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.

property of this model. This mechanistic modelling approach has enabled researchers to understand various ecological phenomena, such as the processes underlying the emergence of prey corridors between predator home ranges (Hamelin & Lewis, 2010;Lewis & Murray, 1993), the effect of disease spread on movement decisions (Potts, Harris, & Giuggioli, 2013), coyote Canis latrans territory rearrangement following the death of an alpha male (Moorcroft, Lewis, & Crabtree, 2006), and fission processes in meerkat Suricata suricatta territories (Bateman, Lewis, Gall, Manser, & Clutton-Brock, 2015). All of these examples make use of explicit, mathematical links between movement and interaction mechanisms and the emergent home range patterns, to make important biological inferences.
Despite the insights gained from mechanistic home range analysis (MHRA) in the context of partial differential equations (PDEs), these analytic techniques have hitherto been used almost exclusively on scent-marking animals (but see Potts, Mokross, & Lewis, 2014), with the exception of the earliest and simplest models, where home ranging arose purely from an attraction to a central place (Holgate, 1971). However, many animals advertise their territory through cues other than scent, such as dominance displays, vocalizations and fighting (collectively known as 'ritualized aggression'). Furthermore, not all animals have a clear 'central point' (such as a den or nest site) which pins their home range in a particular place. The model of Potts and Lewis (2016) was designed to extend MHRA for use with such animals in the context of PDEs. While home range formation with neither a central place nor scent-mark avoidance had previously been modelled using simulations of individual-based models (Moorter et al., 2009;Riotte-Lambert, Benhamou, & Chamaillé-Jammes, 2015;Siniff & Jessen, 1969), the extension to the PDE framework offered by Potts and Lewis (2016) enabled rigorous mathematical analysis of the conditions under which home ranges can form. There, the authors showed mathematically that home ranges can form purely from a mechanism of ritualized aggression, coupled with memory of those aggression events.
In this study, we advance the applicability of PDE-based MHRA further still, by showing that MHRA can be used in situations where there is no explicit territorial behaviour, and also no 'central point' around which animals localize their movement. This is the case for our study species, the long-tailed tit Aegithalos caudatus, outside the breeding season. At these times of year, they live in flocks, each of which has a distinct home range that only overlaps slightly with those of neighbouring flocks (Gaston, 1973;Hatchwell, 2016;Hatchwell, Anderson, Ross, Fowlie, & Blackwell, 2001). They do not maintain a fixed roosting site, so have no clear localization centre, and have rarely been observed to engage in territorial interactions (Napper, Sharp, McGowan, Simeoni, & Hatchwell, 2013).
Nonetheless, it may be that long-tailed tits avoid areas where they have seen neighbouring flocks foraging. This could be to avoid social interaction with other flocks or to strengthen relationships within flocks. We hypothesize that this behaviour acts as a proxy for territoriality, causing distinct home ranges to form without requiring directly observable, aggressive, territorial behaviour. To test this hypothesis, we formulate a mechanistic model, incorporating such non-aggressive avoidance mechanisms and observe whether this can explain the various home range patterns observed in a longtailed tit population across a number of seasons.
Since there is a clear and well-documented effect of habitat type on the space use of long-tailed tits, arising from the structure and composition of woodland (Gaston, 1973), we also incorporate into our study a set of models that are each linked to a distinct hypothesis on the effect of woodland on bird movement. Woodland structure is very complex in our study system, incorporating dozens of genera of trees and shrubs of varying sizes, so we use a simplified approach by viewing woodland as a binary variable: either present or absent.
Then our hypotheses all relate to how flocks move with respect to the presence or absence of trees. We use the resulting model to show that the home range patterns of long-tailed tits can be predominantly explained by a combination of conspecific avoidance and attraction towards woodland areas.
From this starting point, we extend our model to test various hypotheses about more subtle drivers of home range patterns. First, we examine how the relative size of a flock (i.e. number of individuals) affects the extent to which it avoids neighbouring flocks. We hypothesize that smaller flocks will tend to have a stronger avoidance mechanism than larger flocks because if there is competition over space use, smaller flocks are likely to be less competitive than large flocks and so are predicted to avoid potential conflict (Adams & Plowes, 2019;Dyble, Houslay, Manser, & Clutton-Brock, 2019;Port, Kappeler, & Johnstone, 2011). Second, given that there is an apparent correlation between spatial proximity and relatedness (Napper & Hatchwell, 2016), we hypothesize that the relatedness of neighbouring flocks will be inversely related to the strength of avoidance mechanism. We show that these subtle effects of flock size and relatedness cannot be observed using kernel density estimators, a popular statistical model describing a flock's home range, but do emerge from a MHRA approach. This demonstrates the usefulness of mechanistic models of home range for uncovering features of movement that cannot easily be detected using descriptive, statistical models.
Overall, our study makes theoretical advances by demonstrating (a) that MHRA is applicable in a much wider range of situations than previously used and (b) that MHRA can uncover behavioural drivers of movement and space use that are not simple to find using traditional, statistical measures of home range. Furthermore, our study makes important advances in avian behavioural ecology by demonstrating (c) that relatedness and flock size can affect between-flock movement responses and (d) that avoidance mechanisms may exist in species that do not display obvious territorial behaviour, explaining the existence of home range segregation in such species.

