We extended our model, StarDisplay, with two types of escape reactions and a transmission mechanism. We performed our experiments of escape in this extended model.
The model
General outline
Because flying implies movement in all directions, we developed our model in three dimensions. The behaviour of each individual in StarDisplay is based on its cruise speed, its social coordination (depending on the position and heading of its nearby neighbours), its attraction to the roost (site for sleeping), the simplified aerodynamics of flight which includes banking while turning, and reaction time (Hemelrijk and Hildenbrandt 2011). One of the sources of error is that we update the location and heading of each individual at shorter intervals than the interval of the reaction time. For other sources of error, see the random error (Equ S14) in the description of the model in the supplementary material and in our former work (Hemelrijk and Hildenbrandt 2011). The results of the model are robust against such sources of random error. We model social coordination in terms of (social) forces in line with studies by others (Helbing and Molnar 1995; Couzin et al. 2002; Hemelrijk and Hildenbrandt 2008). In the present paper, we have added two types of escape manoeuvre that preserved flock shape: 1) the zigzag-manoeuvre and its halved version, a ‘zig’-manoeuvre of rolling sideward and back (Fig. 1) involving a change of orientation and 2) a manoeuvre involving an acceleration forward in the flock (‘speeding-up-forward’) causing a change of density.
Further, we made flying more natural by increasing the tendency of individuals to pitch and representing head nystagmus as is observed in real birds. Head nystagmus implies that birds stabilise their visual system by keeping their heads still, while banking their body sideward (Warrick et al. 2002). For this, we modelled the head system separately from the body system.
During their normal reaction time of 76 ms (Pomeroy and Heppner 1977) (Table 1), birds do not update their environment while they are flying still. Therefore, in the model this is an important cause of error in their behavioural response.
Table 1 Parameters of escape reactions in the model
We use SI units and choose real parameter values where available (see Parameterization and Suppl. material, Table S1). For details of the model and the basic behavioural rules, see Supplementary material.
Initial condition, escape behaviour and computational experiments
In our earlier simulations, the attraction to return to the site for sleeping (roost) induced many turns of the flock (Hildenbrandt et al. 2010; Hemelrijk and Hildenbrandt 2011, 2012). For studying the agitation wave, we want a flock that does not change its shape and, therefore, does not turn. Therefore, we omitted the attraction to a roost by using a roost or sleeping site of infinite size. The simulation started with a single flock of randomly positioned individuals in a small volume of space, at an approximately default average distance to the nearest neighbours. In order for the normal flocking behaviour to emerge, data collection started after an acclimatisation period of 50 s (Table 1).
Note that the surprise attack is the most common attack strategy of falcons on flocks of starlings (Rudebeck 1950; Zoratto et al. 2010). This is the attack we used.
Since it made no difference in speed of transmission whether we attacked individuals at the rear end of the flock, the side or the front of it, we confined ourselves to attacks from the back because flocks may often face away from the predator when under attack. We need to find individuals at the rear in the model, which is setup in a Euclidian 3D space (based on three perpendicular axes: x, y, and z). The location of the body of each individual relative to the origin is indicated by a vector p. The orientation of the body is given by its forward direction, e
x
, its sideward direction, e
y
, and its upward direction, e
z
, which may change by rotating around these three principal axes, e
x
, e
y
and e
z
(roll, pitch and yaw) (Fig. S2). The individual i at the back of the flock, i
rear (Eq. 3), is found by the lowest value of the dot product between the position of each individual relative to the centre of gravity of the flock (which is the average position of all flock members, Eq. 1) and the average direction of movement of the flock (Eq. 2).
$$ \overline{p}=\frac{1}{N}{\displaystyle {\sum}_i}{\boldsymbol{p}}_i\kern2.12em \mathrm{Flock}\kern0.5em \mathrm{centre}\kern0.5em \mathrm{of}\kern0.5em \mathrm{gravity} $$
(1)
$$ \overline{e_x}={\displaystyle {\sum}_i}{\boldsymbol{e}}_{\boldsymbol{xi}}/\left|{\displaystyle {\sum}_i}{\boldsymbol{e}}_{\boldsymbol{xi}}\right|\kern1.12em \mathrm{Direction}\ \mathrm{of}\ \mathrm{flock}\kern0.5em \mathrm{movement} $$
(2)
$$ {i}_{\mathrm{rear}}=\left\{i\in N;\ {\boldsymbol{p}}_i\overline{e_x}\kern0.5em \mathrm{is}\kern0.5em \mathrm{minimal}\right\}\kern1.12em \mathrm{Hindmost}\kern0.5em \mathrm{individual} $$
(3)
where N is the number of individuals in the flock, p
i
indicates the position of individual i, and e
xi
represents the forward direction of individual i.
We made the hindmost individual, i
rear, escape by one of the three escape manoeuvres, sidewards and back, namely zigzag and zig, or accelerate forward (called speed-up-forward). Because the social coordination (Supplementary material, Equ S5–S12) was modelled by us and others based on social forces (Helbing and Molnar 1995; Couzin et al. 2002; Hemelrijk and Hildenbrandt 2008), we used social forces for the escape manoeuvres also.
