Group chasing tactics: how to catch a faster prey

Group chasing is one of the most important forms of collective behaviour. Its significance arises from the fact that cooperation among predators can notably increase their chances of catching even hard-to-catch prey, which makes these collaborative species much more successful predators. Packs in which the cooperation is more efficient have better chances of survival. This puts evolutionary pressure on the animals to optimize their strategies [15–18].

Previous field studies have reported approximately 50 different species in which increases in group size above a certain point reduce their success rate per capita [15]. A biologically logical explanation is that the group becomes too large and not all the predators were able to feed even if the hunt was successful; therefore the size of chaser groups generally does not grow past a specific threshold [19].

The variety of predator–prey systems is extraordinarily rich within birds, mammals, fish and even insects [3, 17, 20–23]. A surprising finding is that even cheetahs, the fastest animals in the world, show an affinity for hunting in pairs [15, 22]. Additionally, wolves tend to hunt elks, while coyotes have been studied hunting for pronghorns in, e.g., migration corridors using landmarks as a strategic tool (reported in Yellowstone National Park). In both cases the prey is faster than the predator [17, 24]. We can also conclude that even wolves, one of the most unwavering examples in the animal kingdom, give up their pursuit after roughly one or two kilometres (which takes barely two minutes at their top speed).

Cooperation between hunters has different levels. For example, group hunting within bats just emerges completely unintentionally from the similar hunting methods of the individuals, while the study of large carnivores led to the finding that certain predators, while hazing their prey, tend to surround it collectively [26, 27]. This so-called encircling was reported mostly regarding wolves and lions, but even bottlenose dolphins have been observed to encircle their prey [15, 20, 28].

Despite the limited available data, it is clear that for many predator species group hunting has proved to be more beneficial than solitary hunting, even if not every member of the group participates in the pursuit. Evolution seems to have optimized the size of these hunting packs, because in the majority of cases, size of a hunting pack falls within the range 3–10 (figure 1).

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Fleeing is the most common response from animals to their chasers. First intuition would suggest that the only possible strategy for escape is to rush away from the chaser(s) as fast as possible. However, in real biological systems surprisingly different escaping strategies can occur. Based on a concept proposed by Domenici et al we can differentiate between direct escaping and erratic escaping as separate strategies, although other kinds of strange actions such as freezing or retaliating can also occur [29–32].

Direct escaping means that the prey runs straight in the opposite direction from its chaser(s) and relies on its speed and endurance to survive. In other words, under ideal conditions the escaper can outrun its chaser(s), but the prey can be caught if it is slower, if it encounters unexpected obstacles or if the chasers' strategies are more advanced.

When the escaping is erratic, the prey can use a mixture of direct escaping and other, stochastic patterns of motion. This can be zigzagging, a series of sudden changes in direction, or some even more advanced movements such as spinning, looping and whirling (figure 2) [29, 31]. For example, deers use a particular distribution of escaping angles, which is also slightly dependent on the distance from their chaser (figure 3). The effectiveness of these quick and unexpected changes in the escaper's direction relies mostly on the animals' inertia (and the difference between inertia of the prey and the predator) and the different time delays within the animals' sensory and motor systems, even if their reaction time may be very short (of the order of magnitude of 50 ms) in chase and escape situations [33].

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In conclusion, we can say that evaders do not always escape directly from their chaser but are likely to combine direct escaping with some sort of random motion in order to trick their chasers and prevent their tactics (e.g. zigzagging) being learned. The directions of these random segments of their trajectory vary significantly among different species. The prey have typical escaping distances of usually around a few hundred metres and they take into account all the chasers within that range when determining the direction and strategy of the escape (table 1) [34, 35, 37].