| Study system
The data come from a study on long-tailed tits, a small non-territorial bird found across Europe and Asia. Long-tailed tits weigh only 6-8 g (Glen & Perrins, 1988) and their main life-history goal while not focusing on breeding is to forage for food (Gaston, 1973;Hatchwell, 2016).
We studied the behaviour of these birds in the non-breeding season when they live in home ranging flocks of around 5-25 individuals (Napper & Hatchwell, 2016). One or more fledged broods and their parents and helpers are usually the nucleus of a winter flock, which are then joined by failed breeders who may or may not be related to the brood(s) (Napper & Hatchwell, 2016). Thus, the majority of flock members (60%-70%) are typically related (r ≥ 0.25, where r is the coefficient of relatedness, Wright, 1922) to at least one other member of the same flock, although those relatives may be drawn from two or more families . In addition, members of one flock often have relatives in other flocks as a result of dispersal during the non-breeding season (Napper & Hatchwell, 2016). Members of a flock forage together in the day and then sleep together in a communal roost, which often changes location between nights.
The study site is contained within the Rivelin Valley, Sheffield, UK (53°23ʹN, 1°34ʹW), covering approximately 3 km 2 . The population of long-tailed tits that inhabits this site has been studied since 1994.
An observation period began when a flock was first encountered and the first location was recorded. The observation period ended when sight of the flock was lost. We recorded one location every minute to give a trajectory for each observation period. There were 19 ± 2 (mean ± 95% CI) locations per observation period.
For this study, we used location data from six of eight flocks that were followed in the non-breeding season of 2011-2012 in the Fox Hagg woodland. We removed from our analysis two flocks which contained only four and seven locations, collected over one observation period, as we concluded this was not enough data to estimate home

| Mathematical models
The trajectories from each observation period are too short to estimate covariates of stepwise movement decisions dependent upon the presence of other flocks, and therefore fit a stepwise movement kernel to the data as in, for example, Avgar et al. (2015) and Avgar, Potts, Lewis, and Boyce (2016). Additionally, it would be difficult to infer any inter-flock interaction behaviour since the trajectories of different flocks are not recorded at the same time. Instead, we infer the movement processes of a flock by fitting a mechanistic model of space use to locations collected over the entire non-breeding season (May-February). This method is known as MHRA . We use a system of advection-diffusion equations, each of which models a flock's utilization distribution, u i (x, t), which To solve (1) numerically, we must pick a domain, Ω, and enforce boundary conditions on the boundary ∂Ω. A biologically reasonable condition is to assume zero flux across the boundary, meaning that the number of birds exiting the domain at a boundary point is, on average, the same as the number entering at that point. In Equation 1, the flux is

