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

Step selection functions (SSFs) are flexible statistical models used to jointly describe animals' movement and habitat preferences. The popularity of SSFs has grown rapidly, and various extensions have been developed to increase their utility, including the ability to use multiple statistical distributions to describe movement constraints, interactions to allow movements to depend on local environmental features, and random effects and latent states to account for within‐ and among‐individual variability. Although the SSF is a relatively simple statistical model, its presentation has not been consistent in the literature, leading to confusion about model flexibility and interpretation. We believe that part of the confusion has arisen from the conflation of the SSF model with the methods used for statistical inference, and in particular, parameter estimation. Notably, conditional logistic regression (CLR) can be used to fit SSFs in exponential form, and this model fitting approach is often presented interchangeably with the actual model (the SSF itself). However, reliance on CLR reduces model flexibility, and suggests a misleading interpretation of step selection analysis as being equivalent to a case–control study. In this review, we explicitly distinguish between model formulation and inference technique, presenting a coherent framework to fit SSFs based on numerical integration and maximum likelihood estimation. We provide an overview of common numerical integration techniques (including Monte Carlo integration, importance sampling and quadrature), and explain how they relate to popular methods used in step selection analyses. This general framework unifies different model fitting techniques for SSFs, and opens the way for improved inferential methods. In this approach, it is straightforward to model movement with distributions outside the exponential family, and to apply different SSF model formulations to the same data set and compare them with AIC. By separating the model formulation from the inference technique, we hope to clarify many important concepts in step selection analysis.

Many species frequently return to previously visited foraging sites. This bias towards familiar areas suggests that remembering information from past experience is beneficial. Such a memory-based foraging strategy has also been hypothesized to give rise to restricted space use (i.e. a home range). Nonetheless, the benefits of empirically derived memory-based foraging tactics and the extent to which they give rise to restricted space use patterns are still relatively unknown. Using a combination of stochastic agent-based simulations and deterministic integro-difference equations, we developed an adaptive link (based on energy gains as a foraging currency) between memory-based patch selection and its resulting spatial distribution. We used a memory-based foraging model developed and parameterized with patch selection data of free-ranging bison Bison bison in Prince Albert National Park, Canada. Relative to random use of food patches, simulated foragers using both spatial and attribute memory are more efficient, particularly in landscapes with clumped resources. However, a certain amount of random patch use is necessary to avoid frequent returns to relatively poor-quality patches, or avoid being caught in a relatively poor quality area of the landscape. Notably, in landscapes with clumped resources, simulated foragers that kept a reference point of the quality of recently visited patches, and returned to previously visited patches when local patch quality was poorer than the reference point, experienced higher energy gains compared to random patch use. Furthermore, the model of memory-based foraging resulted in restricted space use in simulated landscapes and replicated the restricted space use observed in free-ranging bison reasonably well. Our work demonstrates the adaptive value of spatial and attribute memory in heterogeneous landscapes, and how home ranges can be a byproduct of non-omniscient foragers using past experience to minimize temporal variation in energy gains.

Uncovering the mechanisms behind territory formation is a fundamental problem in behavioural ecology. The broad nature of the underlying conspecific avoidance processes are well documented across a wide range of taxa. Scent marking in particular is common to a large range of terrestrial mammals and is known to be fundamental for communication. However, despite its importance, exact quantification of the time-scales over which scent cues and messages persist remains elusive. Recent work by the present authors has begun to shed light on this problem by modelling animals as random walkers with scent-mediated interaction processes. Territories emerge as dynamic objects that continually change shape and slowly move without settling to a fixed location. As a consequence, the utilisation distribution of such an animal results in a slowly increasing home range, as shown for urban foxes (Vulpes vulpes). For certain other species, however, home ranges reach a stable state. The present work shows that stable home ranges arise when, in addition to scent-mediated conspecific avoidance, each animal moves as a central place forager. That is, the animal's movement has a random aspect but is also biased towards a fixed location, such as a den or nest site. Dynamic territories emerge but the probability distribution of the territory border locations reaches a steady state, causing stable home ranges to emerge from the territorial dynamics. Approximate analytic expressions for the animal's probability density function are derived. A programme is given for using these expressions to quantify both the strength of the animal's movement bias towards the central place and the time-scale over which scent messages persist. Comparisons are made with previous theoretical work modelling central place foragers with conspecific avoidance. Some insights into the mechanisms behind allometric scaling laws of animal space use are also given.

