- Review
- Open Access
Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality
- Luca Giuggioli^{1}Email author and
- V M Kenkre^{2}Email author
https://doi.org/10.1186/s40462-014-0020-7
© Giuggioli and Kenkre; licensee BioMed Central Ltd. 2014
Received: 23 April 2014
Accepted: 8 August 2014
Published: 3 September 2014
Abstract
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.
Keywords
- Animal spacing
- Confinement
- Movement ecology
- Interacting random walks
Introduction
Investigations on the establishment and maintenance of animal spacing patterns and confinement have intrigued researchers for more than a century. Causes and consequences of space use are among the most intensely studied aspects of animal behaviour as is clear from very early reports on territorial behaviour [1-4]. Observations spanning vertebrate and invertebrate phyla have resulted in a consensus that spacing is an innate feature of animal behaviour rooted in basic physiological needs and that it constitutes a fundamental structure through which individuals in social groups live and function (see e.g. [5-15]). Spatial confinement is a crucial feature. Despite its importance, unraveling the mechanisms through which territories, home ranges or other forms of spatial segregation arise, has proved elusive. Spacing patterns can result from the actions of one or more individuals, and serve a variety of functions [16] including the defense of resources and mates (see e.g. [17]), offspring [18,19] and den sites [20]. Other factors are the need to maintain a familiar area to reduce predation risks [21] and a response to intruder pressure [22]. The intermeshing of these different functionalities have made it difficult to define a territory or home range in a way that would embrace all behavioural manifestations of segregation.
A distinction between territories and home ranges put forward by Burt [23] defines a territory as a defended area, that is an area whose exclusive ownership, or at least priority of use, individuals attempt to maintain. A home range, on the other hand, merely represents how an animal has occupied a given region of space; there can be various territorial areas within a home range (see [24] for a more extensive discussion). Burt’s definition represents a conceptual advance in distinguishing territories and home ranges. But it does not provide for a clear delineation of home range boundaries in other contexts. The lack of a rigorous definition for home ranges has caused operational difficulties in determining their size quantitatively. Although major improvements beyond the early estimates of minimum convex polygons [25] have occurred over the years (see e.g. the latest studies in [26-32]), detecting the shape and size of home ranges from the movement of individuals is heavily affected by the sampling frequency as well as the time span to integrate animal locations—daily, seasonally or over the entire lifetime [33,34]. These arbitrary choices determine in different ways whether rarely visited or peripheral areas are included in the size of a home range [35,36]. Current views recognize the intrinsically dynamic nature of a home range, particularly its outer boundaries, and associate its origin to foraging strategies in renewable and patchy resources [37] as well as to an animal’s spatial memory of its environment [16,38].
In the present era of interdisciplinary investigations, it behooves the researcher in any field of science to attempt to profit from insights gained in other fields. The current article subscribes strongly to this philosophy. Characterising random spatial structures has a long tradition in other areas of research such as condensed matter (solid state) physics. Challenges facing solid state researchers are similar to those facing ecologists in this regard [39]. A territory or home range lacks a well-defined periodicity and the resulting structure may depend on the specificity of the underlying substrate, e.g. food distribution, geography of the terrain, vegetation cover, and the initial configuration of the animals in the population. As is the case with observations on amorphous condensed matter aggregates such as glassy and gel systems, ecological experiments require carefully prepared samples and repeated observations to make possible the characterisation of the emerging structures and their comparison to one another. Ensuring repeatability is a great deal more difficult in an ecological setting than in condensed matter systems in the laboratory, mainly because of the deep influence of the heterogeneity of the environment on animal behaviour, for which specific tools such as neutral landscape models have been developed in the last two decades [40-44]. This heterogeneity is partly at the root of intraspecific variability on territory and home range size dependence on resource distribution [14]. In addition, the large number of competing interactions in an animal population—attraction towards resource-rich areas, avoidance of locations visited by a neighbour or a predator, defense of other regions of space, etc.—may make it impossible to reach stationary states that are common place in laboratory systems: ecological systems often tend to remain in long-lived metastable states [45-47]. Metastability is not unknown in condensed matter systems—it occurs in glassy systems (see e.g. [48]) wherein the movement of the individual components can be contingent upon large scale collective rearrangements of the surrounding components. When there is departure from orderly spatial arrangement in a system, special methods of description suited to the treatment of disorder become necessary. When this departure arises from time-dependent dynamics of the interactions between its components, rather than from fixed heterogeneity of the environment, the disorder is called dynamic rather than static [49].
Dynamic disorder has been shown to have profound effects on transport dynamics of quasiparticles in organic crystals. In those systems, crystal vibrations impart to electrons or electronic excitations, the so-called excitons, movement properties quite different from those observed in more rigid solid systems such as inorganic crystals. When the coupling mechanism is strong, the traveling quasiparticle itself becomes heavier and deforms the crystal lattice affecting its own mobility. It thus interacts indirectly with itself trough a dynamic modification of the crystal lattice. These physical phenomena have profound analogy with ecological counterparts resulting in animal spacing patterns. Indirect mechanisms of interaction between animals have been known since 1950s. Pierre-Paul Grassé [50] (see also [51,52]) coined the term stigmergy to represent indirect interactions involving the response of insects to changes in the environments made by other insects. That early observation was prompted by mound construction by termites that dropped pellets of chewed earth at various locations until the pellet-dropping activity concentrated at one location to form a column. A much more recent study [53] has also investigated avoidance behaviour with similar lines of reasoning. We find the similarities of the stigmergy concept with dynamic disorder and polaron phenomena [49,54-56] in condensed matter physics to be so obvious that it would be totally inefficient not to utilize in ecology the enormous body of technique and insights that have accumulated in that field.
The integrative nature of animal space use has resulted in the study of spacing patterns blossoming into a truly interdisciplinary endeavour. A need has been felt for a useful paradigm of organism movement [57,58]. Mathematical modelling tools have been increasingly used to analyse movement data (see e.g. [59] and [60]). Experimental devices of lighter weight and higher resolution, such as movement sensors, trackers and data loggers, have continued to be introduced [61,62]. All this has pushed forward the study of animal space use considerably. Motivated by these latest developments, the co-authors of the present article felt that this was the right time for a synthesis on recent investigations on animal spacing patterns.
Given the vast literature on the topic, we also found it appropriate to focus our review on a small subset, in particular on the latest mathematical approaches to study animal home range and territoriality. These have witnessed a great deal of theoretical progress in the last twenty years since the important and recognised work of Lewis, Moorcroft and Murray [63,64]. Our present review will bring our own perspective on the study of spacing patterns which, while it builds on the previous progress, specializes on tools and concepts borrowed from statistical physics and non-equilibrium phenomena. Our interest is less in presenting an overall review of available treatments in the literature and more in providing an account natural to the thinking of the present coauthors.
The rest of the review is organised as follows. In Section “Fundamental considerations for mathematical development” we present our general approach to model animal movement and interactions and introduce some essentials of the mathematical tools we use. In Section “Interactions with the environment: the emergence of home ranges” we discuss home ranges and show in detail how their quantitative extents may be deduced from observations on animal displacements. In Section “Mutual interactions leading to territoriality: the effects of time scale disparity” we focus on territoriality, providing a simplified picture and practical mathematical procedures to study effects of scent-mediated interactions. In Section “Applications to observations in the field” we explain methodologies to extract biological parameters from movement data. Section “Conclusions” contains concluding remarks and thoughts on future directions.
Fundamental considerations for mathematical development
The problem of free will complicates the description of the motion of animals in comparison to that of inanimate objects. However, observations of the locations of animals exhibit stochastic properties overlaid on consequences of deterministic laws. The stochastic element can be often understood as arising from well defined distributions associated with random walks. A priori discussions of a philosophical nature regarding whether this viewpoint is tenable are of little practical use. The test of the validity of such a starting point ultimately lies in a comparison of the observed distribution with the predictions of a random force treatment. Comparisons of this kind have generally shown that in most cases the stochastic method is successful in addressing experimental observations. Our approach in this article is based on such a combination of random walk theory with classical equations of motion constructed with known biological facts and tendencies firmly in mind.
In such a formulation, specification of the particular realization of the random force R(t) is not required, its properties being reflected in quantities such as the diffusion constant D and the fact that what the evolution provides us is probability densities at any given time.
Equation (4) represents the many-body version of the Fokker-Planck description in Eq. (2), which requires a summation over the animals i since each individual may perceive a different force, the one-body interaction F _{ i }(x _{ i })=∇U _{ i }(x _{ i }), due to the spatial heterogeneity in the environment. The second term accounts for the possibility of different diffusion constants for each animal, and the third term with the second i-summation describes the many-body interaction among the animals through the interaction term \(\mathcal {U}(\mathbf {x}_{1},\mathbf {x}_{2},\cdots \mathbf {x}_{i}, \cdots \mathbf {x}_{N})\).
In systems that one encounters in physics, many-body interactions are almost always taken to be constructed from pairwise pieces. In such a case, the last i-summation above would take the form \(\sum _{i} \nabla _{i} \cdot \left [\sum _{j\ne i} \mathcal {U}(\mathbf {x}_{i},\mathbf {x}_{j})P\right ]\). In most cases in this article, we will begin with a similar starting point but end up in an effective interaction which is not necessarily pairwise. In describing processes such as transmission of infection in an epidemic, we may further include in the description a label for the state of infection. Eq. (4) represents the general framework with which we will describe macroscopic movement patterns, sometimes termed collective phenomena, that emerge from ‘microscopic’ interactions at the individual level [67].
However, such is rarely necessary in the description of animal motion.
While memory functions and the associated generalized master equations that employ them [72] are often convenient for analysis, there are situations in which, instead of a description in terms of time-nonlocal kernels in an integro-differential equation, time-dependent transport coefficients such as D(t) are found to be more natural and useful. A subtlety in such contexts is that it is often then necessary to consider nonlocality in space as well [73]. However, an equivalence between convolution equations with memory and non-convolution but time-dependent equations has been established and a usable prescription provided to go from one formalism to the other [73]. We will see a use of this mathematical viewpoint in Section “Mutual interactions leading to territoriality: the effects of time scale disparity” and Section “Applications to observations in the field”.
We will see that home ranges that emerge from animal interactions with the environment naturally introduce Smoluchowski considerations [74] in the analysis (tethering to an attractive centre) rather than simple diffusion, and that convenient modified pictures arise for the study of territorial behavior stemming from scent-mediated interactions [75]. Considerations such as time scale disparity between processes, an example being the movements of the boundaries of territories relative to the movements of the animals within them [76], will prove to be of crucial importance in our description.
The spatial dependence of the various interaction potentials is gentle in some cases and sharp in others. When the latter is true, the field nature of the potentials (the fact that they are defined at every point in space) may give rise, conveniently, to the dynamics of localized walls or spatial partitions. This can result in mild effects in methodology as when (see Section “Interactions with the environment: the emergence of home ranges” involving repulsive interactions with scent-marking animals, where the analysis is facilitated particularly because of time scale disparity between the wall motion and the animal motion.
Some of the other related tools for the descriptions of interacting random walkers are associated with simple repulsion [77-79], considerations on a discrete lattice [80], when exclusion has a finite range [81], when repulsion occurs within a confined domain [82], in presence of movement in a force field [83,84] and when the walkers have an additional attraction towards each other [85]. When the interactions between animals are attractive rather than repulsive, new phenomena such as flocking or herding arise and have fascinating consequences of their own. We will not deal with them in this article except to point to some exciting recent developments [86-93].
