Out of all the complex phenomena displayed in the behaviour of animal groups, many are thought to be emergent properties of rather simple decisions at the individual level. Some of these phenomena may also be explained by random processes only. Here we investigate to what extent the interaction dynamics of a population of wild house mice (Mus domesticus) in their natural environment can be explained by a simple stochastic model. We first introduce the notion of perceptual landscape, a novel tool used here to describe the utilisation of space by the mouse colony based on the sampling of individuals in discrete locations. We then implement the behavioural assumptions of the perceptual landscape in a multi-agent simulation to verify their accuracy in the reproduction of observed social patterns. We find that many high-level features – with the exception of territoriality – of our behavioural dataset can be accounted for at the population level through the use of this simplified representation. Our findings underline the potential importance of random factors in the apparent complexity of the mice' s social structure. These results resonate in the general context of adaptive behaviour versus elementary environmental interactions.
In a famous passage of his book The Sciences of the Artificial [1] the sociologist Herbert Simon considers the winding, weaving path of an ant as it makes its journey home across the rugged landscape of a wind- and wave-beaten sandy beach. He notes that, whilst the homebound ant has a clear destination, its progression along the path that leads to it is far from a straight line, due to the numerous obstacles encountered on the way. The example inspires in him this startling observation: An ant, viewed as a behaving system, is quite simple. The apparent complexity of its behavior over time is largely a reflection of the complexity of the environment in which it finds itself. More than fifty years after the first mention of the parable of the ant, the general question is still very much alive. There is indeed an ongoing debate today as to what aspects of the behaviour, and especially the social behaviour, observed in an animal species can be explained as a specific adaptation versus an emergent property of simple behavioural rules when individuals interact with their environment. If some emerging properties of a group' s social structure result from more simple behavioural mechanisms, it may be that what is thought to be an explicitly social behaviour does not require specific selective adaptations. Although animal groups often display astonishing emergent patterns in their collective behaviour [2], recent research has shown that much of the complexity of some natural phenomena can be directly attributed to the collective dynamics of simple, self-organised processes and individuals [3]–[5]. In recent years, assumptions of random behaviour have been discussed at length in contexts such as animal movement and foraging [6]. There have also been specific investigations on the description of collective motion in biological systems based on stochastic processes, as exemplified by the concept of Brownian agents [7], [8]. This framework has proven to be a versatile and practical approach to describe collective patterns of movement at different organisational levels, for systems ranging from bacteria [9] to crustaceans [10] and social insects [11]. However, clearly the behaviour of animal groups is not limited to collective motion, but also includes complex social interactions between individuals within the group. These interactions are often of a high-order and individually-differentiated nature, as illustrated by the long history of studying social relationships in animal species with high levels of cognitive development [12], [13]. Despite the efforts mentioned above to describe collective behaviour by means of stochastic forces, little attention has been paid to the importance of randomness for the emergence of complex social patterns in animal groups. Here we use as a case study data from a population of wild house mice (Mus domesticus), social rodents characterised by cooperative breeding, polygynandry, territorial defence against non-group members, high skew in reproductive success and rather short mean life expectancy in both sexes [14]–[16]. This may have led to high flexibility in behaviour and social organisation. Yet exactly what aspects of the social structure of house mice can be explained by the self-emergent properties of collective random behaviour is unknown. In this paper, we first develop the assumption of random behaviour by creating a perceptual landscape that mixes purely diffusive motion and advective-diffusive Brownian bridges to describe the movement of individual animals. We then proceed to build a simple stochastic model that implements the assumed movement dynamics, and discuss its accuracy in accounting for specific characteristics of the social structure of a population of house mice. The novelty of our approach lies in the application of a dynamical perspective on habitat use to the investigation of social structure in animal groups as an emergent property of random interactions.