Habitat selection is a multi-level, hierarchical process that should be a key component in the balance between food acquisition and predation risk avoidance (food–predation trade-off). However, to date, studies have not fully elucidated how fine- and broad-scale habitat decisions by individual prey can help balance food versus risk. We studied broad-scale habitat selection by Newfoundland caribou Rangifer tarandus, focusing on trade-offs between predation risk versus access to forage during the calving and post-calving period. We improved traditional measures of habitat availability by incorporating fine-scale movement patterns of caribou into the availability kernel, thus enabling separation of broad and fine scales of selection. Remote sensing and field surveys served to create a spatio-temporal model of forage availability, whereas GPS telemetry locations from 66 black bears Ursus americanus and 59 coyotes Canis latrans provided models of predation risk. We then used GPS telemetry locations from 114 female caribou to assess food–predation trade-offs through the prism of our refined model of caribou habitat availability. We noted that migratory movements of caribou were oriented mainly towards habitats with abundant forage and lower risk of bear and (to a lesser extent) coyote encounter. These findings were generally consistent across caribou herds and would not have been evident had we used traditional methods instead of our refined model when estimating habitat availability. We interpret these findings in the context of stereotypical migratory behaviour observed in Newfoundland caribou, which occurs despite the extirpation of wolves Canis lupus nearly a century ago. We submit that caribou are able to balance food acquisition against predation risk using a complex set of factors involving both finer and broader scale selection. Accordingly, our study provides a strong argument for using refined habitat availability estimates when assessing food–predation trade-offs.

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

A fundamental goal of ecology is to understand the spatial distribution of species. For moving animals, their location is crucially dependent on the movement mechanisms they employ to navigate the landscape. Animals across many taxa are known to exhibit directional correlation in their movement. This work explores the effect of such directional correlation on spatial pattern formation in a model of between-population taxis (i.e., movement of each population in response to the presence of the others). A telegrapher-taxis formalism is used, which generalises a previously studied diffusion-taxis system by incorporating a parameter T, measuring the characteristic time for directional persistence. The results give general criteria for determining when changes in T will drive qualitative changes in the predictions of linear pattern formation analysis for N ≥ 2 populations. As a specific example, the N = 2 case is explored in detail, showing that directional correlation can cause one population to ‘chase’ the other across the landscape while maintaining a non-constant spatial distribution. Overall, this study demonstrates the importance of accounting for directional correlation in movement for understanding both quantitative and qualitative aspects of species distributions.

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection–diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

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.

Aggregations are emergent features common to many biological systems. Mathematical models to understand their emergence are consequently widespread, with the aggregation–diffusion equation being a prime example. Here we study the aggregation–diffusion equation with linear diffusion in one spatial dimension. This equation is known to support solutions that involve both single and multiple aggregations. However, numerical evidence suggests that the latter, which we term ‘multi-peaked solutions’ may often be long-transient solutions rather than asymptotic steady states. We develop a novel technique for distinguishing between long transients and asymptotic steady states via an energy minimisation approach. The technique involves first approximating our study equation using a limiting process and a moment closure procedure. We then analyse local minimum energy states of this approximate system, hypothesising that these will correspond to asymptotic patterns in the aggregation–diffusion equation. Finally, we verify our hypotheses through numerical investigation, showing that our approximate analytic technique gives good predictions as to whether a state is asymptotic or transient. Overall, we find that almost all twin-peaked, and by extension multi-peaked, solutions are transient, except for some very special cases. We demonstrate numerically that these transients can be arbitrarily long-lived, depending on the parameters of the system.