In case of the zigzag-manoeuvre, the individual moves sideward (by rolling), back and sideward to the other side again (Fig. 1a).
$$ \begin{array}{ll}{\boldsymbol{f}}_{\mathrm{zz}}={w}_{\mathrm{zig}}{\boldsymbol{e}}_{\boldsymbol{y}};\hfill & \hfill 0<t<{T}_{\mathrm{zside}}\hfill \\ {}{\boldsymbol{f}}_{\mathrm{zz}}=-{w}_{\mathrm{zig}}{\boldsymbol{e}}_{\boldsymbol{y}};\hfill & \hfill {T}_{\mathrm{zside}}<t<{T}_{\mathrm{zside}}+{T}_{\mathrm{zback}}\hfill \\ {}{\boldsymbol{f}}_{\mathrm{zz}}={w}_{\mathrm{zig}}{\boldsymbol{e}}_{\boldsymbol{y}};\hfill & {T}_{\mathrm{zside}}+{T}_{\mathrm{zback}}<t<{T}_{\mathrm{zside}}+{T}_{\mathrm{zback}}+{T}_{\mathrm{zside}}\hfill \end{array} $$
(4)
where f
zz is the force. As a consequence of it, the bird moved T
zside seconds to the side and T
zback seconds back again and again T
zside seconds to the other side and T
zback seconds back again (Table 1).
The zig-manoeuvre represents only half of the zigzag, thus, rolling sideward T
zside seconds and T
zback seconds back again (Table 1). This causes a small sideward shift of the individual to the left (Fig. 1b).
The manoeuvre of speeding-up-forward is modelled by
$$ {\boldsymbol{f}}_{\mathrm{sf}}={w}_{\mathrm{sf}}{\boldsymbol{e}}_{\boldsymbol{x}};\kern2em 0<t < {T}_{\mathrm{sf}} $$
(5)
and involves the force f
sf that causes the individual to accelerate for T
sf seconds forwards (Table 1). After each escape event, the individual recovered during a short refractory period of T
rp seconds (Table 1).
We investigated whether transmission of information about escape in the model happened either by individual adjustment of their movement to a close-by escape manoeuvre of another individual or happened by individual recognition of the escape manoeuvre followed by repeating it (Potts 1984). Such recognition and identification of an escape manoeuvre takes time, which we called cue identification time T
cue. In line with studies of others (Bode et al. 2010), we assumed this cue identification time to be shorter than the normal reaction time. The number of neighbours that an individual scanned for a potential escape manoeuvre is labelled as the range of repetition, RangeRep (Table 1).
Parameterization
We represented birds in the model by an ‘arrowhead’ of similar aspect ratios of wing span versus length and height as the starling (Fig. 2a, Table S1) (Videler 2005).
We have parameterized individuals in the model to realistic data of birds (weight, cruise speed, etcetera), especially of starlings, see supplementary material Table S1 and our earlier version of StarDisplay (Hildenbrandt et al. 2010; Hemelrijk and Hildenbrandt 2011). Roll rate and banked turns were tuned to those observed in movies of starlings in that they rolled into the turn faster than that they rolled back (Gillies et al. 2011), roll rate is within the range measured for other species (Gillies et al. 2008, 2011) and banked turns resemble empirical data in that individuals lose height during turns (Pomeroy and Heppner 1992; Gillies et al. 2011).
Because agitation waves have particularly been observed in flocks of large sizes (Procaccini et al. 2011), as a default flock size we used 2000 individuals (Fig. 3) (Ballerini et al. 2008b) with an average distance to their nearest neighbours of 1.3 m resembling empirical data (Major and Dill 1978; Ballerini et al. 2008b). When an individual observed in its range of repetition another individual displaying an escape manoeuvre, it was made to repeat this manoeuvre. We choose a topological range of six to seven closest neighbours to repeat the escape manoeuvre from because this is also the topological range observed empirically during coordination in a flock in the absence of predation (Ballerini et al. 2008a). Here, the topological range included all six to seven nearest neighbours outside the blind angle at the back (Table S1). Empirically, in large-scale stereometric analyses, this number has been established during normal coordination in a flock (in the absence of a predator) as being the number of influential neighbours (Ballerini et al. 2008a). As to the average reaction time during coordination, we used the only empirical data available of 76 ms, which concerns the startle reaction to a light stimulus (Pomeroy and Heppner 1977). We represented variation in reaction time by drawing values from a normal distribution with a standard deviation σ
u
of 10 ms (Table 1). The shortest time needed to recognise an escape manoeuvre is 0.05 s after its start, a value which was inspired by measurements of fish to react to a predator threat (Domenici and Batty 1997). This delay, we label the cue identification time (Table 1). The actual start of the repetition of the escape manoeuvre depends also on the reaction interval (of 0.076 s). Thus on average, individuals repeated an escape manoeuvre after 0.05 s + (0.076/2) = 0.043 s.
When studying flock size (between 500 and 8000 individuals, see Table 1), reaction time and cue identification time, we kept the distance to the nearest neighbours constant by adjusting the separation radius, r
sep (Table 1) (Hildenbrandt et al. 2010; Hemelrijk and Hildenbrandt 2015). To investigate the effects of the density of the flock (measured as average distance to the nearest neighbours, NND), we tuned density with the separation radius, r
sep (Table 1).
Observations and measurements
To detect potential waves of agitation, we have recorded in the model with a virtual camera the flock from the side and from below resembling the setting of the real camera in the empirical study (Procaccini et al. 2011).
We measured the wave speed in the model by starting from the time that the wave has arrived at the centre of gravity of the flock (and is thus clearly visible) by calculating for each escaping individual its spatial distance to the first bird escaping and the time interval between the escape of the first individual and itself. The wave speed is the average of all these measurements. We took the average rather than the median because the variation in reaction time has been drawn from a normal distribution.
For each parameter value of NND, range of repetition, reaction time, cue identification time and flock size, we have run 30 replicas.