A quintessential problem of animal-animal interactions that has very high human relevance as well is that of the transmission of infection in epidemics. Research on this topic was launched quite early on in the seminal contributions of Anderson and May [94,95] and others [96-98], involving concepts such as mass action, SIR compartmental models, and the basic reproductive rate R _{0}. Spatial considerations were introduced into these and related areas of study independently by various authors [72,96,99-108] giving the studies a kinetic equation flavor. Missing from some of these studies were confinement features that arise in animal motion from home ranges and yet are clear and compelling in the light of field observations [109,110]. These confinement issues have now been introduced in a natural and mathematically tractable manner in recent investigations [111,112]. We are thus in possession of a usable framework capable of a detailed fundamental study of the transmission of infection in terms of interacting random walks specially under confinement. While of general applicability, the theory has particular relevance to animal movement in zoonotic diseases such as the Hantavirus [113], plague [114,115] and bovine tuberculosis [116,117] as it allows one to understand these disease systems by studying a set of confined random walkers moving on the terrain and transmitting infection on encounter.
Interactions with the environment: the emergence of home ranges
where the initial value of the MSD has been suppressed on the extreme right side.
A natural development is to augment the basic motion equation by logistic terms to describe sustenance and competition as well as birth and death of the animals, distinguishing, when necessary, between the so-called floaters [120] and resident individuals, and by aggression terms to describe the transmission of infection if it is a matter of concern. Such a detailed framework was constructed and used for the description of the Hantavirus [99,105-107,121,122]. Important to such analysis was the measurement of the quantities employed, in particular the diffusion constant D.
With the simple assumption that we are making in most of the present article that the movement of animals is an uncomplicated random walk, i.e., that the governing equation for the motion is a simple diffusion equation, it appears straightforward to measure the diffusion constant D from observations of the movements of the animals. The basic theoretical tool is the Einstein relation Eq. (6) between the animal MSD and time. Data considered for this purpose are often, although not always, of the mark-recapture kind, i.e., collected by capturing, tagging, and recapturing the animals in a prescribed (finite) region of space. As will be explained in detail in Section “Applications to observations in the field”, an application to observations on rodents in Panama and New Mexico led to the problem that the MSD, initially indeed linear in t, saturates for larger t, introducing L, a saturation length into the description. What is the significance of this length? One way of understanding it is to ascribe it to the fact that the rodents typically move near fixed burrows for reasons of security and food [123-125]. However, another relatively prosaic explanation is also possible. The mark-recapture observations employ a limited region of space where the traps are laid out. This observational feature itself introduces a grid length G independently of any characteristics of the animal motion. Either of these two factors could lead to the observed saturation of the MSD. A study of the interplay of the two length scales and a demonstration of how the home ranges of the animals involved can be extracted from the observations, despite their mutual interference, is presented below. For simplicity, we begin our explanation in 1d.
the center of attraction being the location of the burrow. It is clear that a characteristic spatial extent will emerge in the MSD from features of the potential U(x), viz., the home range L as shown later in Eq. (10) and (12). The mark-recapture method consists of capturing animals at locations x _{0} and then recapturing them later at other locations x. The aim is to deduce the unknown length L with the help of, and despite the interference of, G. The latter is provided by the observation technique and known a priori. Note that L is unknown and is characteristic of the moving rodents. We provide, first a succinct explanation of the procedure given by Kenkre [104], and follow it up with a detailed analysis as given by Giuggioli et al. [126].
Notice that the density of the burrow locations, the probe length G, and the home range length L which is intrinsic to U(x) are all represented in Eq. (8).
Extensions of Eq. (11) and (12) in 2d are straightforward and allow for observation windows of rectangular shape and for a potential along the orthogonal axis which can be different from U(x).
From Eq. (13) and (14) one notices that the parameters ζ=L/G and G ^{2} completely determine the saturation value of the MSD, which is expected given the presence of only two spatial scales L and G. To isolate the dependence on ζ, it is more convenient to normalise the MSD to the size of the observation domain G ^{2}/6, the factor 6 being present to make the right hand-side of Eqs. (13) and (14) reach 1 for \(\zeta \rightarrow +\infty \).
From the knowledge of G and the value of 〈Δ x ^{2}〉_{ss} obtained from movement data, one can then graphically extract the home range extent L [109,110]. In Figure 1 we display three horizontal black segments representing the value of 〈Δ x ^{2}〉_{ss} and its errors obtained from hypothetical movement observations. From the intercepts of these segments with the sigmoidal curve corresponding to the particular selected potential (the red one in this example), one can draw three vertical segments whose intercepts with the horizontal axis yield the value of L/G and its errors. Following this graphical inversion procedure, Giuggioli et al. [109] and Abramson et al. [110] have deduced the values of the home ranges of two different kinds of mice, Zygodontomys brevicauda in Panama and Peromyscus maniculatus in New Mexico, respectively (see more details in Section “Applications to observations in the field”).
Once home ranges are quantitatively measured in the manner explained, important questions arise: how to describe their consequences on animal dynamics, and what measurable effects these consequences have on known phenomena involving the animals. The first question is easily answered in that Smoluchowski equations such as Eq. (7) must be considered. Their propagators, obtained for instance through Ornstein-Uhlenbeck arguments are well-known [66] and yield interesting consequences [128] on the motion of random walkers under confinement. The second question has been answered in a recent investigation of the transmission of infection when animals move under confinement [111]. A surprising result has been found that the existence of finite home ranges can have unexpected effects on the efficiency of the transmission. A change in the diffusion constant or the strength of confinement (the latter being inversely related to L) has non-monotonic consequences on infection: an increase in D or a decrease in L might tend to increase the infection efficiency but only up to a point. Beyond an extremum, the changes have the opposite effect. What this means is that optimum values of D or L exist, departures from which always cause a decrease in the efficiency of infection transmission. Home range confinement is, thus, a nontrivial characteristic of animals in ecological investigations.
Mutual interactions leading to territoriality: the effects of time scale disparity
Many-body problems with mutual interactions are always much more difficult to solve, in any field of science, than those involving non-interacting individuals subjected to external fields. The present section deals with this aspect of our study and is consequently the largest in the article. An understanding of the collective feature of territory emergence is the aim. We focus on a particular approach that one of the coauthors of this article has taken along with his collaborators [75], by constructing the so-called territorial random walker (TRW) model, but describe also alternatives that have been proposed earlier [63,64,129-131].
The TRW model bears similarities to the autocatalytic model developed in 1989 by Deneubourg and co-workers [132,133] used to represent foraging Argentine ants that explore new areas following the pheromone trails left by others. A large literature on models of movement whereby individuals choose directions and steps according to the signals present in the environment have also appeared later, and are often referred to as reinforced [134] or active random walkers [135]. Examples include the formation of dendritic foraging patterns by ants [136], aggregation of myxobacetria [137], movement with preferential relocations to places visited in the past, [138] and many others (see e.g. [59]).
The TRW model is a stochastic computational model representing a set of random walkers, the animals, moving on a discrete lattice (with periodic boundary conditions) in continuous time. As an individual moves on the lattice it deposits a mark, which remains active for a finite time \(\mathcal {T}_{A}\). Upon the encounter of a foreign mark an individual interacts by retreating in a random direction away from foreign marks. Interestingly these reaction rules do not require any information retrieval on the part of the individuals because recollection of the locations visited by others is held in the environment rather than in the animal. Non-overlapping territories or marked areas are generated at each instant of time by the exclusion dynamics [77] of the TRW model [75].
where (m _{ c },n _{ c }) is the centroid position of an animal marked area at time t, \(\kappa _{l}(m,m_{c},n,n_{c})=(m-m_{c})/ \sqrt {(m-m_{c})^{2} + (n-n_{c})^{2}}\), \(\kappa _{u}(m,m_{c},n,n_{c})=(n-n_{c})/\sqrt {(m-m_{c})^{2} + (n-n_{c})^{2}}\). The the retreat bias, or avoidance response, p(τ) is a function of the age τ of the encountered mark. When p=1/2 for all τ, the walkers ignore the scent produced by others, whereas the choice p(τ)=1 for \(\tau \leq \mathcal {T}_{A}\), and equal to 1/2 for all other τ values, was used in the original TRW model—with random bias away from foreign scent rather than through Eq. (15). It is possible to show that a Master equation in discrete space governed by rates given by Eq. (15) with p(τ) independent of τ reduces to the 2d Holgate-Okubo localising tendency model in the continuum limit [140].
The spatial patterns emerging from the dynamics of conspecific avoidance depend on initial locations and scent mark profiles as well as the specific random realisation of the movement paths. Although these various degrees of stochasticity produce a rich repertoire of shape and size of the resulting home ranges, two important characteristic scales of the territorial random walk model can be identified: the movement rate that defines how quickly an animal covers the available territory and the rate of decay \(\mathcal {T}_{A}^{-1}\) of the deposited marks. If an animal succeeds in revisiting past locations before the time \(\mathcal {T}_{A}\) has elapsed, it maintains those locations as part of its territory by refreshing the old marks. An animal thus needs to return to a scented location to maintain that location as part of its territory. The mean return time to a system subset can be calculated for discrete stochastic models, using the Kac recurrence lemma [141] as the ratio of all possible configurations of the system divided by the number of configuration of the subset. In our case, for a discrete random walker on a confined 2d lattice with n sites, the mean return time to a specific site is thus equal to n, i.e., the area of the confined space. Taking the average territory size as the inverse of the population density ρ, the mean return time is simply obtained by rescaling ρ ^{−1} with 4D, where the multiplicative factor 4 is chosen for convenience so that \(4D\mathcal {T}_{A}\) represents the average area explored by a 2d random walker within time T _{ A }. The ratio between the time for the marks to remain active and the mean return time (4D ρ)^{−1} is thus \(Z=4D\rho \mathcal {T}_{A}\).
This parameter Z has also an intuitive interpretation in terms of spatial scales [53]: it is the ratio between the average area that a diffusing animal would cover in a time \(\mathcal {T}_{A}\) and the average size ρ ^{−1} that each animal would occupy if the terrain was equally divided into exclusive regions among the individuals of the population. When the probability of retreat upon the encounter of foreign marks is high, from the perspective of a focal individual an increase in \(\mathcal {T}_{A}\) makes its marks on the terrain last longer. Similarly an increase in ρ, reduces the space available to each animal and causes the focal individual to encounter the edge of its marked area more frequently. As a result Z has been named the spatial competition parameter [53] because an increase of either \(\mathcal {T}_{A}\) or ρ makes territorial marks persist longer, either preventing others from acquiring additional space or limiting the depth of intrusion into foreign territories.
Replacement of the interaction field with moving walls: a simplified picture
The spatial competition parameter is important for the construction of a simplified picture of the emergence of spacing patterns in the territorial random walk model. Small and large Z correspond respectively to fast and slow dynamics of territory boundaries. In the regime of slow territory dynamics, fluctuations of mark locations are limited by their extended persistence and the dynamics of the territories reduce mainly to those of the boundary marks. It is this regime that represents more closely ecological scenarios in which an animal moves relatively quickly within a region whose boundaries are not static but fluctuate over slower time scales [75].
In the regime with strong spatial competition, it is possible to obtain a simplified mathematical description that links the reaction response of the individual animals to the formation of territorial patterns. This simplification is made possible because of the time-scale disparity between the movement rate of the animals and the territory boundaries. Time-scale disparity arguments are commonly employed in interpreting physical problems [142], but they have also been used extensively in the ecological literature, e.g. in studies of intra- and inter-patch dynamics [143,144].
