A unifying framework for quantifying the nature of animal interactions

The role of memory-based movements in the formation of animal home ranges

Most animals live in spatially-constrained home ranges. The prevalence of this space-use pattern in nature suggests that general biological mechanisms are likely to be responsible for their occurrence. Individual-based models of animal movement in both theoretical and empirical settings have demonstrated that the revisitation of familiar areas through memory can lead to the formation of stable home ranges. Here, we formulate a deterministic, mechanistic home range model that includes the interplay between a bi-component memory and resource preference, and evaluate resulting patterns of space-use. We show that a bi-component memory process can lead to the formation of stable home ranges and control its size, with greater spatial memory capabilities being associated with larger home range size. The interplay between memory and resource preferences gives rise to a continuum of space-use patterns-from spatially-restricted movements into a home range that is influenced by local resource heterogeneity, to diffusive-like movements dependent on larger-scale resource distributions, such as in nomadism. Future work could take advantage of this model formulation to evaluate the role of memory in shaping individual performance in response to varying spatio-temporal resource patterns.

Analyzing tiger interaction and home range shifts using a time-geographic approach

BACKGROUND

Interaction through movement can be used as a marker to understand and model interspecific and intraspecific species dynamics, and the collective behavior of animals sharing the same space. This research leverages the time-geography framework, commonly used in human movement research, to explore the dynamic patterns of interaction between Indochinese tigers (Panthera tigris corbeti) in the western forest complex (WEFCOM) in Thailand.

METHODS

We propose and assess ORTEGA, a time-geographic interaction analysis method, to trace spatio-temporal interactions patterns and home range shifts among tigers. Using unique GPS tracking data of tigers in WEFCOM collected over multiple years, concurrent and delayed interaction patterns of tigers are investigated. The outcomes are compared for intraspecific tiger interaction across different genders, relationships, and life stages. Additionally, the performance of ORTEGA is compared to a commonly used proximity-based approach.

RESULTS

Among the 67 tracked tigers, 42 show concurrent interactions at shared boundaries. Further investigation of five tigers with overlapping home ranges (two adult females, a male, and two young male tigers) suggests that the mother tiger and her two young mostly stay together before their dispersal but interact less post-dispersal. The male tiger increases encounters with the mother tiger while her young shift their home ranges. On another timeline, the neighbor female tiger mostly avoids the mother tiger. Through these home range dynamics and interaction patterns, we identify four types of interaction among these tigers: following, encounter, latency, and avoidance. Compared to the proximity-based approach, ORTEGA demonstrates better detects concurrent mother-young interactions during pre-dispersal, while the proximity-based approach misses many interactions among the dyads. With larger spatial buffers and temporal windows, the proximity-based approach detects more encounters but may overestimate the duration of interaction.

CONCLUSIONS

This research demonstrates the applicability and merits of ORTEGA as a time-geographic based approach to animal movement interaction analysis. We show time geography can develop valuable, data-driven insights about animal behavior and interactions. ORTEGA effectively traces frequent encounters and temporally delayed interactions between animals, without relying on specific spatial and temporal buffers. Future research should integrate contextual and behavioral information to better identify and characterize the nature of species interaction.

Optimal foraging strategies for mutually avoiding competitors

Many animals are known to exhibit foraging patterns where the distances they travel in a given direction are drawn from a heavy-tailed Lévy distribution. Previous studies have shown that, under sparse and random resource conditions, solitary non-destructive (with regenerating resources) foragers perform a maximally efficient search with Lévy exponent μ equal to 2, while for destructive foragers, efficiency decreases with μ monotonically and there is no optimal μ. However, in nature, there also exist situations where multiple foragers, displaying avoidance behavior, interact with each other competitively. To understand the effects of such competition, we develop a stochastic agent-based simulation that models competitive foraging among mutually avoiding individuals by incorporating an avoidance zone, or territory, of a certain size around each forager which is not accessible for foraging by other competitors. For non-destructive foraging, our results show that with increasing size of the territory and number of agents the optimal Lévy exponent is still approximately 2 while the overall efficiency of the search decreases. At low values of the Lévy exponent, however, increasing territory size actually increases efficiency. For destructive foraging, we show that certain kinds of avoidance can lead to qualitatively different behavior from solitary foraging, such as the existence of an optimal search with 1<μ<2. Finally, we show that the variance among the efficiencies of the agents increases with increasing Lévy exponent for both solitary and competing foragers, suggesting that reducing variance might be a selective pressure for foragers adopting lower values of μ. Taken together, our results suggest that, for multiple foragers, mutual avoidance and efficiency variance among individuals can lead to optimal Lévy searches with exponents different from those for solitary foragers.