Agents. In our model there are ${N}_{{\rm{c}}}$ chasers and N_e escapers with different top speeds (${v}_{\max ,{\rm{c}}}$ ${v}_{\max ,{\rm{e}}}$) but a similar top acceleration (${a}_{\max }$). Let ${{\bf{r}}}_{{\rm{c}},i},{{\bf{v}}}_{{\rm{c}},i}$ and ${{\bf{r}}}_{{\rm{e}},j},{{\bf{v}}}_{{\rm{e}},j}$ be the positions and velocities of the ith chaser and jth escaper. An actively escaping prey will become inactive (out of the game) if the distance between it and the nearest chaser decreases below the catching distance (${r}_{\mathrm{cd}}$); otherwise the agent remains active. In this model the agents are represented by a circular/spherical object of radius ${r}_{\mathrm{cd}}/2$. The experiment is terminated when all of the escapers are caught.

Arena. We aimed to build a bio-inspired model, and therefore we study group chasing within finite boundaries, because pursuits in nature have boundaries in both space and time. However, many previously published models use either periodic boundary conditions or infinite space. Although both can be very useful, e.g. in statistical physics, in the modelling of group chasing they can lead to very strange and unrealistic situations (for example, a prey 'coming back' from the other side of the cell 'falling into the arms' of chasers wandering at the edge of the cell). For the sake of simplicity and to avoid edge-effects as much as possible, our simulation runs within a circular arena of radius ${r}_{{\rm{a}}}$. We focus on the pursuit, and therefore we assume that each and every agent can see all the other agents—we do not examine (group) foraging methods. In this constellation, the origin is the center of the arena.

The arena is surrounded by a soft wall [44]. In this construction if any of the agents gets outside the arena, a virtual agent starts to repulse it towards the center of the arena:

$$\begin{eqnarray}&&{{\bf{v}}}_{i}^{{\rm{a}}}=s({r}_{i},{r}_{{\rm{a}}},{r}_{\mathrm{wall}})\ \left({v}_{\max ,k}\ \displaystyle \frac{{{\bf{r}}}_{i}}{{r}_{i}}+{{\bf{v}}}_{i}\right)\qquad (k={\rm{e}},\ {\rm{c}}),\end{eqnarray} \tag{ 5 }$$

where

$$\begin{eqnarray}s({r}_{i},{r}_{{\rm{a}}},{r}_{\mathrm{wall}})=\left\{\begin{array}{ll}0 & \mathrm{if}\ {r}_{i}\in [0,{r}_{{\rm{a}}}],\\ -\,\displaystyle \frac{1}{2}\ \sin \left(\displaystyle \frac{\pi }{{r}_{\mathrm{wall}}}({r}_{i}-{r}_{{\rm{a}}})-\displaystyle \frac{\pi }{2}\right)-\displaystyle \frac{1}{2} & \mathrm{if}\ {r}_{i}\in [{r}_{{\rm{a}}},{r}_{{\rm{a}}}+{r}_{\mathrm{wall}}],\\ -\,1 & \mathrm{if}\,{r}_{i}\gt {r}_{{\rm{a}}}+{r}_{\mathrm{wall}}\end{array}\right.\end{eqnarray} \tag{ 6 }$$

where ${r}_{\mathrm{wall}}$ is the width of the wall and ${r}_{i}=| {{\bf{r}}}_{i}|$.

Collision avoidance. Between each pair of agents of the same type there is a short-range repulsive interaction term to avoid collisions:

$$\begin{eqnarray}&&{{\bf{v}}}_{k,i}^{\mathrm{coll}}={v}_{\max ,k}\ { \mathcal N }\,\left[\displaystyle \sum _{j}^{{N}_{k}}\ \displaystyle \frac{| {{\bf{d}}}_{{ij}}| -{r}_{\mathrm{cd}}}{| {{\bf{d}}}_{{ij}}| }\ {{\bf{d}}}_{{ij}}\ {\rm{\Theta }}({r}_{\mathrm{cd}}-| {{\bf{d}}}_{{ij}}| )\right],\quad \mathrm{where}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{{\bf{d}}}_{{ij}}={{\bf{r}}}_{k,i}-{{\bf{r}}}_{k,j}\qquad (k={\rm{e}},\ {\rm{c}}),\end{eqnarray} \tag{ 8 }$$

${r}_{\mathrm{cd}}$ is the catching distance, defined as twice the radius of the agents, Θ is the Heaviside step function, ${{\bf{d}}}_{{ij}}$ is the displacement vector between the ith and jth agents, and the operator ${ \mathcal N }[.]$ normalizes its argument and returns it as a dimensionless vector.