Characterising the movement of the animal
The second line of Eq. (17) represents the no-flux boundary conditions indicating that an animal cannot escape from its territory with L _{1x } and L _{2x }, respectively, the leftmost and rightmost territory edge along the x-axis, and L _{1y } and L _{2y } along the y-axis. The so-called memory ϕ(t) characterises the degree of correlation or anti-correlation of the animal steps. For the case of a persistent walk, the choice ϕ(t)=(v ^{2}/D)e ^{−t/T } represents an animal that moves at speed v in the same direction without turning for an average time T. In this case, the solution of Eq. (17) is separable along each axis and reduces to the product \(W(\vec {x},t)=\mathcal {W}_{x}(x,t)\mathcal {W}_{y}(y,t)\), whose mathematical expressions can be found in [146].
Describing the movement of the boundaries
Having specified \(W(\vec {x},t|\vec {L})\), we now turn to the time dependence of the probability distribution of the boundary positions \(B(\vec {L},t)\) which biologically represents two competing effects: the acquisition of new territory by the resident animal and the pressure of the neighbours for territorial takeover. In the adiabatic regime, a mean field prescription to describe the displacement of the boundaries is to ignore the detailed dynamics of the neighbours and their marked areas, accounting only for the slow movement statistics of the boundaries. As the boundary locations have a tendency to move and reduce (increase) the territory size when larger (smaller) than the equilibrium value, which is the inverse of the population density, a useful approximation is to impose that the boundaries are forced by an assumed spring that maintains the equilibrium territory size.
where K is the boundary diffusion constant, ∇^{2} is the Laplacian operator in the Cartesian coordinates \(\vec {L}=(L_{1x},L_{2x},L_{1y},L_{2y})\), γ is the phenomenological spring constant expressed in units of inverse length, and \(\bar {L}_{z}\) is the average territory length along each trajectory (\(\bar {L}_{x}\bar {L}_{y}\) is equal to the inverse of the population density). Notice that in Eq. (19) the interaction term \(\mathcal {U}(\vec {L})\), described in general terms in Eq. (4), is of the pairwise form \(\mathcal {U}(L_{2z},L_{1z})=L_{2z}-L_{1z}-\bar {L}_{z}\) and \(\mathcal {U}(L_{1z},L_{2z})=-(L_{2z}-L_{1z}-\bar {L}_{z})\)—as discussed in the introduction rare are the situations where interaction is considered to occur as resulting from non-pairwise events. These interaction terms in Eq. (19) represent the derivative of a harmonic potential centered around \(\bar {L}_{z}\) in both directions and for that reason there is a summation along the two axes. When φ(t) is time-independent, \({\int _{0}^{t}}\mathrm {d}s\,\varphi (s)\) is linear in t, and Eq. (19) reduces to a diffusive case. On the other hand, choices with sub-linear \({\int _{0}^{t}}\mathrm {d}s\,\varphi (s)\) reproduce the sub-diffusive scaling in the MSD observed in the full-blown stochastic simulations of territorial random walkers. In 1d, the boundary MSD scales as \(\sqrt {t}\) [76], a characteristic feature of tagged particle dynamics in single file systems (see e.g. [84]). In a 2d lattice, stochastic simulations for intermediate times display logarithmic corrections proportional to \(t/\ln (t)\) [75], which have been used to represent the long-time dependence of the MSD.
The solution of Eq. (19), supplemented by the boundary condition that prevents the left and right boundary along each axis to exchange order, can be obtained by variable separation in the centroid and separation distance [76]. The resulting probability distribution for each axis is given by the product \(\mathcal {R}_{z}(\lambda _{z},t)Q_{z}(\mathcal {L}_{z},t)\), with controlling the dynamics of the boundary edge separation and that of the boundary centroid. The computation of the MSD in this case shows that the long-time dependence is controlled only by \({\int ^{t}_{0}}\mathrm {d}s\,\varphi (s)\), which results from the exclusion statistics of the territory boundary. This long time dependence provides the means to link quantitatively the outcome of the stochastic simulations with the microscopic mechanism of mark avoidance, in particular the relation between the value \(\mathcal {T}_{A}\) during which deposited marks remain active and the diffusion constant K (see Section “Applications to observations in the field”).
Comparison to an earlier approach
The model by Lewis, Moorcroft and Murray (LMM), successfully applied to movement data on wolves [63] and coyotes [149], represents the avoidance interaction between animal pairs by coupling the animal occupation probability of one individual to the distribution profile of the scent of the other. As a result, the movement bias (retreat) in the LMM model is due to a spatially extended interaction potential as compared to sharply peaked walls implemented in the TRW model.
The scent deposition is implemented differently in the two models: with a constant rate independent of the motion in the LMM model, and at regularly spaced intervals whenever an animal moves in the TRW model. This implies that the former model is more suited to model animals that leave consecutive marks further apart as they move quicker over the terrain, whereas the latter model is more suited to model animals that leave no gaps in the terrain between marks. Furthermore, over-marking (increase in marking rate when animals encounter foreign scent) is present so far only in the LMM model [150].
In the LMM model the presence of an attractive potential towards the den site [151] forces the animal occupation probability to reach a steady state. The TRW model, on the other hand, does not possess a steady state. Fluctuations in the boundary locations are always present, except for extremely large values of the active scent time. A comparison of the models in this regard is therefore not straightforward. However, the addition of a bias towards a burrow in the TRW model, e.g. to represent animals that display site fidelity [152], destroys the dynamic nature of the TRW model forcing the occupation probability of each individual to reach a steady state. One can then compare this steady state profile when no over-marking occurs in the LMM model. A further discussion can be found in [140].
In summary, although a microscopic description of the movement of the animals is present in both models, a representation of the discreteness of the events [153] of (avoidance) interaction are present only in the TRW model [153]. In the latter, tracking when and where scent is deposited allows one to define a territory as the set of locations visited by an individual within the time over which animals respond to the encounter of foreign scent. The difficulties in identifying the ever-changing locations of territory boundaries are thereby eliminated. The TRW model thus provides the long-sought operational definition of a territory in scent-marking animals.
Applications to observations in the field
Although the estimation of movement patterns from recordings of animal locations has a long history (see e.g. [96,154]), recent years have seen an explosion in the number and quality of field observations [62] due to the rapid development of cheap and easy to use tracking sensors and loggers. A rich platform has thereby been provided for empiricists and theoreticians to help each other answer fundamental questions in animal behaviour [155]. Inspired by the original studies on diffusive and persistent processes [156-159], various approaches that aim at extracting movement features and environmental drivers have emerged: change-point analysis [160], Brownian bridges [161,162], Hidden Markov models and state-space models [163-166], and others such as the partial sum approach [167]. A common feature of these studies is the ability to account for the spatial and temporal heterogeneity in the observations.
These heterogeneities are also of concern in studies on animal home ranges and territories since they affect the patterns of space use [36]. Movement attributes are more difficult to extract when animals roam in confined space. Part of this difficulty is associated with the fact that home range and territory boundaries are not insurmountable barriers for the animal.
where the potential U and V have a minimum, respectively, at x _{ c } and y _{ c }, i.e. the location of the den site. As this procedure relies upon reconstructing the long-time occupation probability of an individual, it requires independent observations of an animal’s positions. When animal fixes are gathered with sufficiently small sampling rate, the movement data can be fitted to reconstruct the spatial dependence of the animal occupation probability with a fit to Eq. (20) [109,110].
Although a parametric fit to \(P(x,y,t\rightarrow +\infty)\) may give a good estimate of the functional dependence of \(\left [U(x-x_{c})+V(y-y_{c})\right ]/D\), it is necessary to obtain an independent measurement of D. In other words, to disentangle the randomness of the movement from the determinism inherent in the animal drift towards the den site, it is necessary to quantify the stochasticity of the animal trajectories. To perform this task, a useful quantity to identify the statistical features of a movement process from recordings of animal locations is the time dependence of the squared displacement. It has been employed in many contexts including, for instance, for the study of exciton transport in organic crystals [168,169]. More recently it has become the subject of a variety of investigations in animal movement studies [71,170-174] since it provides a synthetic measure of the stochastic features of the individual trajectories. By averaging multiple observations of different individuals [109,110] or by performing a time-window average (see e.g. [175,176]) over a single animal trajectory when tracking occurs over a sufficiently long time, information about the MSD can be obtained.
Aggregate measurements from multiple trajectories
Studying the time-dependence of the MSD at sufficiently short time, such that the animal displacements are not affected by the home range boundaries, also allows the detection of non-diffusive features of the trajectories when present. By considering a general time-dependence of the form 〈Δ x ^{2}〉∼t ^{2H } where Δ x indicates the displacement in 2d of an individual from the initial location x _{0} and H is the so-called Hurst exponent [177], one can extract the anomalous exponent 2H and its associated fractal dimension δ of the animal trajectory through the relation δ=2−H [178]. It is possible to do that by extending the analytical methods presented in Section “Interactions with the environment: the emergence of home ranges” to compute the q-th moment of an animal occupation probability.
and \(\mathcal {I}(\textbf {x})=\mathcal {P}_{\textbf {x}_{0}}(\textbf {x},0)\). The convenience of Eq. (23) lies in its flexibility to capture anomalous statistical features of the animal walk through D(t) and the expected steady state solution (20).
The effects of spatially limited observations on the estimation of home range size mentioned already in Section “Interactions with the environment: the emergence of home ranges” also apply here when estimating the Hurst exponent H(q) with the integration limits in Eq. (22) becoming finite. The moments with high q are heavily affected by the presence of a limited sampled area as they contain spatial information about the tail of the probability distribution. In certain cases, e.g. with square observation windows and complete uncertainty about the initial position of the individuals, i.e. when \(\mathcal {I}(\textbf {x}_{0})\) is uniform, the integrals can be computed explicitly. A simple expression for H(q) then emerges. An application of this analytic procedure to mark-recapture experiments was carried out [147] with a population of Peromiscus maniculatus in New Mexico indicating a high degree of correlation in the displacement of the individuals approaching the ballistic limit, possibly due to habitual movement within their home ranges along well defined paths to reduce predation risks.
While the short-time dependence of 〈|Δ x|^{ q }〉(t) allows the characterisation of the statistical features of the movement, the long-time dependence gives information about the size of the home range. In an animal population with limited variability in the size of individual home ranges and knowledge about their locations, the MSD expression (22) at \(t\rightarrow +\infty \) provides information about home range size. If no information about home range locations is available, one proceeds as in Section “Interactions with the environment: the emergence of home ranges” using Eq. (8).
Home range and diffusion constant parameters extracted from mark-recapture observations
Animal species | Geographic region | |
---|---|---|
Zygodontomys brevicauda | Azuero Peninsula, Panama | |
Peromyscus maniculatus | New Mexico, USA | |
Potential shape | Home range | Diffusion constant |
length (m) | (m ^{ 2 } /day) | |
Box shape | 70\({\!~\!}^{+50}_{-20}\) | 200 ± 50 |
Parabola | 100 ± 25 | 475 ± 50 |
Measurements from single trajectories
There has been a realisation in recent times that heterogeneities in the characteristics of the individuals may give rise to spurious interpretation of anomalous movement [179-181]. This has accelerated the development of tools that extract statistical features from individual trajectories, e.g. wavelet analysis [182,183] For animals moving in unbounded domains, recent approaches include the mean-maximal excursion method for subdiffusive processes [184] or the use of Brownian functional maximum likelihood estimators [185,186] for accurate quantification of the diffusion constant for Brownian processes.