How to scale up from animal movement decisions to spatiotemporal patterns: An approach via step selection

Uncovering the mechanisms behind animal space use patterns is of vital importance for predictive ecology, thus conservation and management of ecosystems. Movement is a core driver of those patterns so understanding how movement mechanisms give rise to space use patterns has become an increasingly active area of research. This study focuses on a particular strand of research in this area, based around step selection analysis (SSA). SSA is a popular way of inferring drivers of movement decisions, but, perhaps less well appreciated, it also parametrises a model of animal movement. Of key interest is that this model can be propagated forwards in time to predict the space use patterns over broader spatial and temporal scales than those that pertain to the proximate movement decisions of animals. Here, we provide a guide for understanding and using the various existing techniques for scaling up step selection models to predict broad-scale space use patterns. We give practical guidance on when to use which technique, as well as specific examples together with code in R and Python. By pulling together various disparate techniques into one place, and providing code and instructions in simple examples, we hope to highlight the importance of these techniques and make them accessible to a wider range of ecologists, ultimately helping expand the usefulness of SSA.

An integrated path for spatial capture-recapture and animal movement modeling

Ecologists and conservation biologists increasingly rely on spatial capture-recapture (SCR) and movement modeling to study animal populations. Historically, SCR has focused on population-level processes (e.g., vital rates, abundance, density, and distribution), whereas animal movement modeling has focused on the behavior of individuals (e.g., activity budgets, resource selection, migration). Even though animal movement is clearly a driver of population-level patterns and dynamics, technical and conceptual developments to date have not forged a firm link between the two fields. Instead, movement modeling has typically focused on the individual level without providing a coherent scaling from individual- to population-level processes, whereas SCR has typically focused on the population level while greatly simplifying the movement processes that give rise to the observations underlying these models. In our view, the integration of SCR and animal movement modeling has tremendous potential for allowing ecologists to scale up from individuals to populations and advancing the types of inferences that can be made at the intersection of population, movement, and landscape ecology. Properly accounting for complex animal movement processes can also potentially reduce bias in estimators of population-level parameters, thereby improving inferences that are critical for species conservation and management. This introductory article to the Special Feature reviews recent advances in SCR and animal movement modeling, establishes a common notation, highlights potential advantages of linking individual-level (Lagrangian) movements to population-level (Eulerian) processes, and outlines a general conceptual framework for the integration of movement and SCR models. We then identify important avenues for future research, including key challenges and potential pitfalls in the developments and applications that lie ahead.

Animal tracking moves community ecology: Opportunities and challenges

1. Individual decisions regarding how, why, and when organisms interact with one another and with their environment scale up to shape patterns and processes in communities. Recent evidence has firmly established the prevalence of intraspecific variation in nature and its relevance in community ecology, yet challenges associated with collecting data on large numbers of individual conspecifics and heterospecifics has hampered integration of individual variation into community ecology. 2. Nevertheless, recent technological and statistical advances in GPS-tracking, remote sensing, and behavioral ecology offer a toolbox for integrating intraspecific variation into community processes. More than simply describing where organisms go, movement data provide unique information about interactions and environmental associations from which a true individual-to-community framework can be built. 3. By linking the movement paths of both conspecifics and heterospecifics with environmental data, ecologists can now simultaneously quantify intra- and interspecific variation regarding the Eltonian (biotic interactions) and Grinnellian (environmental conditions) factors underpinning community assemblage and dynamics, yet substantial logistical and analytical challenges must be addressed for these approaches to realize their full potential. 4. Across communities, empirical integration of Eltonian and Grinnellian factors can support conservation applications and reveal metacommunity dynamics via tracking-based dispersal data. As the logistical and analytical challenges associated with multi-species tracking are surmounted, we envision a future where individual movements and their ecological and environmental signatures will bring resolution to many enduring issues in community ecology.