so a zero flux boundary condition means that
where n x is a vector normal to the boundary at x. Because u i (x, t) is a probability density function defined on Ω, we must also impose the following condition Having set up the general modelling framework in Equations 1-3, we now describe specific choices of the vector field A i (x, t), that correspond to different hypotheses about the movement and interaction mechanisms behind observed home range patterns. First, note that each flock tends to reside in a slightly different part of space with only minimal overlap (Figure 2a), suggesting that flocks may be deliberately avoiding areas that they know to be used by other flocks. In the decades that this population has been studied, interactions between flocks occur regularly, but aggression is very rarely involved in these encounters (Napper & Hatchwell, 2016), meaning that this avoidance mechanism is highly unlikely to be a result of aggressive defence. Another hypothesized reason for spatial segregation of animal populations was given by Riotte-Lambert et al. (2015), and involves depletion and renewal of resources. However, for long-tailed tits, resources are abundant and deplete only minimally as the birds forage (B. J. Hatchwell, pers. obs.).
Therefore, instead of these previously used mechanisms, we use a memory-based approach, assuming that a flock has some knowledge of other flocks' space use, due to previous meetings which they remember. This knowledge causes a flock to avoid areas where they believe other flocks may reside. The precise details of interactions between adjacent flocks are not important for the model, but could include one flock seeing another, or hearing their calls. As long as some interaction has occurred between flocks and there is some avoidance mechanism (of places where past interactions have happened) in place, then our model is appropriate.
To model the avoidance mechanism, we introduce the concept of an interaction zone (IZ) for each flock. The IZ of a flock models a cognitive map of places where the flock remembers having previously interacted with other flocks. We assume that individuals within a flock share information, and so have a common IZ. The probability that a location, x, is in the IZ for flock i at time t is denoted by k i (x, t).
The probability k i increases in places where other flocks have a high probability of using that space and decreases as other flocks become less likely to use the space. Thus, the dynamics of k i (x, t) are described by the following equation: where ρ i is the rate at which the IZ is reinforced when two flocks of long-tailed tits are at the same location and β i is the rate of decay of the IZ due to revisiting parts of space without encountering other flocks. Mathematically, the IZ is equivalent to the concept of a 'conflict zone' introduced by Potts and Lewis (2016). For simplicity, and to avoid an unreasonably large number of parameters, we start by assuming that i = and i = for all i so that they are the same for all flocks.
However, in Section 2.4, we relax this assumption.
When making movement decisions, it is not realistic to assume a flock will examine the infinitesimally precise location where it currently happens to be. Rather, it is better to assume the flock will examine a small area around that location. We model this area as a disc with radius δ. This can be thought of as the flock's perceptual radius for which it makes its movement decisions. This idea corresponds, mathematically, to averaging the value of k i (x, t) over this disc. ( We thus define k i (x, t| ) to be a spatial average over all k i (x, t) within a radius δ of x, so thats where B (x) is a disc of radius δ, centred at x. A similar local averaging was also used by Potts and Lewis (2016) to model territories formed by ritualized aggression, where they showed that it is necessary to use k i instead of k i , to ensure the system is mathematically well-behaved.
Long-tailed tits are known to forage predominantly in woodland habitats (Gaston, 1973;Hatchwell, 2016) and our location data suggest that the flocks are almost always inside woodland. We therefore incorporate into our modelling framework a tendency for flocks to move in areas with tree coverage, alongside the tendency to move away from the IZ. This leads to the following definition of A i from Equation 1. Here, c 1 and c 2 describe the relative magnitude of advection away from the IZ and towards woodland, respectively, and M indexes different models of attraction to woodland, which we now describe. We first discretise the landscape and define any cell with more than half tree coverage to be in a woodland area. Each woodland area is defined visually using a procedure detailed in Appendix S4. We define six models as follows:  Figure 1a. We non-dimensionalize the system in Equations 1-6 as follows Immediately dropping the tildes for notational convenience, we arrive at the following dimensionless version of Equations 1-6: We summarize all the notation used in Table 1.

| Model fitting and comparison
We solve Equations 8-11 for u i and k i numerically using a finitedifference approximation (Smith, 1986) we use the Nelder-Mead maximization algorithm (Nelder & Mead, 1965) and select the best model based on their Bayesian information criterion (BIC) scores (Schwarz, 1978).

| Testing for other behavioural effects of movement
As well as testing hypotheses regarding inter-flock interactions and effects of woodland, we also use our modelling approach to test effects where s i (resp. s j ) is the size of flock i (resp. j).
Equation 14 gives a higher value for (1) when s j > s i than when s j < s i , meaning the probability that a location will be considered to be in the IZ of flock i will be higher if flock j is larger. This tests the hypothesis that a flock is less likely to consider a location safe if they have observed it being used by a larger flock than if the same location were observed being used by a smaller flock. Equation 15 gives combines the two hypotheses. We fit the parameters σ 1 , σ 2 and σ 3 using the functions 14-16. We use BIC both to select between the three models, and examine whether they are an improvement on the null model ( = 1 for all i, j).

| The effect of the landscape on kinesis
We have so far considered the effects of woodland and the IZ on  (17) t)), comparing BIC values, we see that for three of the other datasets, the best-fitting model is confirmed to be Model 4, and for one it is Model 2 (Table 3). This gives further indication that the birds prefer to move away from the woodland edge, as these models are the only two which describe this movement (Figure 1). The datasets and their corresponding utilization distributions are shown in Figure 3 and Appendix S6.