Viscous friction-like interaction. To simulate realistic motion, we introduced viscous friction-like interaction terms [45], parametrized with a specific friction coefficient ${C}_{{\rm{f}}}$, with the general form

$$\begin{eqnarray}&&{{\bf{v}}}_{{ij}}={C}_{{\rm{f}}}\ \displaystyle \frac{{{\bf{v}}}_{j}-{{\bf{v}}}_{i}}{{(\max \{{r}_{\mathrm{cd}},| {{\bf{d}}}_{{ij}}| \})}^{2}}.\end{eqnarray} \tag{ 9 }$$

Additionally, viscous friction-like damping is important because it minimizes unwanted oscillations in any delayed system [42].

Chasing rule. Each chaser chases the closest escaper (chasing term). The collective chasing strategy includes a soft repulsion between the chasers and the prediction of their target's position.

Chasing term. The chasing is represented by an attractive interaction between the ith chaser and its target escaper toward the escaper and a viscous friction-like velocity alignment term in the following way:

$$\begin{eqnarray}&&{{\bf{v}}}_{{\rm{c}},i}^{\mathrm{ch}}={v}_{\max ,{\rm{c}}}\ { \mathcal N }\,\left[\displaystyle \frac{{{\bf{r}}}_{{\rm{c}},i}-{{\bf{r}}}_{{\rm{e}}}}{| {{\bf{r}}}_{{\rm{c}},i}-{{\bf{r}}}_{{\rm{e}}}| }-{C}_{{\rm{f}}}^{\prime }\ \displaystyle \frac{{{\bf{v}}}_{{\rm{c}},i}-{{\bf{v}}}_{{\rm{e}}}}{| {{\bf{r}}}_{{\rm{c}},i}-{{\bf{r}}}_{{\rm{e}}}{| }^{2}}\right],\end{eqnarray} \tag{ 10 }$$

where ${C}_{{\rm{f}}}^{\prime }$ is the coefficient of the velocity alignment term relative to the attraction.

Prediction. A part of the chasers' strategy is the cognitive prediction of their prey's future position. This means that the chasers are heading to the point ${\bf{r}}{^{\prime} }_{{\rm{e}}}$ (the estimated future position of the escaper) instead of the escaper's current position ${{\bf{r}}}_{{\rm{e}}}$ (figure 4). Here we use only a simple linear approximation to determine the value of ${\bf{r}}{^{\prime} }_{{\rm{e}}}$, which is the point at which the chaser could catch the escaper if the time needed for this is below a certain threshold time (${\tau }_{\mathrm{pred}}$); if this is not the case, then it is the point that the chaser can reach during time ${\tau }_{\mathrm{pred}}$. Then the value of ${\bf{r}}{^{\prime} }_{{\rm{e}}}$ is the numeric solution of the following equation:

$$\begin{eqnarray}&&{\bf{r}}{^{\prime} }_{{\rm{e}}}={{\bf{r}}}_{{\rm{e}}}+\displaystyle \frac{{{\bf{v}}}_{{\rm{e}}}}{| {{\bf{v}}}_{{\rm{e}}}| }\ {v}_{\max ,{\rm{e}}}\ \min \left[\displaystyle \frac{{\bf{r}}{^{\prime} }_{{\rm{e}}}-{{\bf{r}}}_{{\rm{c}}}}{{v}_{\max ,{\rm{e}}}},{\tau }_{\mathrm{pred}}\right]\end{eqnarray} \tag{ 11 }$$