Conceptual and methodological advances in habitat-selection modeling: guidelines for ecology and evolution

Habitat selection is a fundamental animal behavior that shapes a wide range of ecological processes, including animal movement, nutrient transfer, trophic dynamics and population distribution. Although habitat selection has been a focus of ecological studies for decades, technological, conceptual and methodological advances over the last 20 yr have led to a surge in studies addressing this process. Despite the substantial literature focused on quantifying the habitat-selection patterns of animals, there is a marked lack of guidance on best analytical practices. The conceptual foundations of the most commonly applied modeling frameworks can be confusing even to those well versed in their application. Furthermore, there has yet to be a synthesis of the advances made over the last 20 yr. Therefore, there is a need for both synthesis of the current state of knowledge on habitat selection, and guidance for those seeking to study this process. Here, we provide an approachable overview and synthesis of the literature on habitat-selection analyses (HSAs) conducted using selection functions, which are by far the most applied modeling framework for understanding the habitat-selection process. This review is purposefully non-technical and focused on understanding without heavy mathematical and statistical notation, which can confuse many practitioners. We offer an overview and history of HSAs, describing the tortuous conceptual path to our current understanding. Through this overview, we also aim to address the areas of greatest confusion in the literature. We synthesize the literature outlining the most exciting conceptual advances in the field of habitat-selection modeling, discussing the substantial ecological and evolutionary inference that can be made using contemporary techniques. We aim for this paper to provide clarity for those navigating the complex literature on HSAs while acting as a reference and best practices guide for practitioners.

Learning and Animal Movement

Integrating diverse concepts from animal behavior, movement ecology, and machine learning, we develop an overview of the ecology of learning and animal movement. Learning-based movement is clearly relevant to ecological problems, but the subject is rooted firmly in psychology, including a distinct terminology. We contrast this psychological origin of learning with the task-oriented perspective on learning that has emerged from the field of machine learning. We review conceptual frameworks that characterize the role of learning in movement, discuss emerging trends, and summarize recent developments in the analysis of movement data. We also discuss the relative advantages of different modeling approaches for exploring the learning-movement interface. We explore in depth how individual and social modalities of learning can matter to the ecology of animal movement, and highlight how diverse kinds of field studies, ranging from translocation efforts to manipulative experiments, can provide critical insight into the learning process in animal movement.

Mechanistic home range analysis reveals drivers of space use patterns for a non-territorial passerine

Home ranging is a near-ubiquitous phenomenon in the animal kingdom. Understanding the behavioural mechanisms that give rise to observed home range patterns is thus an important general question, and mechanistic home range analysis (MHRA) provides the tools to address it. However, such analysis has hitherto been principally restricted to scent-marking territorial animals, so its potential breadth of application has not been tested. Here, we apply MHRA to a population of long-tailed tits Aegithalos caudatus, a non-territorial passerine, in the non-breeding season where there is no clear 'central place' near which birds need to remain. The aim is to uncover the principal movement mechanisms underlying observed home range formation. Our foundational models consist of memory-mediated conspecific avoidance between flocks, combined with attraction to woodland. These are then modified to incorporate the effects of flock size and relatedness (i.e. kinship), to uncover the effect of these on the mechanisms of home range formation. We found that a simple model of spatial avoidance, together with attraction to the central parts of woodland areas, accurately captures long-tailed tit home range patterns. Refining these models further, we show that the magnitude of spatial avoidance by a flock is negatively correlated to both the relative size of the flock (compared to its neighbour) and the relatedness of the flock with its neighbour. Our study applies MHRA beyond the confines of scent-marking, territorial animals, so paves the way for much broader taxonomic application. These could potentially help uncover general properties underlying the emergence of animal space use patterns. This is also the first study to apply MHRA to questions of relatedness and flock size, thus broadening the potential possible applications of this suite of analytic techniques.