For movement in confined space, a promising approach is the one developed to characterize animal movement in circular arenas [187]. With the help of extensive stochastic simulations, it is possible to construct an approximate analytic expression for the MSD of the movement of a (positively) correlated random walker in confined space. Fit to observations provides an effective persistence \(\xi =-\mathcal {L}/[\!R\ln (\langle \cos (\theta)\rangle)]\) of the animal, where \(\langle \cos (\theta)\rangle \) is the mean of the turning angle distribution in the absence of any reflecting barrier, is the mean of the step length distribution and R is the radius of the arena. The persistence reduces to ξ=0 with a uniform turning angle distribution, which corresponds to Brownian motion, progressively increasing as the distribution becomes more peaked around zero. The ballistic motion limit is reached when \(\xi \rightarrow +\infty \).
Application of this procedure to a laboratory experiment with rats searching for food pellets appearing at random locations has shown that individual animals move with a directional persistence that minimises the coverage time, i.e. the average time it takes to visit the entire arena [187]. Given its generality and the easy applicability resulting from the use of analytic expressions, this methodology promises to be a useful benchmark to study and interpret foraging processes within home ranges.
Characterization of movement and active scent-time in territorial animals
Although studies to characterise the movement processes in scent-marking territorial animals abound, only a very small number attempt to extract, simultaneously, information about movement and interaction of the individuals. This is the case of the TRW model presented in Section “Mutual interactions leading to territoriality: the effects of time scale disparity” and applied to location data of red foxes (Vulpes vulpes) [188]. Movement data can be fitted to an approximate analytic expression for P(x,y,t|v,T,γ,K,L), the occupation probability for the animal to be at coordinates (x,y) relative to its home range center at time t. The parameters v, T, γ, K, and L, represent, respectively, the average animal speed, the average time an animal moves before turning, the average rate for a territory size to relax toward the inverse of the population density, the territory border diffusion constant, and the average territory width.
The exponential dependence of the territory border diffusion constant as a function of the active scent time displayed in Figure 5 can be understood with a simple reasoning based on a first-passage calculation in 1d. Focusing on an animal starting at a boundary location x _{0}, say the right one, in a territory of size equal to L, the probability for the territory boundaries not to move requires the animal to move from x _{0} to the left boundary and return to x _{0} by time \(t=\mathcal {T}_{A}\). The diffusion constant of the territory border is thus proportional to the probability for the animal not having moved between its edges. Then \(K\propto 1- \int _{0}^{\mathcal {T}_{A}}\mathrm {d}s\mathcal {F}(s)\), where \(\mathcal {F}(t)\) is the first-passage probability to start at x _{0} and reach the left boundary, and subsequently return from the left boundary to x _{0}. The first-passage probability from x _{0} to the left boundary, and similarly its return to x _{0} from the left boundary, is proportional to \(e^{-\pi ^{2} D t/4L^{2}}\) [193] where D is the diffusion constant of the animal. From this, one realises that \(K\propto e^{-\pi ^{2} D\mathcal {T}_{A} /4L^{2}}\). An exponential dependence on \(\mathcal {T}_{A}\) thus results.
Conclusions
The study of animal dynamics and animal interactions is an open subject teeming with activity. Much remains to be understood and much is being done. Below we mention some avenues along which we expect, at least hope, progress to occur in the near future.
An important aspect, not considered here, has to do with the effects of environmental spatial heterogeneities on animal movement and interaction processes. Whereas we have focused in the present article on what we have termed dynamic disorder, such heterogeneities correspond to static disorder. The home range models presented in Section “Interactions with the environment: the emergence of home ranges” possess some ability to include spatial heterogeneities not only through the shape of the confining potential, but also through the distribution of home range centres. Better choices for this distribution can be obtained from more detailed landscape models present in the literature, e.g. [42,43]. Introduction of landscape features has been attempted by linking population spatial distribution to animal spatial memory and landscape persistence [194], as well as to prey distribution and terrain steepness [129,149]. All these approaches, however, lack one fundamental aspect, the coupling of the dynamics of the environment with that of the movement and interaction of the individuals. When the spatial heterogeneity is due to a distribution of resources that gets depleted, a detailed study of the resource-animal system becomes necessary.
Inclusion of resource dynamics on the territorial random walk model might be key to answer many of the unresolved issues on the dependence of territoriality and food availability in scent-marking species. Despite the general hypothesis of the inverted U-shaped relationship between territorial behaviour and food availability [195], an interesting long-term study on the Iberian lynx [196] showed that territorial behaviour was unaffected by prey abundance (wild rabbit). Apparently the unpredictability of rabbit abundance makes it more convenient for lynx to maintain exclusive core areas (territories) limiting the number of contacts with other conspecifics [196]. Such behavioural patterns are also seen for other carnivores [197,198]. A detailed modelling of lynx foraging behaviour in a territorial random walk model might provide a mechanistic explanation linking contact rates to search strategies.
Another aspect not included in the approaches that we have described is a formalism capable of accounting for learning abilities and spatial memory. Attempts along this line have been made by Stamps and Krishnan [199,200]. They have analysed spatially implicit models in which individuals learn about competitive abilities of the neighbouring animals via past successes or failures in agonistic encounters [8]. Incorporating this type of learning in the spatially explicit models presented here promises to be a fruitful direction to test ideas on spatial memory [201]. Spatial games of cooperation and detection have also been well studied [202] and could be imported in more detailed representation of avoidance dynamics.
More generally, cognitive processes represent a research avenue with distinct potential. These processes clearly have a role in animal foraging and the formation of movement patterns. Memory, and learned or evolutionarily acquired expectations about landscape attributes, are used by animals to infer the current state of areas not previously visited. It is believed that this is done on the basis of information remembered from previous visits to neighbouring locations [203]. Recollection of a set of favourable or more profitable locations in the habitat has also been shown to be sufficient. Work has been done on models of home range formation in which a single individual displays both an avoidance response to recently visited resource patches, and an attractive response toward resource patches that have been visited sometime in the past [204]. These and other features of the cognitive skills of an animal have only just started to be incorporated in mechanistic models of movement (see e.g. [205]). It is our expectation that they will acquire a prominent role as home range and territory models begin to treat in detail energy costs of locomotion and foraging strategy.
Advances in this direction are expected with improved representation of animal decision-making. An animal searching for food would in fact make decisions based not only on its internal state and sensory inputs, but also on past knowledge and experience, and possible future outcomes. This implies that speed and direction of movement change continuously depending on past, present and expected circumstances. Realistic representation of these decision making processes in a population of interacting animals might hold the key for an improved understanding of the emergence spacing patterns.
Exploration of new regions not visited previously and exploitation of regions already familiar from earlier visits point in different directions. Accordingly, there are tradeoffs responsible for at least two distinct types of territorial patroling observed in scent-marking species. A hinterland strategy [206], modelled in Section “Mutual interactions leading to territoriality: the effects of time scale disparity Mutual interactions leading to territoriality: the effects of time scale disparity”, ensures that various locations inside the territory are scented regularly, whereas a borderland strategy [207] consists of depositing marks only on the outer boundaries of a territory. Examples of the former can be found in red foxes [208], otters [209] and pine martens [210], whereas those of the latter occur in spotted hyaenas [211], meerkats [212] and badgers [213]. To support the idea that sparsely distributed resources may favour a hinterland strategy [214], one should attempt modifications of the original territorial random walk model that include foraging costs and active border patroling, the latter partially explored already in [75]. We hope that mathematical developments already available in other areas will be used for these issues. Examples are general studies on first-passage problems in confined domains [215], and specific studies on partial confinement [216] and escape problems in cellular domains [217].
Red foxes have provided an example of terrestrial animals in the analysis given in Section “Applications to observations in the field”. Although other vertebrate species, such as wolves [218], squirrels [219], and deer [220], are a testbed for the ideas and predictions on territorial defense presented in that analysis, invertebrate species could also exhibit related behaviour. We believe that well studied marine gastropods that exhibit territorial responses are worth exploring to verify certain predictions or to generate novel and unexplored hypotheses on animal spacing. The complex behaviour of the owl limpet Lottia gigantea [221,222], a marine gastropod mollusc, appears particularly suitable because contacts with other conspecifics result in avoidance behaviour [223]. As the decision to fight or flee is strongly influenced by recent agonistic successes or failures [224], L. gigantea would present an ideal candidate to study how past experiences affect spacing patterns.
The ability to detect mucus of other individuals and its use for territorial marking, as observed in other species [225], could also be explored. The small territory size and slow movements [226] allow one to conduct ecological experiments on marking strategies in a laboratory environment. This permits the identification of whether, and where, individuals interact. The relative ease of manipulation of the food sources and the substrate over which the mollusc moves suggests L. gigantea as a model system. With its help, one could study the effects of the environment and intruder pressure on the choices that animals make between maintaining strong social ties and sharing space with the neighbouring individuals [53,227]. The effects of conspecific interactions and resource abundance on this dichotomy has already been observed both in the field in the African golden-wing sunbird [228] and in laboratory experiments in pygmy sunfish [229].
Other examples of invertebrates to investigate, with focus on avoidance mechanisms, are several species in the taxonomic order Diptera, e.g. flies, mosquito and midges. In swarms, while these insects remain globally bound together within a certain volume around a physical marker, local interactions also occur as individuals appear to correlate their displacements with some of their neighbours. Evidence in that direction has been collected from swarming mosquitos, e.g. Anopheles gambiae [230] and different species of midges, e.g. Dasyhelea flavifrons and Cladotanytarsus atridorsum [231], and Chironomus riparius [232]. Although these insects move freely throughout the available volume, they form groups without apparent polarisation. Ideas on some form of short time alignment based on velocity and exposure angles of nearby individuals [233,234] together with a mechanism of avoidance of locations recently visited by other individuals may help to explain why despite the lack of collective order, insect swarms are strongly correlated over large length scales. The avoidance here clearly would not rely on scenting the space, since the air does not retain a memory of the passage of these insects. However, past locations visited by other individuals may be retained in the memory of each animal providing a mechanism of exclusion analogous to the one presented for scent-marking species. Testing these ideas of memory-induced avoidance should help develop territorial formation in other non-scenting mammals.
The coupling of animal confinement—man-made, e.g. enclosing fences, or inherent in the animal socioecology—with various types of interactions among the animals, attractive or repulsive, results in profound effects on the transmission of infection in the context of various diseases. As infection is transmitted upon contact or proximity between individuals, the degree of social cohesion of the population determines the direct or indirect animal encounter rate and ultimately the speed of spread of a pathogen. Contact events and disease transmission rates are thus fundamentally linked to the way animals move and respond to their neighbours. Control or management interventions to reduce the prevalence of an infection may become ineffective if the social structure of the animal population is being heavily disrupted, e.g. by culling procedures [235] or by a natural disease [188].
Modelling disease transmission between individuals segregated in different regions of space requires the relaxation of traditional assumptions of homogeneity and well-mixing and demands new theoretical tools capable of treating together the movement and interactions of the animals. While some work has appeared earlier [236,237], a powerful new technique that applies to any number of dimensions, has built-in confinement analysis wherever needed, predicts unexpected insights into epidemic spread, and is suited to the unification of model calculations (for low population densities), and kinetic approaches (for high population densities), has appeared recently [111,112]. We give only a skeletal description.
The concept behind the recent Kenkre-Sugaya treatment of infection transmission is to perform an exact model calculation for low animal densities by treating a single pair of animals represented as tethered random walkers moving diffusively on the terrain, extracting an infection rate in ways formally reminiscent of the Fermi Golden Rule prescription in quantum systems [238], and using the rate in a kinetic equation framework valid for denser systems. The model calculation is based on an early treatment of interacting walkers [239] combined with the Smoluchowski equation description for confinement [128] produced by home ranges. The calculation results in some expected consequences but in some surprising phenomena as well, and yields an infection rate determined by initial conditions as well as the system dynamics. The rate is then fed into a kinetic equation framework similar to that of [99] but augmented to include confinement due to home ranges [105,107,112]. A comprehensive theoretical scheme is thus available and work is in progress both for further development of the scheme and for its application to zoonotic diseases of various kinds including bovine tuberculosis and plague.