Animal experimentation was approved by the Swiss Animal Experimentation Commission (Kantonale Tierversuchskommission, no. 26/2002,210/2003,215/2006). We study an established free-living population of wild house mice in a barn outside of Zurich, where mice can freely emigrate and immigrate. In the barn, the mice nest in artificial nest boxes, and are provided with straw as nesting material, and food and water outside the boxes. At four to six week intervals, comprehensive trapping is conducted to monitor the population, and adult mice are implanted with a transponder (RFID tag) so that they are individually identifiable. Transponder readers are installed in the tunnels that provide entrances to the nest boxes (two readers per nest box make it possible to distinguish a mouse that leaves a box from a mouse that enters a box); these readers connect to a computer and continuously track movements into and out of nest boxes. This provides 24-hour information on movements and social affiliations of adult mice (for a detailed description of the barn population and the methods used, see [17]). Data collection started in May 2007 and totals around 29 million individual recordings as of June 2012. Here we analyse the period ranging from Jan. 1,2008 to Dec. 31,2009. Our 2-year dataset covers 11' 259' 557 location records for 508 mice, accounting for 1' 376' 720 stay events in 40 nest boxes, and leading to 1' 064' 695 one-to-one encounters inside nest boxes, whose frequency, context and duration we use as a proxy for the characterisation of social interactions (the full Dataset S1 is available to download from the Supplementary Material page). Figure 1 shows the geographical positioning of the nextboxes, as well as the heterogeneity of their occupation pattern: indeed, the total occupation duration per box ranges from 264 to 22' 332 hours (the lowest figure may, however, be attributed to a malfunctioning RFID antenna; the maximum value is longer than our study period because some stays can overlap when two or more mice stay simultaneously in the same nest box). This aggregated view allows to identify the “hot spots” of the barn and the busiest routes between nest boxes, from a static perspective on the behavioural data. It also shows how geographically clustered the traffic between nest boxes is, as a result of the partitioning and the obstacles of the barn. Evidently the physical environment of the mice affects their movement, which in turn has an impact on their social contact patterns. However, the view presented in Figure 1 is insufficient to characterise the dynamics of movement of individual mice, and the link between these dynamics and their social behaviour. Indeed, it focuses on aggregated properties of the study system rather than dynamical ones.
Through the description of the perceptual landscape, we have developed the assumption of simplistic individual motion to reconstruct the environment of a wild house mouse. We now test this hypothesis by implementing the assumptions of the perceptual landscape in an elementary model of collective behaviour, in which all agents are governed by the principles of random motion we have introduced previously. We make the assumption that through the collective behaviour of those agents, whose complexity lies far below that of real mice, we can reproduce some of the global behavioural patterns we observe in the barn.
In this paper we have developed the assumption of simple stochastic processes as a driving force for social interaction in a population of wild house mice. We introduced the notion of perceptual landscape, which maps the patterns of movement of mice between nest boxes in a barn into the motion of stochastic particles within a potential field. We are well aware that our model ignores the fact that such patterns have resulted from natural selection and adapt mice to their environment. Instead, we are interested in analysing whether and to what degree a general movement pattern alone can reflect important characteristics of the spatial and social behaviour of free-living mice. Our approach integrates two important facts, often neglected when mapping the home range or territory of wild animals: (i) the movement from one sampled location to another ought not to be thought of as a straight line, but instead may be better approximated by planar diffusion, and (ii) when an animal is resting in a safe area like a nest box (or generally visiting an area with a strong potential of attraction) it is less likely to exit and move on than if it were at another point within its home range. The parameters defining the landscape were obtained from the recorded data. The method has only one user-defined parameter, namely the assumed travelling speed of an animal. However, we detail a way to obtain this parameter from the data. Here we assumed a constant speed for a mouse moving along a Brownian bridge between two nest boxes. This may seem rather slow but is based on the fact that house mice spend a considerable part of their time outside nest boxes not moving, but instead feeding/drinking, marking their territory or partaking in social activities or territorial defence. Moreover, the mean radial diffusion coefficient obtained from this value of is; this amounts to exploring an area of slightly less than one centimeter per second in each direction, which is a sensible figure in the case of small animals like house mice. As a tool to study animal behaviour, the perceptual landscape method scales linearly with the number of active paths, i. e. pairs of locations with a large number of transits. Therefore we argue that the method could scale properly to much larger systems, and provide a new way to analyse the spatial behaviour of animals on a large scale. Interestingly, the perceptual landscape contains several of the structural features observed in the real landscape of the mice. This confirms previous observations [30], [31] that house mice use these structures to build up their own representation of the environment and navigate across it. From a formal point of view, it also reveals that the assumptions we made on the movement of random particles across the landscape yield meaningful behavioural patterns. Indeed, these patterns integrate important aspects of the decision rules guiding mice when they use their environment and defend their territory. Figure 3 illustrates the use of such a perspective for the study of individual home ranges (panels B and C), as well as the movement patterns of the