cluster, a cell, or even an animal such as a mouse carrying an epidemic. Theoretical considerations have almost always assumed that the particle motion is translationally invariant. We study here the case when that assumption is relaxed, in that the particle is additionally subjected to a harmonic potential. This tethering to a center modifies the reaction-diffusion phenomenon. Using a Smoluchowski equation to describe the system, we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions. Interesting features emerge depending on the relative location of the trap, the attractive center, and the initial placement of the diffusing particle.

Spatial extent of an outbreak in animal epidemics

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, and therefore, fluctuations around their average are important: then, it is commonly assumed that the susceptible-infected-recovered mechanism can be described by a stochastic birth-death process of Galton-Watson type. The displacements of the infected individuals can be modeled by resorting to brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, like in the case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and we compare them with Monte Carlo simulations.

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.

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.

Linking animal movement to site fidelity

Site fidelity, the recurrent visit of an animal to a previously occupied area is a wide-spread behavior in the animal kingdom. The relevance of site fidelity to territoriality, successful breeding, social associations, optimal foraging and other ecological processes, demands accurate quantification. Here we generalize previous theory that connects site fidelity patterns to random walk parameters within the framework of the space-time fractional diffusion equation. In particular, we describe the site fidelity function in terms of animal movement characteristics via the Lévy exponent, which controls the step-length distribution of the random steps at each turning point, and the waiting time exponent that controls for how long an animal awaits before actually moving. The analytical results obtained will provide a rigorous benchmark for empirically driven studies of animal site fidelity.

Animal interactions and the emergence of territoriality

Inferring the role of interactions in territorial animals relies upon accurate recordings of the behaviour of neighbouring individuals. Such accurate recordings are rarely available from field studies. As a result, quantification of the interaction mechanisms has often relied upon theoretical approaches, which hitherto have been limited to comparisons of macroscopic population-level predictions from un-tested interaction models. Here we present a quantitative framework that possesses a microscopic testable hypothesis on the mechanism of conspecific avoidance mediated by olfactory signals in the form of scent marks. We find that the key parameters controlling territoriality are two: the average territory size, i.e. the inverse of the population density, and the time span during which animal scent marks remain active. Since permanent monitoring of a territorial border is not possible, scent marks need to function in the temporary absence of the resident. As chemical signals carried by the scent only last a finite amount of time, each animal needs to revisit territorial boundaries frequently and refresh its own scent marks in order to deter possible intruders. The size of the territory an animal can maintain is thus proportional to the time necessary for an animal to move between its own territorial boundaries. By using an agent-based model to take into account the possible spatio-temporal movement trajectories of individual animals, we show that the emerging territories are the result of a form of collective animal movement where, different to shoaling, flocking or herding, interactions are highly heterogeneous in space and time. The applicability of our hypothesis has been tested with a prototypical territorial animal, the red fox (Vulpes vulpes).

How animals succeed in sharing and occupying space in an efficient way has always fascinated biologists. When occupying space involves marking and defending a given area, the animal is said to be territorial. By scent marking the locations that an animal visits, it conveys to a potential intruder that the area is claimed by another animal. Once an intruder encounters a foreign scent, it typically retreats from it to avoid an aggressive response by the resident animal. This is the so-called mechanism of conspecific avoidance. By considering this mechanism and the movement of the individual animals, we predict how territorial patterns are formed and maintained. Data and information on the red fox has served as a benchmark to test our predictions and has provided the experimental support to our theory. The implications of our results reach far beyond behavioural ecology, encompassing fields from epidemiology and conservation biology to social and state boundary dynamics in human society and ‘divide and conquer’ approaches to collective robotics.

Home range evolution and its implication in population outbreaks

We investigated the phenomenon of population outbreaks in a spatial predator-prey model, and we found that pattern formation and outbreaks occur if the predators have a limited neighbourhood of interaction with the preys. The outbreaks can display a scale-invariant power-law tail, indicating self-organized criticality. We have also studied the system from an evolutionary point of view, where the predator home range is a hereditary trait subjected to mutations. We found that mutation drives the predator home range area to an optimal value where pattern formation and outbreaks are still present, but the latter are much less frequent. We developed analytical approximations using mean field and pair correlation techniques that indicate that the predation strategy is crucial for existence of this optimal home range area.