F I G U R E 2
Using data collected in 2018-2019, we extended the model selection procedure to test for an effect of flock size and relatedness between flocks on home range utilization. In the absence of this additional mechanism, Model 4 was the best-fitting model (Table 3;   (Table 4). More specifically, the results showed that smaller flocks avoided large flocks and large flocks were less likely to avoid small flocks while avoidance decreased as inter-flock relatedness increased (Table 4). It should also be noted that these effects of kinship and flock size on avoidance behaviour were not found when home ranges were analysed using KDE, a relatively simple statistical model (Worton, 1989; Appendix S7).
Finally, when considering the effect of a landscape-varying kinesis mechanism on the space use of the flocks, we find no improvement in the model fit: indeed, the BIC values are larger when we include the effect of landscape on kinesis into the best-fit models from the study without such a kinesis effect (Table 5; Appendix S8).
Furthermore, when fitting models where landscape only affects kinesis and not advection, the fit is very poor compared with models with landscape-driven advection (Appendix S8).

| D ISCUSS I ON
We have used MHRA  to show that memory of past conspecific interactions and movement towards woodland combine to model home range patterns of long-tailed tits with good accuracy (Figure 2). This study has extended the application of MHRA to non-breeding, non-territorial passerines, thereby showing that an understanding of space use patterns can be gained from this approach in the absence of either territorial scent-marking or advection towards a central place .
Past research on the foraging behaviours of long-tailed tits suggests that they are attracted to woodland (Gaston, 1973). This idea agrees visually with the location data shown in Figures 2a and 3.
Our best-fit model, Model 4, also suggests that the flocks have a tendency to move from the edges of woodland towards the central parts of the woodland area. There are various possible reasons for this. One hypothesis is that core areas of woodland are better than edge habitats for avoidance of predators, as reported in several  Sedláček, & Albrecht, 2010) and a meta-analysis suggests little consistency across habitat types and landscapes (Vetter, Rücker, & Storch, 2013). We are currently investigating the effects of habitat types on nest predation rate, although their impact on survival of juveniles and adults is much harder to quantify because of long-tailed tits' extensive ranges that typically encompass both core and edge habitats. Alternatively, core areas of woodland may differ from peripheral areas in their food availability (e.g. Rosli, Zakaria, & Rajpar, 2018;Terraube et al., 2016), thereby influencing forager movements.
This possibility is also hard to test in long-tailed tits because of the difficulty of sampling food availability for a canopy-feeding generalist insectivore. Teasing apart these hypotheses would require new field studies, particularly tailored to this question. Thus, our results demonstrate the role of mechanistic modelling in determining potentially fruitful ideas for future empirical research.
Our model also suggests that flocks avoid places they remember interacting with other flocks in the past. These interactions could be visual or vocal, and therefore may take place at a distance (incorporated in the averaging kernel, Equation 5). The memory capacity of small passerines is hard to test biologically as little is known about the cognitive abilities of small birds in general, regardless of species (Emery, 2006). However, the avoidance behaviour and such indirect inference is not without precedent (Avgar et al., 2015;Fagan et al., 2013;Merkle, Fortin, & Morales, 2014;Merkle, Potts, & Fortin, 2017).
Although woodland and memory of past interactions affected advective movement, we found no measurable effect on diffusive movement (i.e. kinesis). This is in contrast with MHRA studies on coyote home ranges  where the diffusive aspect of movement was found to be highly dependent on prey density.
Although the central aim of our study was to select between models, it is also worth commenting briefly on the parameter values of the best-fit models, as they can give some additional insight.
Looking at Table 3, the first thing to notice is that, of the four datasets where Model 4 is the best-fit model, there is not a great amount of variation between seasons in γ (7.7-10) which controls the advection away from the IZ, ζ (12.9-25.9) which controls the advection towards central woodland, or δ (0.047-0.096) which is the birds perceptual radius. However, there is a large variation in b (1.14-13.3), the parameter controlling the decrease in the IZ due to safe visits, across seasons. The outlier is the 2010-2011 season (b = 1.14).
Here, there were fewer flocks than in other seasons, so we hypothesize this might have affected the best-fit value of b. The reason for this is that the first term on the right-hand side of Equation 9 is a sum that increases with the number of flocks. Thus, one would expect the best-fit value of b to increase with the number of flocks in the study. While this is unlikely to account for all of the variation, it perhaps gives a partial explanation for this anomaly. It is also interesting to note that the mean value of δ, when converted into metres is 81 m. This means that our models suggest birds are, on average, considering an area with a radius of 81 m around their present location when making movement decisions. Here, for simplicity, we have assumed that perception is a binary quantity: perceived within the δ-disc and not perceived outside this disc. However, it would also be possible to consider other non-local formalisms, such as exponential decay (Avgar et al., 2015).