Declarations
Acknowledgements
We have learnt much over the years from numerous collaborators and colleagues. LG wishes to thank in particular Stephen Harris for his biological insights about territorial systems. LG also acknowledges the support of various funding sources: an RDF World University Network grant on ‘The impact of climate change on the socio-ecology of animals: effects on epidemic disease spread and species invasion’, the Engineering and Physical Sciences Research Council (EPSRC) grant EP/I013717/1 and the Medical Research Council grant MR/J002097/1. VMK acknowledges in particular his collaborations with Bob Parmenter and the late Terry Yates, and the financial support in different recent periods by the National Science Foundation, the Los Alamos National Laboratory, the Mexico-USA Foundation, DARPA, the Howard Hughes Medical Institute, and the PIBBS interdisciplinary program of the University of New Mexico. One of us (VMK) thanks Matthew Chase, Anastasia Ierides and Satomi Sugaya for comments on the ms.
Authors’ Affiliations
References
- Altum JBT: Der Vogel und Sein Leben, Münster, Germany: Niemann; 1875.Google Scholar
- Mayr E: Bernard Altum and the territory theory. Proc Linn Soc NY1935, 45(46):24–38.Google Scholar
- Moffat CB: The spring rivalry of birds. Irish Nat1903, 12:152–166.Google Scholar
- Howard HE: Territory in Bird Life, New York: Dutton; 1920.View ArticleGoogle Scholar
- Carpenter CR: A field study of the behavior and social relations of howling monkeys ( Alouatta palliata ). Comp Psychol Monogr1934, 10(2):1–168.Google Scholar
- Nice MM: The role of territory in bird life. Am Nat1941, 26(3):441–487.View ArticleGoogle Scholar
- Hinde RA: The biological significance of the territory of birds. Ibis1956, 98:340–369.View ArticleGoogle Scholar
- Brown JL, Orians GH: Spacing patterns in mobile animals. Ann Rev Ecol Syst1970, 1:239–262.View ArticleGoogle Scholar
- Wilson EO: Sociobiology: the New Synthesis: 25th edn, Cambridge: Harvard Uni. Press; 2000.Google Scholar
- Davies NB, Houston AI: Territory economics. In Behavioural Ecology: an Evolutionary Approach: 2nd edn. Edited by Krebs JR, Davies NB. Blackwell Sci.; 1984:148–169.Google Scholar
- Newton I: Experiments on the limitation of bird numbers by territorial behaviour. Biol Rev1992, 67:129–173.View ArticleGoogle Scholar
- Stamps JA: Territorial behavior: testing the assumptions. Adv Study Behav1994, 23:173–232.View ArticleGoogle Scholar
- Gordon DM: The population consequences of territorial behavior. Trends Ecol Evol1997, 12:63–66.View ArticleGoogle Scholar
- Adams ES: Approaches to the study of territory size and shape. Ann Rev Ecol Syst2001, 32:277–303.View ArticleGoogle Scholar
- Börger L, Dalziel BD, Fryxell JM: Are there general mechanisms of animal home range behaviour? A review and prospects for future research. Ecol Lett2008, 11:637–650.View ArticleGoogle Scholar
- Powell RA, Mitchell MS: What is a home range? J Mammal2012, 93(4):948–958.View ArticleGoogle Scholar
- Krebs JR, Davies NB: An Introduction to Behavioral Ecology, Oxford: Blackwell Scientific Publications; 1987.Google Scholar
- Wolff JO: Why are female small mammals territorial? Oikos1993, 68:364–370.View ArticleGoogle Scholar
- Wolff JO, Peterson JA: An offspring-defense hypothesis for territoriality in female mammals. Ethol Ecol Evol1998, 10:227–239.View ArticleGoogle Scholar
- Doncaster CP, Woodroffe R: Den site can determine shape and size of badger territories: implications for group-living. Oikos1993, 66:88–93.View ArticleGoogle Scholar
- Stamps JA: Motor learning and the value of familiar space. Am Nat1995, 146:41–58.View ArticleGoogle Scholar
- Hyman J, Hughes M, Searcy WA, Nowicki S: Individual variation in the strength of territory defense in male song sparrows: correlates of age, territory tenure, and neighbor aggressiveness. Behaviour2004, 141:15–27.View ArticleGoogle Scholar
- Burt WH: Territoriality and home range concepts as applied to mammals. J Mammal1943, 24:346–352.View ArticleGoogle Scholar
- Maher CA, Lott DF: Definitions of territoriality used in the study of variation in vertebrate spacing systems. Anim Behav1995, 49:1581–1597.View ArticleGoogle Scholar
- Mohr CO: Table of equivalent populations of North American mammals. Am Midland Nat1947, 37:223–249.View ArticleGoogle Scholar
- Randon-Furling J, Majumdar SN, Comtet A: Convex hull of N planar brownian motions: exact results and an application to ecology. Phys Rev Lett2009, 103:140602.View ArticleGoogle Scholar
- Majumdar SN, Comtet A, Randon-Furling J: Random convex hulls and extreme value statistics. J Stat Phys2010, 138:955–1009.View ArticleGoogle Scholar
- Benhamou S, Cornelis D: Incorporating movement behavior and barriers to improve kernel home range space use estimates. J Wildlife Manage2010, 74:1353–1360.View ArticleGoogle Scholar
- Downs JA, Horner MW, Tucker AD: Time-geographic density estimation for home range analysis. Annals GIS2011, 17(3):163–171.View ArticleGoogle Scholar
- Long JA, Nelson TA: Time geography and wildlife home range delineation. J Wildlife Manage2012, 76(2):407–413.View ArticleGoogle Scholar
- Steiniger S, Hunter AJS: A scaled line-based kernel density estimator for the retrieval of utilization distributions and home ranges from gps movement tracks. Ecol Inform2013, 13:1–8.View ArticleGoogle Scholar
- Lyons AJ, Turner WC, Getz WM: Home range plus: a space-time characterization of movement over real landscapes. Move Ecol2013, 1(2):14.Google Scholar
- Börger L, Fraconi N, Michele GD, Gatnz A, Meschi F, Manica A, Lovari S, Coulson T: Effects of sampling regime on the mean and variance of home range size estimates. J Anim Ecol2006, 75(6):1393–1405.View ArticleGoogle Scholar
- Fieberg J, Börger L: Could you please phrase “home range” as a question? J Mammal2012, 93(4):890–902.View ArticleGoogle Scholar
- Harris S, Cresswell WJ, Forde PG, Trewhella WJ, Woollard T, Wray S: Home-range analysis using radio-tracking data: a review of problems and techniques particularly as applied to the study of mammals. Mammal Rev1990, 20:97–123.View ArticleGoogle Scholar
- Powell RA: Animal home ranges and territories and home range estimators. In Research Techniques in Animal Ecology: Controversies and Consequences. Edited by Boitani L, Fuller TK. New York: Columbia University Press; 2000:65–110.Google Scholar
- Abramson G, Kupermannd MN, Morales JM, Miller JC: Space use by foragers consuming renewable resources. Eur Phys J B2014, 87:100.View ArticleGoogle Scholar
- Moorcroft PR: Mechanistic approaches to understanding and predicting mammalian space use: recent advances, future directions. J Mammal2012, 93(4):903–916.View ArticleGoogle Scholar
- Weaire D, Rivier N: Soap, cells and statistics—random patterns in two dimensions. Contemp Phys1984, 25:59–99.View ArticleGoogle Scholar
- Forman RTT, Gordon M: Landscape Ecology, New York: John Wiley & Sons; 1986.Google Scholar
- With KA, King AW: The use and misuse of neutral landscape models in ecology. Oikos1997, 79:219–229.View ArticleGoogle Scholar
- Gustafson EJ: Quantifying landscape spatial pattern: what is the state of the art? Ecosystems1998, 1:143–156.View ArticleGoogle Scholar
- Burel F, Baudry J: Landscape Ecology: Concepts, Methods, and Applications, Enfield: Science Publishers; 2003.Google Scholar
- Gaucherel C, Fleury D, Auclair A, Dreyfus P: Neutral models for patchy landscapes. Ecol Model2006, 197:159–170.View ArticleGoogle Scholar
- Turner MG, Romme WH, Gardner RH, O’Neill OV, Kratz TK: A revised concept of landscape equilibrium: disturbance and stability on scaled. Landscape Ecol1993, 8:213–227.View ArticleGoogle Scholar
- Zimmerman JK, Comita LS, Thompson J, Uriarte M, Brokaw N: Patch dynamics and community metastability of a subtropical forest: compound effects of natural disturbance and human land use. Landscape Ecol2010, 25:1099–1111.View ArticleGoogle Scholar
- Dick JM, Shock EL: A metastable equilibrium model for the relative abundances of microbial phyla in a hot spring. PLoS ONE2013, 8(9):72395.View ArticleGoogle Scholar
- Shintani H, Tanaka H: Frustration on the way to crystallization in glass. Nat Phys2006, 2:200–206.View ArticleGoogle Scholar
- Kenkre VM, Dunlap DH: Charge transport in molecular solids: dynamic and static disorder. Phil Mag1992, 65:831–841.View ArticleGoogle Scholar
- Grassé P-P: La reconstruction du nid et les coordinations interindividuelles chez Bellicositermes natalensis et Cubitermes sp. La théorie de la stigmergie: essai dÕinterprétation du comportement des termites constructeurs. Insec Soc1959, 6(1):41–83.View ArticleGoogle Scholar
- Theraulaz G, Bonabeau E: A brief history of stigmergy. Artif Life1999, 5(2):97–116.View ArticleGoogle Scholar
- Holland O, Melhuish C: Stigmergy, self-organisation and sorting in collective robotics. Artif Life1999, 5(2):173–202.View ArticleGoogle Scholar
- Giuggioli L, Potts JR, Rubenstein DI, Levin SA: Stigmergy, collective actions, and animal social spacing. Proc Natl Acad Sci USA2013, 110(42):16904–16909.View ArticleGoogle Scholar
- Dunlap DH, Kenkre VM: Disordered polaron transport: a theoretical description of the motion of photoinjected charges in molecularly doped polymers. Chem Phys1993, 178:67–75.View ArticleGoogle Scholar
- Kenkre VM: Nonlinear dynamics of polarons. In Polarons and Applications. Edited by Lakhno VD. Leeds: Wiley & Sons; 1994:383–403.Google Scholar
- Kenkre VM: What do polarons owe to their harmonic origins? Physica D1998, 113:233–241.View ArticleGoogle Scholar
- Nathan RM: An emerging movement ecology paradigm. Proc Natl Acad Sci USA2008, 105:19050–19051.View ArticleGoogle Scholar
- Nathan RM, Giuggioli L: A milestone for movement ecology research. Move Ecol2013, 1(1):3.View ArticleGoogle Scholar
- Codling EA, Planck MJ, Benhamou S: Random walk models in biology. J Roy Soc Interface2008, 95(5):813–834.View ArticleGoogle Scholar
- Viswanathan GM, da Luz MGE, Raposo EP, Stanley HE: The Physics of Foraging: an Introduction to Random Searches and Biological Encounters, Cambridge: Cambridge Univ. Press; 2011.View ArticleGoogle Scholar
- Cooke SJ, Hinch SG, Wikelski M, Andrews RD, Kuchel LJ, Wolcott TG, Butler PJ: Biotelemetry: a mechanistic approach to ecology. Trends Ecol Evol2004, 19:334–343.View ArticleGoogle Scholar