Wolf habitat selection when sympatric or allopatric with brown bears in Scandinavia

Habitat selection of animals depends on factors such as food availability, landscape features, and intra- and interspecific interactions. Individuals can show several behavioral responses to reduce competition for habitat, yet the mechanisms that drive them are poorly understood. This is particularly true for large carnivores, whose fine-scale monitoring is logistically complex and expensive. In Scandinavia, the home-range establishment and kill rates of gray wolves (Canis lupus) are affected by the coexistence with brown bears (Ursus arctos). Here, we applied resource selection functions and a multivariate approach to compare wolf habitat selection within home ranges of wolves that were either sympatric or allopatric with bears. Wolves selected for lower altitudes in winter, particularly in the area where bears and wolves are sympatric, where altitude is generally higher than where they are allopatric. Wolves may follow the winter migration of their staple prey, moose (Alces alces), to lower altitudes. Otherwise, we did not find any effect of bear presence on wolf habitat selection, in contrast with our previous studies. Our new results indicate that the manifestation of a specific driver of habitat selection, namely interspecific competition, can vary at different spatial-temporal scales. This is important to understand the structure of ecological communities and the varying mechanisms underlying interspecific interactions.

How range residency and long-range perception change encounter rates

Encounter rates link movement strategies to intra- and inter-specific interactions, and therefore translate individual movement behavior into higher-level ecological processes. Indeed, a large body of interacting population theory rests on the law of mass action, which can be derived from assumptions of Brownian motion in an enclosed container with exclusively local perception. These assumptions imply completely uniform space use, individual home ranges equivalent to the population range, and encounter dependent on movement paths actually crossing. Mounting empirical evidence, however, suggests that animals use space non-uniformly, occupy home ranges substantially smaller than the population range, and are often capable of nonlocal perception. Here, we explore how these empirically supported behaviors change pairwise encounter rates. Specifically, we derive novel analytical expressions for encounter rates under Ornstein-Uhlenbeck motion, which features non-uniform space use and allows individual home ranges to differ from the population range. We compare OU-based encounter predictions to those of Reflected Brownian Motion, from which the law of mass action can be derived. For both models, we further explore how the interplay between the scale of perception and home-range size affects encounter rates. We find that neglecting realistic movement and perceptual behaviors can lead to systematic, non-negligible biases in encounter-rate predictions.

Spatial Memory and Taxis-Driven Pattern Formation in Model Ecosystems

Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales, where births and deaths are negligible. This phenomenon is particularly prevalent in populations of larger, vertebrate animals who often reproduce only once per year or less. To understand spatial arrangements of animal communities on such timescales, we use a class of diffusion-taxis equations for modelling inter-population movement responses between [Formula: see text] populations. These systems of equations incorporate the effect on animal movement of both the current presence of other populations and the memory of past presence encoded either in the environment or in the minds of animals. We give general criteria for the spontaneous formation of both stationary and oscillatory patterns, via linear pattern formation analysis. For [Formula: see text], we classify completely the pattern formation properties using a combination of linear analysis and nonlinear energy functionals. In this case, the only patterns that can occur asymptotically in time are stationary. However, for [Formula: see text], oscillatory patterns can occur asymptotically, giving rise to a sequence of period-doubling bifurcations leading to patterns with no obvious regularity, a hallmark of chaos. Our study highlights the importance of understanding between-population animal movement for understanding spatial species distributions, something that is typically ignored in species distribution modelling, and so develops a new paradigm for spatial population dynamics.

Metrics for describing dyadic movement: a review

In movement ecology, the few works that have taken collective behaviour into account are data-driven and rely on simplistic theoretical assumptions, relying in metrics that may or may not be measuring what is intended. In the present paper, we focus on pairwise joint-movement behaviour, where individuals move together during at least a segment of their path. We investigate the adequacy of twelve metrics introduced in previous works for assessing joint movement by analysing their theoretical properties and confronting them with contrasting case scenarios. Two criteria are taken into account for review of those metrics: 1) practical use, and 2) dependence on parameters and underlying assumptions. When analysing the similarities between the metrics as defined, we show how some of them can be expressed using general mathematical forms. In addition, we evaluate the ability of each metric to assess specific aspects of joint-movement behaviour: proximity (closeness in space-time) and coordination (synchrony) in direction and speed. We found that some metrics are better suited to assess proximity and others are more sensitive to coordination. To help readers choose metrics, we elaborate a graphical representation of the metrics in the coordination and proximity space based on our results, and give a few examples of proximity and coordination focus in different movement studies.