Aside from avoidance of other flocks and attraction to woodland, we have also shown that movement decisions in response to adjacent flocks depend on the relatedness between the two flocks.
We saw a negative relationship between avoidance mechanisms and flocks with more kin-connections. Other things being equal, one would expect this to cause a positive relationship between home range overlap and kinship, a phenomenon observed in long-tailed tits , as well as in several other taxa, including mammals (Sera & Gaines, 1994;Støen et al., 2005;Walker, Taylor, & Sunnucks, 2008), lizards (While, Uller, & Wapstra, 2009) and fish (Griffiths & Armstrong, 2002). Our study thus reveals plausible mechanisms behind such observations.
Our results also suggest that the relative size of each flock has an effect on their movement away from other flocks. Specifically, smaller flocks were less likely to move to places where they had interacted with larger flocks in the past and larger flocks were less likely to avoid places where they had encountered smaller flocks, suggesting greater avoidance of larger flocks. This effect of group size on the use or avoidance of overlapping ranges of adjacent social groups appears to be very unusual among social vertebrates and we are not aware of any equivalent findings, presumably because social species typically defend exclusive territories. However, this situation is captured in the theoretical models of 'battle dynamics' between social insect colonies, where the outcome of conflicts over space may be determined by relative colony size (Adams & Plowes, 2019;Adler, Quinonez, Plowes, & Adams, 2018). Testing avoidance of larger flocks directly would require analysis of synchronous observations of many flocks, which is a difficult task for field work.
MHRA provides a way of making such inferences with much less data, providing there is sufficient data to capture the home range.
The question of why flocks may benefit from avoiding one another remains open. One possibility is that it is related to avoidance of antagonistic social interactions (Sharp, McGowan, Wood, & Hatchwell, 2005). Although long-tailed tits do not defend territories and escalated conflicts are observed very rarely, simulated intrusions of individuals into flocks using playback experiments result in brief bouts of aggression that could deter interactions (Napper & Hatchwell, 2016). This would be consistent with the extended version of Model 4, which found that small flocks were more likely to avoid large flocks and large flocks were less likely to avoid small flocks. A second hypothesis is that separation into flocks with segregated space use is an anti-predator tactic, with flocks avoiding each other to prevent total flock size exceeding some optimum at which the benefits of group-living are maximized (Pulliam & Caraco, 1984).
A final explanation concerns the social benefit of flocking with a consistent set of conspecifics. Long-tailed tits are cooperative breeders in which helping behaviour is kin-selected (Hatchwell, Gullett, & Adams, 2014), with helpers exhibiting a strong kin preference in their helping behaviour (Leedale, Sharp, Simeoni, Robinson, & Hatchwell, 2018;Russell & Hatchwell, 2001). Kin recognition is achieved through association using learned vocal cues (Sharp et al., 2005) and helping decisions are also influenced by association during the non-breeding season (Napper & Hatchwell, 2016). Therefore, there are substantial fitness benefits to be gained by maintaining contacts with relatives during the winter, and perhaps also by avoiding dilu-  For example, if individuals do not use space exclusively, such as in polar bears Ursus maritimus (Ferguson, Taylor, Born, Rosing-Asvid, & Messier, 1999) and vultures (Coleman & Fraser, 1989), one would alter the advection term to include movement drivers which do not describe avoidance of other individuals of the same species, instead incorporating advection towards prey or desirable environment. Vultures use a central place which depends upon age so this would mean the advection term would include an attraction towards the central place, with the attraction parameter dependent on age. That said, some species have a similar correlation between relatedness and home range structure to long-tailed tits, despite being of rather different taxonomies, for example, bottlenose dolphins Tursiops truncatus (Frère et al., 2010) and giraffes Giraffa camelopardalis (Carter, Seddon, Frère, Carter, & Goldizen, 2013). Therefore, the models one might use in those cases may be very similar to the ones used here.
In summary, our finding that kinship influences space use is consistent with previous statistical home range analysis of our longtailed tit population Napper & Hatchwell, 2016), but here we have provided new insight into the avoidance mechanism from which these space use patterns emerge. Our study has also uncovered further drivers of space use, showing that (a) the distance from the woodland edge influences movement, (b) there is a memory-based avoidance mechanism between flocks and (c) flock size influences inter-flock movement decisions. More broadly, MHRA has potential to provide a wealth of understanding of drivers of movement and home range use of animal species. This study extends the usage of MHRA beyond scent-marking, central-place foragers and paves the way to understand the behaviours of a whole new range of taxa.  led the writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.

DATA AVA I L A B I L I T Y S TAT E M E N T
The datasets supporting this article are available from the Dryad Digital Repository https://doi.org/10.5061/dryad.fqz61 2jqj (Ellison, Hatchwell, Biddiscombe, Napper, & Potts, 2020).