- Cagnacci F, Boitani L, Powell RA, Boyce MS: Animal ecology meets gps-based radiotelemetry: a perfect storm of opportunities and challenges. Proc Roy Soc B2010, 365:2157–2162.Google Scholar
- Lewis MA, Murray JD: Modelling territoriality and wolf-deer interactions. Ecology1993, 366:738–740.Google Scholar
- Moorcroft PR, Lewis MA: Mechanistic Home Range Analysis, Princeton: Princeton University Press; 2006.Google Scholar
- Chandrasekar S: Stochastic problems in physics and astronomy. Rev Mod Phys1943, 15:1–89.View ArticleGoogle Scholar
- Reichl LE: A Modern Course in Statistical Physics, 3rd edn, Weinheim: Wiley & Sons; 2009.Google Scholar
- Levin SA: The problem of pattern and scale in ecology. Ecology1992, 73(6):1943–1967.View ArticleGoogle Scholar
- Kenkre VM, Montroll EW, Shlesinger MF: Generalized master equations for continuous-time random walks. J Stat Phys1973, 9:45–50.View ArticleGoogle Scholar
- Kenkre VM: The generalized master equation and its applications. In Statistical Mechanics and Statistical Methods in Theory and Application. Edited by Landman U. New York: Plenum; 1977:441–461.View ArticleGoogle Scholar
- Hilfer R, Anton L: Fractional master equations and fractal time random walks. Phys Rev E1995, 51:848–851.View ArticleGoogle Scholar
- Giuggioli L, Sevilla FJ, Kenkre VM: A generalised master equation approach to modelling anomalous transport in animal movement. J Phys A: Math Theor2009, 42:434004.View ArticleGoogle Scholar
- Kenkre VM: Memory formalism, nonlinear techniques, and kinetic equation approaches. In AIP Conference Proceedings on Modern Challenges in Statistical Mechanics: Patterns, Noise, and the Interplay of Nonlinearity and Complexity, vol. 658. Edited by Kenkre VM, Lindenberg K. Melville: American Institute of Physics; 2003:63–103.View ArticleGoogle Scholar
- Kenkre VM, Sevilla FJ: Thoughts about anomalous diffusion: time-dependent coefficients versus memory functions. In Contributions to Mathematical Physics: a Tribute to Gerard G. Emch. Edited by Ali TS, Sinha KB. New Delhi: Hindustani Book Agency; 2007:147–160.Google Scholar
- von Smoluchowski M: Über Brownsche molekularbewegung unter einwirkung äu β erer kräfte und deren zusammenhang mit der verallgemeinerten diffusionsgleichung. Ann Phys1916, 353(24):1103–1112.View ArticleGoogle Scholar
- Giuggioli L, Potts JR, Harris S: Animal interactions and the emergence of territoriality. PLoS Comput Biol2011, 7(3):1002008.View ArticleGoogle Scholar
- Giuggioli L, Potts JR, Harris S: Brownian walkers within subdiffusing territorial boundaries. Phys Rev E2011, 83:061138.View ArticleGoogle Scholar
- Harris TE: Diffusion with “collisions” between particles. J Appl Probab1965, 2:323–338.View ArticleGoogle Scholar
- Rödenbeck C, Kärger J, Hahn K: Calculating exact propagators in single-file systems via the reflection principle. Phys Rev E1998, 57(4):4382–4397.View ArticleGoogle Scholar
- Kumar D: Diffusion of interacting particles in one dimension. Phys Rev E2008, 78:021133.View ArticleGoogle Scholar
- Aslangul C: Diffusion of two repulsive particles in a one-dimensional lattice. J Phys A: Math Gen1999, 32(9):3993–4003.View ArticleGoogle Scholar
- Ambjörnsson T, Silbey RJ: Diffusion of two particles with a finite interaction potential in one dimension. J Chem Phys2008, 129(16):165103.View ArticleGoogle Scholar
- Lizana L, Ambjörnsson T: Single-file diffusion in a box. Phys Rev Lett2008, 100:200601.View ArticleGoogle Scholar
- Barkai E, Silbey R: Theory of single file diffusion in a force field. Phys Rev Lett2009, 102:050602.View ArticleGoogle Scholar
- Barkai E, Silbey R: Diffusion of tagged particle in an exclusion process. Phys Rev E2010, 81:041129.View ArticleGoogle Scholar
- Potts JR, Harris S, Giuggioli L: An anti-symmetric exclusion process for two particles on an infinite 1d lattice. J Phys A: Math Theor2011, 44:485003.View ArticleGoogle Scholar
- Mikhailov AS, Zanette DH: Noise-induced breakdown of coherent collective motion in swarms. Phys Rev E1999, 60:4571–4575.View ArticleGoogle Scholar
- Erdmann U, Ebeling W, Mikhailov AS: Trail following in ants: individual properties determine population behaviour. Phys Rev E2005, 71:051904.View ArticleGoogle Scholar
- D’Orsogna MR, Chuang YL, Bertozzi AL, Chayes LS: Self-propelled particles with soft-core interactions: patterns, stability, and collapse. Phys Rev Lett2006, 96:104302.View ArticleGoogle Scholar
- Aldana M, Dossetti V, Huepe C, Kenkre VM, Larralde H: Phase transitions in systems of self-propelled agents and related network models. Phys Rev Lett2007, 98:095702.View ArticleGoogle Scholar
- Dossetti V, Sevilla FJ, Kenkre VM: Phase transitions induced by complex nonlinear noise in a system of self-propelled agents. Phys Rev E2009, 79:051115.View ArticleGoogle Scholar
- Astwood A, Raghib M, Kenkre VM: Orientational Model for Flocking: Memory Description of Collective Motion. University of New Mexico preprint; see also A. Astwood, Ph. D. thesis, 2013.Google Scholar
- Vicsek T, Zafeiris A: Collective motion. Phys Rep2012, 517:71–140.View ArticleGoogle Scholar
- Strandburg-Peshkin A, Twomey CR, Bode NW, Kao AB, Katz Y, Ioannou CC, Rosenthal SB, Torney CJ, Wu H, Levin SA, Couzin ID: Visual sensory networks and effective information transfer in animal groups. Curr Biol2013, 23(17):709–711.View ArticleGoogle Scholar
- Anderson RM, May RM: Population biology of infectious diseases: Part i. Nature1979, 280:361–367.View ArticleGoogle Scholar
- May RM, Anderson RM: Population biology of infectious diseases: Part ii. Nature1979, 280(5722):455–461.View ArticleGoogle Scholar
- Okubo A, Levin SA: Diffusion and Ecological Problems: Modern Perspectives, 2nd edn, New York: Springer; 2001.View ArticleGoogle Scholar
- Hethcote HW: The mathematics of infectious diseases. SIAM Rev1995, 42:599–653.View ArticleGoogle Scholar
- Brauer F, Castillo-Chávez C: Mathematical Models in Population Biology and Epidemiology, New York: Springer; 2001.View ArticleGoogle Scholar
- Abramson G, Kenkre VM: Spatio-temporal patterns in Hantavirus infection. Phys Rev E2002, 66:011912.View ArticleGoogle Scholar
- Dickmann U, Law R, Metz JAJ: The Geometry of Ecological Interactions, Cambridge: Cambridge University Press; 2000.View ArticleGoogle Scholar
- Cantrell RS, Cosner C: Spatial Ecology Via Reaction-diffusion Equations, Chichester: Wiley & Sons; 2003.Google Scholar
- Kenkre VM: Results from variants of the Fisher equation in the study of epidemics and bacteria. Phys A2004, 342:242–248.View ArticleGoogle Scholar
- McKane AJ, Newman TJ: Stochastic models in population biology and their deterministic analogs. Phys Rev E2004, 70:041902.View ArticleGoogle Scholar
- Kenkre VM: Statistical mechanical considerations in the theory of the spread of the Hantavirus. Phys A2005, 356:121–126.View ArticleGoogle Scholar
- Kenkre VM, Giuggioli L, Abramson G, Camelo-Neto G: Theory of Hantavirus infection spread incorporating localized adult and itinerant juvenile mice. Eur Phys J B2007, 55:461–470.View ArticleGoogle Scholar
- Aguirre MA, Abramson G, Bishop AR, Kenkre VM: Simulations in the mathematical modeling of the spread of the Hantavirus. Phys Rev E2002, 66:041908.View ArticleGoogle Scholar
- MacInnis D, Abramson G, Kenkre VM: Effects of Confinement Potentials on Spatial Patterns of Infection in Hantavirus Refugia. University of New Mexico preprint; see also D. MacInnis, Ph. D. thesis, 2008.Google Scholar
- Abramson G, Giuggioli L, Parmenter RR, Kenkre VM: Quasi-one-dimensional waves in rodent populations in heterogeneous habitats: a consequence of elevational gradients on spatio-temporal dynamics. J Theor Biol2013, 319:96–101.View ArticleGoogle Scholar
- Giuggioli L, Abramson G, Kenkre VM, Suzán G, Marcé E, Yates TL: Diffusion and home range parameters from rodent population measurements in Panama. Bull Math Biol2005, 67(5):1135–1149.View ArticleGoogle Scholar
- Abramson G, Giuggioli L, Kenkre VM, Dragoo JW, Parmenter RR, Parmenter CA, Yates TL: Diffusion and home range parameters for rodents: Peromyscus maniculatus in New Mexico. Ecol Complex2006, 3:64–70.View ArticleGoogle Scholar
- Kenkre VM, Sugaya S: Theory of the Transmission of Infection in the Spread of Epidemics: Interacting Random Walkers with and Without Confinement; 2014. [http://arxiv.org/abs/1408.5430]
- Sugaya S, Kenkre VM: Transmission of Infection in Epidemics: Extension of the Theory to Dense Population With and Without Confinement: University of New Mexico preprint; 2014.Google Scholar
- Yates TL, Mills JN, Parmenter CA, Ksiazek TG, Parmenter RR, Castle JRV, Calisher CH, Nichol ST, Abbott KD, Young JC, Morrison ML, Beaty BJ, Dunnum JL, Baker RJ, Salazar-Bravo J, Peters CJ: The ecology and evolutionary history of an emergent disease: Hantavirus pulmonary syndrome. BioScience2002, 52(11):989–998.View ArticleGoogle Scholar
- Samia NI, Kausrud KL, Heesterbeek H, Ageyev V, Begon M, Chan KS, Stenseth NC: Dynamics of the plague-wildlife-human system in central Asia are controlled by two epidemiological thresholds. Proc Natl Acad Sci USA2011, 108(35):14527–14532.View ArticleGoogle Scholar
- Reijniers J, Davis S, Begon M, Heesterbeek JAP, Ageyev VS, Leirs H: A curve of thresholds governs plague epizootics in central Asia. Ecol Lett2012, 15:554–560.View ArticleGoogle Scholar
- Donnelly CA, Woodroffe R, Cox DR, Bourne FJ, Gettinby G, Fevre AML, McInerney JP, Morrison WI: Impact of localized badger culling on tuberculosis incidence in British cattle. Nature2003, 426(6968):834–837.View ArticleGoogle Scholar
- Donnelly CA, Woodroffe R, Cox DR, Bourne FJ, Cheeseman CL, Clifton-Hadley RS, Wei G, Gettinby G, Gilks P, Jenkins H, Johnston WT, Fevre AML, McInerney JP, Morrison MI: Positive and negative effects of widespread badger culling on tuberculosis in cattle. Nature2006, 439(7078):843–846.View ArticleGoogle Scholar
- Montroll EW, Weiss GH: Random walks on lattices II. J Math Phys1965, 6(2):167–181.View ArticleGoogle Scholar
- Montroll EW, West BJ: On an enriched collection of stochastic processes. In Studies in Statistical Mechanics: Vol VII. Fluctuation Phenomena. Edited by Montroll EW, Lebowitz JJ. Amsterdam: North Holland Publishing; 1979:61–175.View ArticleGoogle Scholar