Collective movement in ecology: from emerging technologies to conservation and management

Recent advances in technology and quantitative methods have led to the emergence of a new field of study that stands to link insights of researchers from two closely related, but often disconnected disciplines: movement ecology and collective animal behaviour. To date, the field of movement ecology has focused on elucidating the internal and external drivers of animal movement and the influence of movement on broader ecological processes. Typically, tracking and/or remote sensing technology is employed to study individual animals in natural conditions. By contrast, the field of collective behaviour has quantified the significant role social interactions play in the decision-making of animals within groups and, to date, has predominantly relied on controlled laboratory-based studies and theoretical models owing to the constraints of studying interacting animals in the field. This themed issue is intended to formalize the burgeoning field of collective movement ecology which integrates research from both movement ecology and collective behaviour. In this introductory paper, we set the stage for the issue by briefly examining the approaches and current status of research in these areas. Next, we outline the structure of the theme issue and describe the obstacles collective movement researchers face, from data acquisition in the field to analysis and problems of scale, and highlight the key contributions of the assembled papers. We finish by presenting research that links individual and broad-scale ecological and evolutionary processes to collective movement, and finally relate these concepts to emerging challenges for the management and conservation of animals on the move in a world that is increasingly impacted by human activity.This article is part of the theme issue 'Collective movement ecology'.

The importance of individual variation in the dynamics of animal collective movements

Animal collective movements are a key example of a system that links two clearly defined levels of organization: the individual and the group. Most models investigating collective movements have generated coherent collective behaviours without the inclusion of individual variability. However, new individual-based models, together with emerging empirical information, emphasize that within-group heterogeneity may strongly influence collective movement behaviour. Here we (i) review the empirical evidence for individual variation in animal collective movements, (ii) explore how theoretical investigations have represented individual heterogeneity when modelling collective movements and (iii) present a model to show how within-group heterogeneity influences the collective properties of a group. Our review underscores the need to consider variability at the level of the individual to improve our understanding of how individual decision rules lead to emergent movement patterns, and also to yield better quantitative predictions of collective behaviour.This article is part of the theme issue 'Collective movement ecology'.

How memory of direct animal interactions can lead to territorial pattern formation

Mechanistic home range analysis (MHRA) is a highly effective tool for understanding spacing patterns of animal populations. It has hitherto focused on populations where animals defend their territories by communicating indirectly, e.g. via scent marks. However, many animal populations defend their territories using direct interactions, such as ritualized aggression. To enable application of MHRA to such populations, we construct a model of direct territorial interactions, using linear stability analysis and energy methods to understand when territorial patterns may form. We show that spatial memory of past interactions is vital for pattern formation, as is memory of 'safe' places, where the animal has visited but not suffered recent territorial encounters. Additionally, the spatial range over which animals make decisions to move is key to understanding the size and shape of their resulting territories. Analysis using energy methods, on a simplified version of our system, shows that stability in the nonlinear system corresponds well to predictions of linear analysis. We also uncover a hysteresis in the process of territory formation, so that formation may depend crucially on initial space-use. Our analysis, in one dimension and two dimensions, provides mathematical groundwork required for extending MHRA to situations where territories are defended by direct encounters.

Partial differential equation techniques for analysing animal movement: A comparison of different methods

Recent advances in animal tracking have allowed us to uncover the drivers of movement in unprecedented detail. This has enabled modellers to construct ever more realistic models of animal movement, which aid in uncovering detailed patterns of space use in animal populations. Partial differential equations (PDEs) provide a popular tool for mathematically analysing such models. However, their construction often relies on simplifying assumptions which may greatly affect the model outcomes. Here, we analyse the effect of various PDE approximations on the analysis of some simple movement models, including a biased random walk, central-place foraging processes and movement in heterogeneous landscapes. Perhaps the most commonly-used PDE method dates back to a seminal paper of Patlak from 1953. However, our results show that this can be a very poor approximation in even quite simple models. On the other hand, more recent methods, based on transport equation formalisms, can provide more accurate results, as long as the kernel describing the animal's movement is sufficiently smooth. When the movement kernel is not smooth, we show that both the older and newer methods can lead to quantitatively misleading results. Our detailed analysis will aid future researchers in the appropriate choice of PDE approximation for analysing models of animal movement.

