‘Selfish herders’ finish last in mobile animal groups

Abstract

Predation is a powerful selective pressure and probably a driver of why many animal species live in groups. One key explanation for the evolution of sociality is the ‘selfish herd' model, which describes how individuals who stay close to others effectively put neighbours between themselves and a predator to survive incoming attacks. This model is often illustrated with reference to herds of ungulates, schools of fish or flocks of birds. Yet in nature, when a predator strikes, herds are often found fleeing cohesively in the same direction, not jostling for position in the centre of the group. This paper highlights a critical assumption of the original model, namely that prey do not move in response to position of their predator. In this model, I relax this assumption and find that individuals who adopt ‘selfish herd' behaviour are often more likely to be captured, because they end up at the back of a fleeing herd. By contrast, individuals that adopt a rule of ‘neighbour to neighbour alignment' are able to avoid rearmost positions in a moving herd. Alignment is more successful than selfish herding across much of the parameter space, which may explain why highly aligned fleeing behaviour is commonly observed in nature.

1. Introduction

Predation is a key driver of natural selection across taxa, and behavioural strategies to reduce the risk of predation are thus thought to be under strong selection. If a behavioural strategy to aggregate with others can mitigate predation risk in some way, the impact of predation could even drive the evolution of group living. Indeed, one prominent hypothesis is that aggregating and cover-seeking within the flock can reduce individual risk of predation at the expense of their neighbours [1–10]. Such, ‘selfish herd' behaviour is defined here as attraction toward the centroid of perceivable neighbours (following [6,9,11,12]). Despite the original author W.D. Hamilton's own acceptance that the hypothesis is not ‘universal or unipotent' [10] the current literature on ‘selfish herd' dynamics has been a story of overarching support for the hypothesis (e.g. [1–9], but see [12,13]). Nevertheless, there are two important issues with the current literature. Firstly, previous theoretical works seldom consider spatially explicit prey responses to predators [2–4,7,10,14,15] (but see [11]). Second, empirical work is often conducted in enclosed spaces, such as, but not exclusively, fish tanks [6,8,9,16] (but see [1,5]), where crowding might be the best option given the inability to mount an effective fleeing response. Arguably predator responsiveness and free-ranging space are more widespread in nature, and their consideration should be important in understanding the evolution of adaptive anti-predator behaviour.

A recent analysis of free-flying GPS-tagged pigeon flocks (Columba livia) found no evidence of ‘selfish herd’ behaviour (defined as greater centroid-attraction) in response to a flying robotic peregrine falcon (Falco peregrinus) [12]. Instead, the two largest effects on individual movement decisions were aligning with neighbour's orientations, and heading away from the predator's orientation [12]. These results may reflect a broader picture than just pigeon behaviour, or even bird behaviour. For example, both impala (Aepyceros melampus) and zebra (Equus quagga), two species of terrestrial mammal, respond similarly to predators by fleeing at high speed [17]. Also, in fish, sardines (Sardinella aurita) studied in open ocean environments were found to flee their sailfish predators (Istiophorus platypterus), which led to prey capture at the back of moving groups [18]. These examples indicate that fleeing—and perhaps neighbour to neighbour alignment—is paramount when predators strike, with perhaps little or no additional attraction to the centre of a group.

In order to relax a key and biologically unrealistic assumption from the original ‘selfish herd' model [10], that prey only respond to their neighbours' and not their predators' positions [10], I adopted the simulation model of Wood & Ackland [11] (building on Couzin et al. [19]) who modelled individuals responding to both their neighbours and their predator. The authors demonstrated that centroid-focused (‘selfish herding' flocks) and alignment-focused (‘fluid dynamic' flocks), were both evolutionarily stable strategies [11]. My main research question was to discover how, mechanistically, a centroid-focused (selfish herd) tendency could ever fail to be the fittest strategy—given that they would be expected to head to the group's centre, and benefit from safety relative to individuals which did not. That is, when considering that preferential attack on group edges, known as marginal predation, is well supported [7,15,20]. To this end, I set up models with a wider range of parameters and group compositions to determine whether and when ‘selfish herding' was less successful in response to predation, and the mechanism by which a centroid-focused tendency could possibly fail (see Methods; figure 1; also view the model and edit the parameters on the Shiny app at https://sankey-dan.shinyapps.io/selfishBoids/).

Figure 1.