- Penteriani V, Ferrer M, Delgado MM: Floater strategies and dynamics in birds, and their importance in conservation biology: towards an understanding of nonbreeders in avian populations. Anim Conserv2011, 14(3):233–241.View ArticleGoogle Scholar
- Abramson G, Kenkre VM, Yates TL, Parmenter RR: Traveling waves of infection in the Hantavirus epidemics. Ecol Complex2003, 65:519–534.Google Scholar
- Camelo-Neto G, Silva ATC, Giuggioli L, Kenkre VM: Effect of predators of juvenile rodents on the spread of the Hantavirus epidemic. Bull Math Biol2008, 70(1):179–188.View ArticleGoogle Scholar
- Stickel LF: Home range and travels. In Biology of Peromyscus (Rodentia), Special Publication No. 2. Edited by King JA. Stillwater: The American Society of Mammalogists; 1968:373–411.Google Scholar
- Parmenter RR, MacMahon JA: Factors determining the abundance and distribution of rodents in a shrub-steppe ecosystem: the role of shrubs. Oecologia1983, 59:145–156.View ArticleGoogle Scholar
- Wolff JO: Population regulation in mammals: an evolutionary perspective. J Anim Ecol1997, 66:1–13.View ArticleGoogle Scholar
- Giuggioli L, Abramson G, Kenkre VM, Parmenter RR, Yates TL: Theory of home range estimation from displacement measurements of animal populations. J Theor Biol2006, 240:126–135.View ArticleGoogle Scholar
- Risken H: The Fokker-Planck Equation: Methods of Solution and Applications, Berlin: Springer; 1984.View ArticleGoogle Scholar
- Spendier K, Sugaya S, Kenkre VM: Reaction-diffusion theory in the presence of an attractive harmonic potential. Phys Rev E2013, 88:062142.View ArticleGoogle Scholar
- Moorcroft PR, Lewis MA, Crabtree RL: Home range analysis using mechanistic home range model. Ecology1999, 80(5):1656–1665.View ArticleGoogle Scholar
- Lewis MA, Moorcroft P: Ess analysis of mechanistic models for territoriality: the value of scent marks in spatial resource partitioning. J Theor Biol2001, 210(4):449–461.View ArticleGoogle Scholar
- Hamelin F, Lewis MA: A differential game theoretical analysis of mech- anistic models for territoriality. J Math Biol2010, 61:665–694.View ArticleGoogle Scholar
- Deneubourg JL, Goss S: Collective patterns and decision-making. Ethol Ecol Evol1989, 1:295–311.View ArticleGoogle Scholar
- Deneubourg JL, Aron S, Goss S, Pasteels JM: The self-organizing exploratory pattern of Argentine ant. J Insect Behav1990, 2(3):159–168.View ArticleGoogle Scholar
- Davis B: Reinforced random walks. Probab Theory Rel1990, 84:203–229.View ArticleGoogle Scholar
- Lam L: Active walker models for complex systems. Chaos Soliton Fract1995, 6:267–285.View ArticleGoogle Scholar
- Schweitzer F, Lao K, Family F: Active random walkers simulate trunk trail formation by ants. BioSystems1997, 41:153–166.View ArticleGoogle Scholar
- Othmer HA, Stevens A: Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J Appl Math1997, 57(4):1044–1081.View ArticleGoogle Scholar
- Boyer D, Solis-Salas C: Random walks with preferential relocations to places visited in the past and their application to biology. Phys Rev Lett2014, 112:240601.View ArticleGoogle Scholar
- Gosling LM, Roberts SC: Scent-marking by male mammals: cheat-proof signals to competitors and mates. Adv Stud Behav2001, 30:169–217.View ArticleGoogle Scholar
- Potts JR, Harris S, Giuggioli L: Territorial dynamics and stable home range formation for central place foragers. PLoS ONE2012, 7(3):34033–1013710034033.View ArticleGoogle Scholar
- Kac M: On the notion of recurrence in discrete stochastic processes. Bull Am Math Soc1947, 53:1002–1010.View ArticleGoogle Scholar
- von Born M, Oppenheimer JR: Zur quantentheorie der molekeln. Ann Phys Leipzig1927, 389(20):457–484.View ArticleGoogle Scholar
- Levin SA: Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull Ent Soc Amer1969, 15:237–240.Google Scholar
- Hanski I, Poyry J, Pakkala T, Kuussaari M: Multiple equilibria in metapopulation dynamics. Nature1995, 377:618–621.View ArticleGoogle Scholar
- Kato T: On the adiabatic theorem of quantum mechanics. J Phys Soc Japan1950, 5:435–439.View ArticleGoogle Scholar
- Giuggioli L, Potts JR, Harris S: Predicting oscillatory dynamics in the movement of territorial animals. J Roy Soc Interface2012, 9(72):1529–1543.View ArticleGoogle Scholar
- Giuggioli L, Viswanathan GM, Kenkre VM, Parmenter RR, Yates TL: Effects of finite probing windows on the interpretation of the multifractal properties of random walks. Europhys Lett2007, 77:4004.View ArticleGoogle Scholar
- Kenkre VM: Master-equation theory of the effect of vibrational relaxation on intermolecular transfer of electronic excitation. Phys Rev A1977, 16:766–776.View ArticleGoogle Scholar
- Moorcroft PR, Lewis MA, Crabtree RL: Mechanistic home range models capture spatial patterns and dynamics of coyote territories in yellowstone. Proc Roy Soc B2006, 273:1651–1659.View ArticleGoogle Scholar
- Lewis MA, White KAJ, Moorcroft PR: Analysis of a model for wolf territories. J Math Biol1997, 35:749–774.View ArticleGoogle Scholar
- Holgate P: Random walk models for animal behavior. In Pennsylvania State Statistics - Statistical Ecology: Sampling and Modeling Biological Populations and Population Dynamics, vol. 2. Edited by Patil G, Pielou E, Walters W. University Park: Pennsylvania State University Press; 1971:1–12.Google Scholar
- Giuggioli L, Bartumeus F: Linking animal movement to site fidelity. J Math Biol2012, 64:647–656.View ArticleGoogle Scholar
- Durrett R, Levin SA: The importance of being discrete (and spatial). Theor Pop Biol1994, 46:363–394.View ArticleGoogle Scholar
- Turchin P: Quantitative Analysis of Movement: Measuring and Modelling Population Redistribution in Animals and Plants, Sunderland: Sinauer Associates; 1998.Google Scholar
- Giuggioli L, Bartumeus F: Animal movement, search strategies and behavioural ecology: a cross-disciplinary way forward. J Anim Ecol2010, 79:906–909.Google Scholar
- Skellam JG: Random dispersal in theoretical populations. Biometrika1951, 38:196–218.View ArticleGoogle Scholar
- Skellam JG: The formulation and interpretation of mathematical models of diffusionary processes in population biology. In The Mathematical Theory of the Dynamics of Biological Populations. Edited by Bartlett MS, Hiorns RW. New York: Academic Press; 1973:63–85.Google Scholar
- Okubo A: Diffusion and Ecological Problems: Mathematical Models vol. 10 Biomathematics, Berlin: Springer; 1980.Google Scholar
- Kareiva PM, Shigesada N: Analyzing insect movement as a correlated random walk. Oecologia1983, 56:234–238.View ArticleGoogle Scholar
- Gurarie E, Andrews RD, Laidre KL: A novel method for identifying behavioural changes in animal movement data. Ecol Lett2009, 12(5):395–408.View ArticleGoogle Scholar
- Horne JS, Garton EO, Krone SM, Lewis JS: Analyzing animal movements using Brownian bridges. Ecology2007, 88(9):2354–2363.View ArticleGoogle Scholar
- Kranstauber B, Safi K, Bartumeus F: Bivariate Gaussian bridges: directional factorization of diffusion in Brownian bridge models. Move Ecol2014, 2(5):10.Google Scholar
- Morales JM, Haydon DT, Frair J, Holsiner KE, Fryxell JM: Extracting more out of relocation data: building movement models as mixture of random walks. Ecology2004, 85:2436–2445.View ArticleGoogle Scholar
- Forester JD, Ives AR, Turner MG, Anderson DP, Fortin DP, Beyer HL, Smith DW, Boyce MS: State-space models link elk movement patterns to landscape characteristics in Yellowstone National Park. Ecol Monogr2007, 77:285–299.View ArticleGoogle Scholar
- Patterson TA, Thomas L, Wilcox C, Ovaskainen O, Matthiopolous J: State-space models of individual animal movement. Trends Ecol Evol2008, 23:87–94.View ArticleGoogle Scholar
- Langrock R, King R, Matthiopoulos J, Thomas L, Fortin D, Morales JM: Flexible and practical modeling of animal telemetry data: hidden markov models and extension. Ecology2012, 93:2336–2342.View ArticleGoogle Scholar
- Knell AS, Codling EA: Classifying area-restricted search (ARS) using a partial sum approach. Theor Ecol2012, 5:325–339.View ArticleGoogle Scholar
- Kenkre VM: Coupled wave-like and diffusive motion of excitons. Phys Lett A1974, 47:119–120.View ArticleGoogle Scholar
- Kenkre VM, Knox RS: Generalized-master-equation theory of excitation transfer. Phys Rev B1974, 9:5279–5290.View ArticleGoogle Scholar
- Benhamou S, Bovet P: Distinguishing between elementary orientation mechanisms by means of path analysis. Anim Behav1992, 43(3):371–377.View ArticleGoogle Scholar
- Benhamou S: On the expected net displacement of animals’ random movements. Ecol Model2004, 171(1):207–208.View ArticleGoogle Scholar
- Kölzsch A, Blasius B: Theoretical approaches to bird migration. Eur Phys J Special Topics2008, 157(1):191–208.View ArticleGoogle Scholar
- Nouvellet P, Bacon JP, Waxman D: Fundamental insights into the random movement of animals from a single distance-related statistics. Am Nat2009, 174(4):506–514.View ArticleGoogle Scholar
- Bunnefeld N, Börger L, van Moorter B, Rolandsen CM, Dettki H, Solberg EJ, Ericsson G: A model-driven approach to quantify migration patterns: individual, regional and yearly differences. J Anim Ecol2011, 80:466–476.View ArticleGoogle Scholar
- Grigolini P, Leddon D, Scafetta N: Diffusion entropy and waiting time statistics of hard-x-ray solar flares. Phys Rev E2002, 65(4):046203.View ArticleGoogle Scholar
- Scafetta N, Grigolini P: Scaling detection in time series: diffusion entropy analysis. Phys Rev E2002, 66(3):036130.View ArticleGoogle Scholar
- Hurst HE, Black RP, Simaika YM: Long-term Storage: an Experimental Study, London: Constable; 1965.Google Scholar
- West BJ, Bologna M, Grigolini P: Physics of Fractal Operators, New York: Springer; 2003.View ArticleGoogle Scholar
- Hapca S, Crawford JW, Young IM: Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level. J R Soc Interface2009, 6(30):111–122.View ArticleGoogle Scholar
- Plank MJ, Codling EA: Sampling rate and misidentification of lévy and non-lévy movement paths. Ecology2009, 90:3546–3553.View ArticleGoogle Scholar
- Petrovskii S, Mashanova A, Jansen VAA: Variation in individual walking behavior creates the impression of a Lévy flight. Proc Natl Acad Sci USA2011, 108(21):8704–8707.View ArticleGoogle Scholar