Extinction of refugia of hantavirus infection in a spatially heterogeneous environment

We predict an abrupt observable transition, on the basis of numerical studies, of hantavirus infection in terrain characterized by spatially dependent environmental resources. The underlying framework of the analysis is that of Fisher equations with an internal degree of freedom, the state of infection. The unexpected prediction is of the sudden disappearance of refugia of infection in spite of the existence of supercritical (favorable) food resources, brought about by reduction of their spatial extent. Numerical results are presented and a theoretical explanation is provided on analytic grounds on the basis of the competition of diffusion of rodents carrying the hantavirus and nonlinearity present in the resource interactions.

Theory of possible effects of the Allee phenomenon on the population of an epidemic reservoir

We investigate possible effects of high-order nonlinearities on the shapes of infection refugia of the reservoir of an infectious disease. We replace Fisher-type equations that have been recently used to describe, among others, the Hantavirus spread in mouse populations by generalizations capable of describing Allee effects that are a consequence of the high-order nonlinearities. After analyzing the equations to calculate steady-state solutions, we study the stability of those solutions and compare to the earlier Fisher-type case. Finally, we consider the spatial modulation of the environment and find that unexpected results appear, including a bifurcation that has not been studied before.

Molecular motion in cell membranes: analytic study of fence-hindered random walks

A theoretical calculation is presented to describe the confined motion of transmembrane molecules in cell membranes. The study is analytic, based on Master equations for the probability of the molecules moving as random walkers, and leads to explicit usable solutions including expressions for the molecular mean square displacement and effective diffusion constants. One outcome is a detailed understanding of the dependence of the time variation of the mean square displacement on the initial placement of the molecule within the confined region. How to use the calculations is illustrated by extracting (confinement) compartment sizes from experimentally reported published observations from single particle tracking experiments on the diffusion of gold-tagged G -protein coupled mu -opioid receptors in the normal rat kidney cell membrane, and by further comparing the analytical results to observations on the diffusion of phospholipids, also in normal rat kidney cells.

Analytic steady-state space use patterns and rapid computations in mechanistic home range analysis

Mechanistic home range models are important tools in modeling animal dynamics in spatially complex environments. We introduce a class of stochastic models for animal movement in a habitat of varying preference. Such models interpolate between spatially implicit resource selection analysis (RSA) and advection-diffusion models, possessing these two models as limiting cases. We find a closed-form solution for the steady-state (equilibrium) probability distribution u using a factorization of the redistribution operator into symmetric and diagonal parts. How space use is controlled by the habitat preference function w depends on the characteristic width of the animals' redistribution kernel: when the redistribution kernel is wide relative to variation in w, u proportional, variant w, whereas when it is narrow relative to variation in w, u proportional, variant w (2). In addition, we analyze the behavior at discontinuities in w which occur at habitat type boundaries, and simulate the dynamics of space use given two-dimensional prey-availability data, exploring the effect of the redistribution kernel width. Our factorization allows such numerical simulations to be done extremely fast; we expect this to aid the computationally intensive task of model parameter fitting and inverse modeling.

Effect of predators of juvenile rodents on the spread of the hantavirus epidemic

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.

Determining landscape use of Holocene mammals using strontium isotopes

The use of the landscape by animals is predicted to be a function of their body size. However, empirical data relating these two variables from an array of body sizes within a single mammalian community are scarce. We tested this prediction by assessing landscape use of mammals by analyzing strontium (Sr) isotope signatures found in mammalian hard tissues representing a 3,000-year record. We examined: (1) the Sr-determined landscape area of small (approximately 100 g), medium (approximately 1,500 g) and large (approximately 100,000 g) mammals, and; (2) whether the area used by these mammals varied during periods of environmental change. Strontium isotope values were obtained from 46 specimens from the Holocene paleontological deposits of Lamar Cave and Waterfall Locality in Wyoming, USA, as well as from 13 modern ungulate specimens from the same area. Our data indicate that medium- and large-sized species use larger percentages of the landscape than do species of small body size. The isotope values for specimens from each of the paleontological sites are similar across all stratigraphic levels, suggesting no change in home range over the last 3,000 years, even though climate is known to have fluctuated at these sites over this time period. Further, our study verifies that the fossil localities represent the local community. Where bedrock geology is appropriate, the use of strontium isotope analyses provides a valuable tool for discerning landscape use by vertebrate communities, an important though generally difficult aspect of an ancient species niche to identify.