(a) At prey capture, density of individuals in front-back and left-right distance to centroid (density scale defined in [21]). The group which consists of 50–50% centroid-focused to alignment-focused individuals is headed from left to right. Top (centroid-focused individuals), middle (alignment-focused individuals), bottom (predator). Coloured lines below are median x-axis values. (b) Alignment benefit represents how much alignment-focused individuals are safer than centroid-focused individuals; dotted line p = 0.05. Negative values represent centroid-focused being safer. Each data point (N = 81) represents N = 10 000 simulations for each set of parameters (see results split by parameter in figure 2). For visual communication, χ² results are transformed, where ‘alignment benefit' = sgn(x)·log(|x| + 1) (see Methods: Statistics). (c) Distribution of N = 10 000 group speeds (median centroid speed) for each composition at default parameters (see Methods). (Online version in colour.)

2. Results and discussion

Generally, in the model, centroid-focused individuals, which attempted to head to the centre of the flock, ended up at the back, because they lost ground on individuals which aligned with the groups' direction (figure 1a). Being at the back also brought greater risk—as the predator, more often than not, attacked from behind (figure 1a; also see [18]). "Finishing last", referred to in the title, therefore, reflects both the ‘selfish herders’ greater risk (figure 1b), and their positions at the back of the moving flock. Importantly, the modelled predator focused their movements toward their nearest target [11] (for empirical validation, see [1,22]), not toward the centroid of the group, which would presumably bias results towards disfavouring centroid-focus further.

Centroid-focused individuals were not always more heavily predated. In group compositions with 90% centroid-focused individuals, most of the parameter space shows significantly negative ‘alignment benefit’ (figure 1b; also see [11] where centroid-focus was evolutionarily stable). The reason centroid-focus can be favoured in highly attracted groups is likely due to the lower relative speed of these highly attracted groups (figure 1c), as it can help to crowd to the centre in slow-moving groups [4,10,11]. Thus, alignment-focus can only trump centroid-focus in groups which flee the predator at speed, which I have argued, in nature may be a more common response to predation (e.g. in birds [12], mammals [17] and fish [18]). The high variance of ‘alignment benefit' within different group compositions (figure 1b) also highlights the importance that we consider species' ecology—and parameterise models accordingly—when moving forward on this topic [13].

I now move my focus to tests of discrepancies between my results and the current literature. A first discrepency is that ‘selfish herd’ dynamics were observed in a free ranging (and thus ‘borderless') system, in wild fiddler crabs (Uca pugilator). I suggest this result could reflect that fiddler crabs are slower than their clapper rail (Rallus longirostris) predators. Although data are unknown for speeds, it is highly likely that a 37 cm bird [23] can outpace a 0.7–1.6 cm crab [24], given body size to speed scaling principles [25]. I suggest that where predators are much faster, and/or more manoeuvrable [26], prey cannot produce an effective fleeing response. This is consistent with the model presented here, as greater predator speeds decreased the benefit of ‘alignment-focused' movement rules (figure 2a).

Figure 2.

Alignment benefit for compositions with different (a) predator speeds, (b) border avoidance conditions and (c) all other parameters not directly referenced in main text. All parameters set at default except the parameter explored on the x-axis of each panel (table 1 and Methods). For visual communication, χ² results are transformed, where ‘alignment benefit' = sgn(x) · log(|x| + 1) (see Methods: Statistics). (Online version in colour.)

I suggested that many findings validating ‘selfish herd' were found in enclosed spaces, limiting the ability of alignment-focused individuals to flee. I tested this assertion by including impassable borders at the edges of the grid space used in the model, with simple border avoidance rules (see Methods). I did find a drop in ‘alignment benefit' in groups composed of 10% centroid-focused individuals (figure 2b). Yet, in other conditions alignment was favoured in systems with borders. Therefore, the border avoidance rules I employed cannot explain the discrepancy between the model presented here and observations of ‘selfish herding' in captive animals [6,9]. Observationally, the modelled prey ‘circle' the grid space when border avoidance rules are employed (D.W.E.S. 2022, personal observation), and return to a group resembling their state in a borderless model, so further work using more realistic border avoidance strategies [27,28] would be useful to elucidate the effect, if any, of borders. Nevertheless, identification of ‘selfish herd' dynamics from previous studies in enclosed spaces should be interpreted carefully. The question empiricists should ask is whether their animals can effectively ‘align and flee’ within the boundaries provided, or whether highly aligned fleeing behaviours are precluded by the environmental constraints.