whole population studied (panel A). We observed that the perceptual landscape is similar, but not identical, to the physical environment of the real mice. For example, the perceptual “wall” corresponding to the divider between segments A and B of the mouse barn is only weakly expressed. This may indicate that some mice regularly use nest boxes on both sides of the divider, and travel between them. Conversely to some physical features of the environment disappearing, some are overly expressed. For example, the entrance to the barn (see inlay of Figure 3A) is an area that mice could use, as it is open, but that they tend to avoid due to the presence of experimental equipment and the absence of protected nesting sites. As a result, the whole area appears in the perceptual landscape as a raised plateau, demonstrating little effective utilisation of that space. Generally, such discrepancies between the perceptual and the physical landscape may result from differences in the micro-environment of the animals. In practice, the mouse barn is not a homogeneous environment, but differs locally in humidity and temperature, in the degree of protection perceived by the mice (suitable hides or other spatial structures outside the nest boxes), in the exposure to popular traffic routes used by many individuals, in the availability of food and water in close proximity to a nest box, or in the amount of light (mice tend to avoid bright areas [14]), etc. In addition, the movement of mice between nest boxes will be influenced by their social environment. Since the perceptual landscape integrates all such factors, it may be seen as a cartographic tool of a much higher precision than a standard schematics or map of the environment of an animal population. In other words, this landscape is the combination of all dimensions that animals perceive in their environment (be them physical boundaries, temperature, humidity or presence of conspecifics). This is especially important when considering that many animals view their environment in a way that is different from the way we see it. Indeed, house mice have poor visual acuity and their world is “dominated by smell” [14]. The representation of an animal' s environment by simply mapping it as we would map our environment may thus be misleading. It is interesting to note that the perceptual landscape we described in this paper is a static construction, which represents the collective behaviour of the animals from a quasi-stationary perspective. As such, it results from the aggregate behaviour of the individuals of a population rather than adapts to it. However, the construction of this landscape by the animals is arguably a dynamical process: house mice, for example, alter their home range in response to the nearby presence of social partners [31]. In order to account for this dynamical aspect of the formation of the landscape, the Brownian agent framework [7], [8] may be more suitable than simpler stochastic particles: indeed, Brownian agents move across an adaptive landscape, which builds up over time as a result of their behaviour. The Brownian agent paradigm constitutes a natural framework to study non-equilibrium systems, such as a population of interacting individuals. As such, this framework could be a logical extension to the perceptual landscape technique. In a second step, we tested the accuracy of the assumptions of our landscape. To this purpose, we implemented an agent-based model of the behaviour based on the landscape assumptions and tested its results against real observational data. As we assumed a very simple individual behaviour (stochastic motion) for the mice moving across the landscape, a simple approach was sufficient to reproduce it. We described the transitions between nest boxes by a set of stochastic processes, or zero-order Markov processes, effectively representing the set of nest boxes as a Markov chain (this comparison is developed in Text S1). In this description, the escape rates from each state (inside a nest box or moving between two nest boxes) were calculated from the real aggregated data. The underlying assumption of this Markov approach is that the process has no memory, the transition probability being only dependent on the current state. In this paradigm, individual mice are particles travelling along the Markov chain and all follow the same rules of motion. Each particle can be thought of as representing the “average mouse”, an individual who behaves as a composite of all the mice from the real population, without particular individual characteristics. We aimed to study how such an approximation performs in a social context, or how well it may reproduce the observed patterns of social encounters in a wild house mouse population. We used in our simulation as many average mice as the average number of RFID-tagged mice in the barn over the two-year study period. It should be noted that this approach implies no a priori assumption on the importance of the social interactions that can occur each time two mice meet inside or outside of a nest box, although it is well-known that social aspects play a crucial role in the behaviour of house mice, especially female [30], [32]–[35]. Remarkably, we found that this simplistic, randow walk-like approach is sufficient to reproduce some features of the nest box occupation patterns, both at the population and the individual levels. In other words, the collective dynamics of the population as a whole, with its intrinsic fluctuations (birth and death cycle) and interindividual differences, may be well approximated by the behaviour of the average mouse. This is obvious from the match between experimental and simulated data in occupation density and transit frequencies between nest boxes. This accurate match should, however, come as little surprise: indeed, these features are aggregated observations on the behaviour of the whole population, and precisely those whose estimate we used to calibrate the stochastic model. Of more interest is the comparison between the model output and the observational data with regard to higher-level social features. The statistical test we ran on the simulation results amounts to asking whether the average mouse, moving randomly across the perceptual landscape, has a social behaviour (defined as its pattern of encounters inside nest boxes) consistent with the social behaviour of a real mouse from our study population. In agreement with the results obtained at the population level, we found that at