- Gaucherel C: Wavelet analysis to detect regime shifts in animal movement. Comput Ecol Softw2011, 1:69–85.Google Scholar
- Riotte-Lambert L, Benhamou S, Chamaillé-Jammes S: Periodicity analysis of movement recursions. J Theor Biol2013, 317:238–243.View ArticleGoogle Scholar
- Tejedor V, Schad M, Bénichou O, Voituriez R, Metzler R: Encounter distribution of two random walkers on a finite one-dimensional interval. J Phys A: Math Theor2011, 44:395005.View ArticleGoogle Scholar
- Boyer D, Dean DS: On the distribution of estimators of diffusion constants for Brownian motion. J Phys A: Math Theor2011, 44:335003.View ArticleGoogle Scholar
- Boyer D, Dean DS, Mejia-Monasterio C, Oshanin G: Optimal estimates of the diffusion coefficient of a single brownian trajectory. Phys Rev E2012, 85:031136.View ArticleGoogle Scholar
- Giuggioli L, Robles AH, Templey S, Zinyuk LE, Jones MW: Efficient Foraging Strategies in Confined Space: Coverage Time and the Effective Persistence of an Animal’s Walk: University of Bristol preprint; 2014.Google Scholar
- Potts JR, Harris S, Giuggioli L: Quantifying behavioral changes in territorial animals caused by sudden population declines. Am Nat2013, 182(3):E73—E82.View ArticleGoogle Scholar
- Millar RB: Statistics in Practice, in Maximum Likelihood Estimation and Inference: with Examples in R, SAS and ADMB, Chichester: John Wiley & Sons; 2011.View ArticleGoogle Scholar
- Nelder JA, Mead R: A simplex method for function minimization. Comput J1965, 7:308–313.View ArticleGoogle Scholar
- Lagarias JC, Reed JA, Wright MH, Wright PE: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J Optimiz1998, 9:112–147.View ArticleGoogle Scholar
- Wasserman L: All of Statistics: a Concise Course in Statistical Inference, 2nd edn, New York: Springer; 2004.View ArticleGoogle Scholar
- Redner S: A Guide to First-passage Processes, Cambridge: Cambridge University Press; 2001.View ArticleGoogle Scholar
- Berbert JM, Fagan WF: How the interplay between individual spatial memory and landscape persistence can generate population distribution patterns. Ecol Complex2012, 12:1–12.View ArticleGoogle Scholar
- Maher CR, Lott DF: A review of ecological determinants of territoriality within vertebrate species. Am Midl Nat2000, 143:1–29.View ArticleGoogle Scholar
- López-Bao JV, Rodríguez A, Delibes M, Fedriani JM, Calzada J, Ferreras P, Palomares F: Revisiting food-based models of territoriality in solitary predators. J Anim Ecol2014, 83(4):934–942.View ArticleGoogle Scholar
- Hemker TP, Lindzey FG, Ackerman BB: Population characteristics and movement patterns of cougars in southern Utah. J Wildlife Manage1984, 48:1275–1284.View ArticleGoogle Scholar
- Mattisson J, Persson J, Andrén H, Segerström P: Temporal and spatial interactions between an obligate predator, the Eurasian lynx ( Lynx lynx ): and a facultative scavenger, the wolverine ( Gulo gulo ). Can J Zoolog2011, 89:79–89.View ArticleGoogle Scholar
- Stamps JA, Krishnan VV: A learning-based model of territory establishment. Q Rev Biol1999, 74:291–318.View ArticleGoogle Scholar
- Stamps JA, Krishnan VV: How territorial animals compete for divisible space: a learning-based model with unequal competitors. Am Nat2001, 157:154–169.View ArticleGoogle Scholar
- Fagan WF, Lewis MA, Auger-Méthé M, Avgar T, Benhamou S, Breed G, LaDage L, Schlägel UE, Tang W, Papastamatiou YP, Forester J, Mueller T: Spatial memory and animal movement. Ecol Lett2013, 16:1316–1329.View ArticleGoogle Scholar
- Nowak MA, Sigmund K: Games on grids. In The Geometry of Ecological Interactions: Simplifying Spatial Complexity. Edited by Dieckmann U, Law R, Metz JAJ. New York: Cambridge University Press; 2000:135–150.View ArticleGoogle Scholar
- Fawcett TW, Fallenstein B, Higginson AD, Houston AI, Mallpress DEW, Trimmer PC, McNamara JM: Using first-passage time in the analysis of area-restricted search and habitat selection. Trends Cogn Sci2014, 18(3):153–161.View ArticleGoogle Scholar
- van Moorter B, Visscher D, Benhamou S, Börger L, Boyce MS, Gaillard J-M: Memory keeps you at home: a mechanistic model for home range emergence. Oikos2009, 118:641–652.View ArticleGoogle Scholar
- Fronhofer EA, Hovestadt T, Poethke HJ: From random walks to informed movement. Oikos2013, 122:857–866.View ArticleGoogle Scholar
- Gorman ML, Trowbridge BJ: The role of odors in the social lives of carnivores. In Carnivore Behavior, Ecology, and Evolution. Edited by Gittleman JL. Cornell: Cornell University Press; 1989:57–88.View ArticleGoogle Scholar
- Hutchings MR, Service KM, Harris S: Defecation and urination patterns of badgers Meles meles at low density in southwest England. Acta Theriol2001, 46:87–96.Google Scholar
- Macdonald DW: Patterns of scent marking with urine and faeces amongst carnivore communities. Symp Zool Soc Lond1980, 45:107–139.Google Scholar
- Kruuk H: Scent marking by otters ( Lutra lutra :) signalling the use of resources. Behav Ecol1992, 3:133–140.View ArticleGoogle Scholar
- Pulliainen E: Scent-marking in the pine marten ( Martes martes ) in Finnish forest Lapland in winter. Z Saugetierkd1982, 47:91–99.Google Scholar
- Mills MGL, Gorman ML: The scent-marking behaviour of the spotted hyaena Crocuta crocuta in the southern Kalahari. J Zool2009, 212(3):483–497.View ArticleGoogle Scholar
- Jordan NR, Cherry MI, Manser MB: Latrine distribution and patterns of use by wild meerkats: implications for territory and mate defence. Anim Behav2007, 73:613–622.View ArticleGoogle Scholar
- Kilshaw K, Newman C, Buesching C, Bunyan J, Macdonald DW: Coordinated latrine use by European badgers, Meles meles : potential consequences for territory defense. J Mammal2009, 90:1188–1198.View ArticleGoogle Scholar
- Hutchinson JMC, White PCL: Mustelid scent-marking in managed ecosystems: implications for population management. Mammal Rev2000, 30(3):157–169.Google Scholar
- Bénichou O, Voituriez R: From first-passage times of random walks in confinement to geometry-controlled kinetics. Phys Rep2014, 539:225–284.View ArticleGoogle Scholar
- Kenkre VM, Giuggioli L, Kalay Z: Molecular motion in cell membranes: analytic study of fence-hindered random walks. Phys Rev E2008, 77:051907.View ArticleGoogle Scholar
- Marten F, Tsaneva-Atanasova K, Giuggioli L: Bacterial secretion and the role of diffusive and subdiffusive first passage processes. PLoS ONE2012, 7(8):41421.View ArticleGoogle Scholar
- Sillero-Zubiri C, Macdonald DW: Scent-marking and territorial behavior of Ethiopian wolves Canis simensis. J Zool1998, 245:351–361.View ArticleGoogle Scholar
- Vache M, Ferron J, Gouat P: The ability of red squirrels ( Tamiasciurus hudsonicus ) to discriminate conspecific olfactory signatures. Can J Zoolog2001, 79:1296–1300.View ArticleGoogle Scholar
- Miller KV, Jemiolo B, Gassett JW, Jelinek I, Wiesler D, Novotny M: Putative chemical signals from white-tailed deer ( Odocoileus virginianus ): social and seasonal effects on urinary volatile excretion in males. J Chem Ecol1998, 24:673–683.View ArticleGoogle Scholar
- Stimpson J: Territorial behavior of the owl limpet, Lottia gigantea. Ecology1970, 51:113–118.View ArticleGoogle Scholar
- Stimpson J: The role of territory on the ecology of the intertidal limpet, Lottia gigantea. Ecology1973, 54:1020–1030.View ArticleGoogle Scholar
- Wright WG: Ritualized behavior in a territorial limpet. J Exp Mar Biol Ecol1982, 60:245–251.View ArticleGoogle Scholar
- Wright WG, Shanks AL: Previous experience determines territorial behavior in an archaeogastropod limpet. J Exp Mar Biol Ecol1993, 166:217–229.View ArticleGoogle Scholar
- Bretz DD, Dimock RV Jr: Behaviorally important characteristics of the mucous trail of the marine gastropod ilyanassa obsoleta (say). J Exp Mar Biol Ecol1983, 71:181–191.View ArticleGoogle Scholar
- Shanks AL: Previous agonistic experience determines both foraging behavior and territoriality in the limpet Lottia gigantea (soweby). Behav Ecol2002, 13:467–471.View ArticleGoogle Scholar
- Waser PM: Sociality or territorial defense? The influence of resource renewal. Behav Ecol Sociobiol1981, 8:231–237.View ArticleGoogle Scholar
- Pyke GH: The economics of territory size and time budget in the golden-winged sunbird. Am Nat1979, 114:131–145.View ArticleGoogle Scholar
- Rubenstein DI: Individual variation and competition in the everglades pygmy sunfish. J Anim Ecol1981, 50:337–350.View ArticleGoogle Scholar
- Butail S, Manoukis N, Diallo M, Ribeiro JM, Lehmann T, Paley DA: Reconstructing the flight kinematics of swarming and mating in wild mosquitoes. J Roy Soc Interface2012, 9:2624–2638.View ArticleGoogle Scholar
- Attanasi A, Cavagna A, Castello LD, Giardina I, Melillo S, Parisi L, Pohl O, Rossaro B, Shen E, Silvestri E, Viale M: Collective behaviour without collective order in wild swarms of midges2014. arXiv preprint arXiv:1307.5631.Google Scholar
- Kelley DH, Ouellette NT: Emergent dynamics of laboratory insect swarms. Sci Rep2013, 3:1073.View ArticleGoogle Scholar
- Giuggioli L, McKetterick TJ, Hoderied M: Delayed Response and Biosonar Perception Explain Movement Coordination in Trawling Bats; 2014.Google Scholar
- McKetterick TJ, Giuggioli L: Exact Dynamics of Stochastic Linear Delayed Systems: Application to Spatio-Temporal Coordination of Co-moving Agents: University of Bristol preprint; 2014.Google Scholar
- McDonald RA, Delahay RJ, Carter SP, Smith GC, Cheesman CL: Perturbing implications of wildlife ecology for disease control. Trends Ecol Evol2007, 23(2):53–56.View ArticleGoogle Scholar
- Giuggioli L, Pérez-Becker S, Sanders DP: Encounter times in overlapping domains: application to epidemic spread in a population of territorial animals. Phys Rev Lett2013, 110:058103.View ArticleGoogle Scholar
- Dumonteil E, Majumdar SN, Rosso A, Zoia A: Spatial extent of an outbreak in animal epidemics. Proc Natl Acad Sci USA2013, 1110(11):4239–4244.View ArticleGoogle Scholar
- Cohen-Tannoudji C, Diu B, Laloë F: Quantum Mechanics Vol. II, Paris: John Wiley & Sons; 1977.Google Scholar
- Kenkre VM: Theory of exciton annihilation in molecular crystals. Phys Rev B1980, 22:2089–2098.View ArticleGoogle Scholar
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