In this study and in [11], predators were often less successful when hunting (i) alignment-focused groups than (ii) centroid-focused groups (electronic supplementary material, figure S1), perhaps owing to the faster group speeds in the former (figure 1c). This suggests a role for both between-group and within-group selective forces: between-group selection favouring individuals in highly aligned groups, and within-group selection favouring either phenotype, depending on the within-group composition (as in figure 1b). Of course, this requires that migration between groups is—at least partially—limited [29], but stable groups are common in nature [30–33]. In such a multi-level selection framework, it is possible for alignment-focus to evolve even in populations of centroid-focused individuals, when it is disfavoured within the group. Here, despite their lower fitness at the within-group level, alignment-focus could evolve via its greater fitness at the between-group level. Simulations, as utilised here, but with between-group and within-group evolutionary components could prove useful towards an understanding of how these (potentially opposing) selective forces collide.

In sum, I have provided evidence which partially deconstructs the ‘selfish herd' argument and provides a mechanism by which it can sometimes fail. Furthermore, these results suggest that when assessing whether centroid-focus or alignment-focus would be a more adaptive strategy, the ecology of the system matters (e.g. predator/prey speed ratios). Regarding the evolution of group living, this work does not refute the role of ‘selfish herding' in the formation of groups, and even supports its evolutionary stability once formed. Nevertheless, to reinforce these conclusions, it is important for future work to also consider using more advanced or realistic interaction rules [34,35], state-based decision making: using different interaction rules based on flock position or ecological context (e.g. see [36]), and a multi-level selection framework to disentangle differences which stem from between-group and within-group selective forces. Altogether, the present model is a simplistic yet explanatory mechanism to better describe the cohesive fleeing behaviour commonly observed in nature [12,17,18], while additionally challenging assumptions of prey motives in the field of predator–prey dynamics.

3. Methods

(a) . Model

Firstly, N prey individuals, consisting of proportion p·N centroid-focused and (1 – p)·N alignment-focused individuals, are set up in infinite grid space (x and y = 0−100, with periodic boundary conditions). Each individual starts on the grid with a random position chosen from a uniform distribution, speed (v) = 1 unit per timestep (τ), and with directional heading chosen from a uniform distribution of all possible angles. Before the predator appears (at τ = 300), individuals respond to (1) their current velocity, with noise around this velocity (µ = current velocity, s.d. = σ; see [11,19]), and (2) desired directional movements based on other group mates' positions and/or velocities within a perception distance range R_s (see definitions below). After the predator appears, individuals also (3) repel from the position of the predator, if the predator is not outside of the prey's maximum perception range R_s (following [11]).

A predator starts at τ = 300 with position uniformly distributed in x and y coordinates (following [10]), a random directional heading, and a speed of ν_p. As predators are generally faster than prey, but less manoeuvrable [17,26], I set values where ν_p > ν and φ_p < φ. Predators attract towards the nearest individual (see [22] for empirical evidence), following attraction rules detailed below. Predators do not have a perception range in this model, which was a decision to reduce parameter space for investigation. Thus, the predator's inclusion at τ = 300 can be seen as the moment the predator had spotted prey and started the chase. The predator is said to have caught the prey when the distance from predator to prey is less than the radius of repulsion R_r (following [11]). I applied less noise to the movements of predators relative to prey (i.e. σ_p << σ). This was based on initial fine-tuning of simulations, whereby with a larger σ_p the predator did not catch prey easily despite close proximity.

(i) . Movements with respect to neighbours

Following [11,19], the highest priority neighbour-to-neighbour interaction for all individuals was to move away from others within a ‘zone of repulsion' z_r, demarcated by radius R_r. This is achieved by setting a desired heading with respect to neighbours' (d_i) that always equals desired repulsion d_r if any neighbours were in this zone. Desired repulsion d_r is the angle directly opposite to the mean centre of mass of all neighbours within radius R_r; this can be seen as a collision avoidance strategy. Note, this repulsion is not necessarily more desirable than avoiding a predator, as thus far, this strategy only includes neighbour movements. If no neighbours are within z_r, individuals' sample for neighbours within larger radius R_s which is their maximum sensory range. Between R_r and R_s I define a desired heading d_a toward the centre of mass, and desired heading d_o towards the mean velocity, of all sampled neighbours. If no neighbours are found in the radius R_r, alignment-focused individuals' desired direction d_i = 0.2d_a + 0.8d_o and for centroid-focused individuals' d_i = 0.8d_a + 0.2d_o. If d_i is greater than the maximum turn angle φ, then d_i = φ.