Habitat and social factors shape individual decisions and emergent group structure during baboon collective movement

For group-living animals traveling through heterogeneous landscapes, collective movement can be influenced by both habitat structure and social interactions. Yet research in collective behavior has largely neglected habitat influences on movement. Here we integrate simultaneous, high-resolution, tracking of wild baboons within a troop with a 3-dimensional reconstruction of their habitat to identify key drivers of baboon movement. A previously unexplored social influence - baboons' preference for locations that other troop members have recently traversed - is the most important predictor of individual movement decisions. Habitat is shown to influence movement over multiple spatial scales, from long-range attraction and repulsion from the troop's sleeping site, to relatively local influences including road-following and a short-range avoidance of dense vegetation. Scaling to the collective level reveals a clear association between habitat features and the emergent structure of the group, highlighting the importance of habitat heterogeneity in shaping group coordination.

Modeling interdependent animal movement in continuous time

This article presents a new approach to modeling group animal movement in continuous time. The movement of a group of animals is modeled as a multivariate Ornstein Uhlenbeck diffusion process in a high-dimensional space. Each individual of the group is attracted to a leading point which is generally unobserved, and the movement of the leading point is also an Ornstein Uhlenbeck process attracted to an unknown attractor. The Ornstein Uhlenbeck bridge is applied to reconstruct the location of the leading point. All movement parameters are estimated using Markov chain Monte Carlo sampling, specifically a Metropolis Hastings algorithm. We apply the method to a small group of simultaneously tracked reindeer, Rangifer tarandus tarandus, showing that the method detects dependency in movement between individuals.

ECOLOGY. Terrestrial animal tracking as an eye on life and planet

Moving animals connect our world, spreading pollen, seeds, nutrients, and parasites as they go about the their daily lives. Recent integration of high-resolution Global Positioning System and other sensors into miniaturized tracking tags has dramatically improved our ability to describe animal movement. This has created opportunities and challenges that parallel big data transformations in other fields and has rapidly advanced animal ecology and physiology. New analytical approaches, combined with remotely sensed or modeled environmental information, have opened up a host of new questions on the causes of movement and its consequences for individuals, populations, and ecosystems. Simultaneous tracking of multiple animals is leading to new insights on species interactions and, scaled up, may enable distributed monitoring of both animals and our changing environment.

Topological data analysis of biological aggregation models

We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.

Territorial pattern formation in the absence of an attractive potential

Territoriality is a phenomenon exhibited throughout nature. On the individual level, it is the processes by which organisms exclude others of the same species from certain parts of space. On the population level, it is the segregation of space into separate areas, each used by subsections of the population. Proving mathematically that such individual-level processes can cause observed population-level patterns to form is necessary for linking these two levels of description in a non-speculative way. Previous mathematical analysis has relied upon assuming animals are attracted to a central area. This can either be a fixed geographical point, such as a den- or nest-site, or a region where they have previously visited. However, recent simulation-based studies suggest that this attractive potential is not necessary for territorial pattern formation. Here, we construct a partial differential equation (PDE) model of territorial interactions based on the individual-based model (IBM) from those simulation studies. The resulting PDE does not rely on attraction to spatial locations, but purely on conspecific avoidance, mediated via scent-marking. We show analytically that steady-state patterns can form, as long as (i) the scent does not decay faster than it takes the animal to traverse the terrain, and (ii) the spatial scale over which animals detect scent is incorporated into the PDE. As part of the analysis, we develop a general method for taking the PDE limit of an IBM that avoids destroying any intrinsic spatial scale in the underlying behavioral decisions.

How do animal territories form and change? Lessons from 20 years of mechanistic modelling

Territory formation is ubiquitous throughout the animal kingdom. At the individual level, various behaviours attempt to exclude conspecifics from regions of space. At the population level, animals often segregate into distinct territorial areas. Consequently, it should be possible to derive territorial patterns from the underlying behavioural processes of animal movements and interactions. Such derivations are an important element in the development of an ecological theory that can predict the effects of changing conditions on territorial populations. Here, we review the approaches developed over the past 20 years or so, which go under the umbrella of 'mechanistic territorial models'. We detail the two main strands to this research: partial differential equations and individual-based approaches, showing what each has offered to our understanding of territoriality and how they can be unified. We explain how they are related to other approaches to studying territories and home ranges, and point towards possible future directions.