the individual level most of the social features in the average mouse' s behaviour were compatible with the behaviour of a real mouse, with the exception of the territorial aspect (expressed in the number of nest boxes used per day). This is especially interesting when considering that we excluded any influence of conspecifics on an individual' s probability to enter and/or stay in a nest box. Yet, the patterns of social interaction (number of social encounters per individual, number of social partners, or duration of a social encounter) did not differ significantly from those observed in the population of real house mice. It is important to note, however, that mice tagged in the study population are only adults, which typically had already established their territory and integrated in a social group. The behaviour of young, dispersing individuals that still move between groups or territories is thus underrepresented, although it may differ. Once integrated in a social group, however, mice seem to regularly meet with all group members, without pronounced individual preferences. Nevertheless, many real mice had fewer social partners and fewer social encounters than the average mouse from our model (although the differences were not statistically significant from the global population). These discrepancies may, again, reflect territoriality, social preferences and differences in reproductive dominance [17]. Indeed, from a behavioural perspective, the assumption made for the average mouse, whose behaviour is the average of that of all other conspecifics, explicitly ignores any variability among individuals. However, it is well established – not only for house mice – that individuals within species and populations vary in their behaviour according to their sex, age, dominance or reproductive status, or personality. Even within individuals, behavioural performances can change over time due to individual experiences and modifications in physiology (hunger, hormones, etc.). All such individual variability can explain differences in competitiveness, aggressiveness, social tolerance, or boldness, which will affect individual preferences towards conspecifics as well as towards nest boxes or other spatial structures. Furthermore, hindering non-group members from entering own nest boxes is very important due to the tendency of mice of both sexes to kill non-offspring. Evidently, the omission of all such factors in the model leads to different movement and social interaction patterns, changing the structure of the social network. Of course, it is highly implausible that a random particle may faithfully mimic a living mouse, and the simple modelling approach we have presented could not pretend to fully reproduce the complex, dynamical social network of a population of wild house mice. Yet, it points to the fundamental importance of simple, universal processes in the establishment of such a structure, and generally shows that the collective dynamics of stochastic processes are sufficient to reproduce some properties of a social system. This is clearly encouraging in the search for a more advanced cross-species model that could lead to a broader understanding of animal sociality. We proposed the perceptual landscape as a framework in which the complexity of the individual interactions is transferred to that of the environment (the landscape). In doing so, we effectively simplified the analysis of collective social behaviour by moving from the study of many interacting individuals to the study of a single landscape object, whose properties can be quantified against those of the external environment (such as its physical structure). Moreover, this approach provided us with a null model for the social behaviour of the individuals in the landscape, which we could use to characterise the social network built by a population of average mice. In this regard, the results of the simulations presented in Figure 4 represent a good null assumption for sociality in animal groups. Notwithstanding the apparent performance of such a simplifying approach, it should be noted that there are key differences at the individual level between the average mouse and an individual from the real population. This points to the existence of more sophisticated rules governing animal behaviour than merely random principles, as can be expected from complex creatures such as social mammals. We have applied methods from statistical physics to the understanding of the randomness underlying seemingly complex spatial and social animal behaviour. It appears that at least some elements of animal social behaviour can emerge from the collective dynamics of independent random processes. This complements recent work [36] which showed that such methods can be efficiently used to study the emergence of territoriality in animal populations. Such findings ultimately may parallel other examples of self-organisation in contexts as diverse as evolution [37], speciation [38] and even human economic behaviour [39].
From the synchronised beauty of fish schools to the rigorous hierarchy of ant colonies, animals often display awe-inspiring collective behaviour. In recent years, principles of statistical physics have helped to unveil some simple mechanisms behind the emergence of such collective dynamics. Among the most elementary tools used to explain group behaviour are random processes, a typical example being the so-called “random walk”. In this paper, we have developed a framework based on such random assumptions to study the spatial and social structure of a population of wild house mice. We introduce the concept of perceptual landscape to describe the spatial behaviour of animals, whilst including all sensory and social constraints they are subject to: the perceptual landscape effectively maps the environment of animals as they perceive it. By applying our assumptions to a multi-agent model, we are able to reveal that much of the high-level social behaviour observed in the mouse population can indeed be explained through the many interactions of randomly moving individuals. This raises the question of how much of what we often regard as complex natural phenomena may, in fact, be the result of exceedingly simple forces.
physics
stochastic processes
animal behavior
mathematics
statistical mechanics
ecology
evolutionary biology
biology
probability theory
behavioral ecology
How Random Is Social Behaviour? Disentangling Social Complexity through the Study of a Wild House Mouse Population