(ii) . Movements with respect to the predator

Repulsion from the predator d_p is defined as a desired turn away from the position of the predator. Desired direction after spotting a predator d_{i + p} is given by ((Ω − 1) x d_i) + (Ω x d_p), where Ω is the weight of predator avoidance. E.g. if Ω = 0.5, equal attention will be paid to group mates and the predator. A value of 0.5 was chosen as the default parameter, following roughly equal strength of neighbour-focused and predator-focused rules in [12].

(iii) . Movements with respect to borders

To test the hypothesis that borders disfavour alignment rules, I ran simulations with and without impassable borders. I code this as individuals having a desired turn away from borders d_b. The strength of this desired turn is dependent on the distance to the border. If an individual is in contact with a border, their previous desired direction d_i+p is undermined, so d_i+p+b = d_b. Outside of a border avoidance distance γ, individuals pay no attention to the border, so d_i+p+b = d_i+p. Between these distances (0 – γ units), the weight of d_b on d_i+p+b decreases linearly. If any individuals including the predator passes borders, I apply reflective boundary movements, with rules that if an individual would have passed the x border, the x component of their velocity was multiplied by −1, and similarly if they would have passed the y border, the y component of their velocity was multiplied by −1. These rules avoid collision and lead to movements away from the border.

(iv) . Predicting future movements

I also consider that both predators and prey might head to a future position i.e. where they expect their prey, or neighbours will be in a future timestep [37]. I included the opportunity for both predators and prey to update desired directions into a projected image of where individuals would be if velocity and turning angle of perceived individuals remained constant over the next λ or λ_p iterations of τ, for prey and predators, respectively (see [12] for validation of this method).

(b) . Statistics

Statistical analyses were conducted on predator captures. A results variable of either (i) no individual caught, (ii) centroid-focused caught or (iii) alignment-focused caught, was recorded for each simulation in each parameter set. There were N = 10 000 simulations run for each unique set of parameters (N = 81; table 1), totalling N = 810 000 simulations across the whole study. Using only the simulations that ended in capture (5.0% ended in escape), I ran a two-tailed χ² test to determine which variant was safer. The null hypothesis being that proportion p centroid-focused individuals would be expected to be caught if there was no difference between the variants. χ² is reported, but converted to an ‘alignment benefit' metric, whereby χ² is multiplied by −1 if centroid-focused individuals were the fitter variant, and further transformed by a log modulus function as follows: y = sgn(x)·log(|x| + 1), to minimize extreme positive and negative values for clearer graphical interpretation (e.g. see figures 1 and 2).

Table 1.

Parameter values, default values are bold. Other values in non-bold are tested against default values for all other variables. Only p is tested against all other parameters (all are bold). In total, there are 81 distinct sets of parameters using this method. Parameters are arbitrary, based on values in [11,19], and initial runs to produce visually realistic flocks; researchers are encouraged to explore outcomes under different parameter values using the R-Shiny [38] web app available at https://sankey-dan.shinyapps.io/selfishBoids/.

parameter	shorthand	values
number of prey individuals	N	10, 20, 30
proportion of centroid-focused individuals	p	0.1, 0.5, 0.9
proportion of alignment-focused individuals	N.A.	(1 – p)
radius of repulsion (units)	Rr	1
maximum sensory range	Rs	20, 25, 30
maximum turn angle for prey (rad / timestep)	φ	0.1, 0.5, 1
maximum turn angle for predator (rad / timestep)	φp	0.01, 0.1, 0.2
speed of prey (units / τ)	ν	1
speed of predator (units / τ)	νp	1, 1.5, 2, 3, 4, 5
noise—prey movement (s.d.)	σ	0.05, 0.1, 0.2
noise—predator movement (s.d.)	σp	0.005, 0.01, 0.02
anti-predator preference	Ω	0.1, 0.5, 0.9
prey future prediction of movements (τ)	λ	1, 2, 5
predator future prediction of movements (τ)	λp	1, 2, 5
border avoidance distance (units)	ϴ	NA, 2, 5, 10

Data accessibility

The data are provided in electronic supplementary material [39]. Code is freely available to download from https://github.com/sankeydan/selfishBoids.

Authors' contributions

D.W.E.S.: conceptualization, data curation, formal analysis, investigation, methodology, visualization, writing—original draft, writing—review and editing.

Conflict of interest declaration

I declare I have no competing interests.

Funding

No funding has been received for this article.