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Interaction between chasers. Let us assume that a group of chasers is chasing the same escaper. If there is no interaction between the chasers they will eventually get too close to each other because their driving forces are similar. This results in a tail of chasers behind the escaper. This is neither a logical predatory behaviour nor one observable in nature. What we can see in nature, though, is that chasers do not get too close to each other as they pursue their prey, but the distance between them decreases as they get closer to (and finally catch) the escaper. We modelled this with a repulsive interaction term that has a characteristic length ${r}_{\mathrm{inter}}$, a magnitude factor ${C}_{\mathrm{inter}}$ and a magnitude ${C}_{\mathrm{inter}}\ {v}_{\max ,{\rm{c}}}$. Consequently, ${C}_{\mathrm{inter}}=0$ means that there is no interaction and ${C}_{\mathrm{inter}}=1$ means that the repulsion between chasers is just as strong as their attraction towards the escaper. The mathematical definition of this interaction is

$$\begin{eqnarray}&&{{\bf{v}}}_{{\rm{c}},i}^{\mathrm{inter}}={C}_{\mathrm{inter}}{v}_{\max ,{\rm{c}}}\ { \mathcal N }\,\left[\displaystyle \sum _{j}\left({{\bf{d}}}_{{ij}}\ \displaystyle \frac{| {{\bf{d}}}_{{ij}}| -{r}_{\mathrm{inter}}}{| {{\bf{d}}}_{{ij}}| }-{C}_{{\rm{f}}}^{\prime\prime }\ \displaystyle \frac{{{\bf{v}}}_{{\rm{c}},i}-{{\bf{v}}}_{{\rm{c}},j}}{| {{\bf{d}}}_{{ij}}{| }^{2}}\right)\right],\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{{\bf{d}}}_{{ij}}={{\bf{r}}}_{{\rm{c}},i}-{{\bf{r}}}_{{\rm{c}},j},\end{eqnarray} \tag{ 13 }$$

where ${C}_{{\rm{f}}}^{\prime\prime }$ is the coefficient of the viscous friction-like term.

The driving force of the ith chaser is the sum of the previously introduced terms (interaction with the wall of the arena, collision-avoiding short-term repulsion, direct chasing force and the long-term repulsion between chasers):

$$\begin{eqnarray}&&{{\bf{f}}}_{{\rm{c}},i}={{\bf{v}}}_{i}^{{\rm{a}}}+{{\bf{v}}}_{{\rm{c}},i}^{\mathrm{coll}}+{{\bf{v}}}_{{\rm{c}},i}^{\mathrm{ch}}+{{\bf{v}}}_{{\rm{c}},i}^{\mathrm{inter}}.\end{eqnarray} \tag{ 14 }$$

Escaping rule. The escaper moves in the free direction that is furthest away from all the chasers within its sensitivity range. At the wall, the escaper aligns its velocity to the wall. If the escaper can slip away between the nearest two chasers, then it returns to the field. We defined a panic parameter that depends on the distance between the escaper and its nearest chaser and controls the erratic behaviour.

Direct escaping. An escaping agent takes into account every chaser within its range of sensitivity (${r}_{\mathrm{sens}}$), while it weights this effect by the distance of each chaser. This represents the biological observation that the prey is concerned about all the chasers that are too close, and the closer a predator is, the more dangerous it is. This interaction term for the ith escaper is

$$\begin{eqnarray}&&{{\bf{v}}}_{{\rm{e}},i}^{\mathrm{esc}}={v}_{\max ,{\rm{e}}}\ { \mathcal N }\,\left[\displaystyle \sum _{j}\left(\displaystyle \frac{{{\bf{r}}}_{{\rm{e}},i}-{{\bf{r}}}_{{\rm{c}},j}}{| {{\bf{r}}}_{{\rm{e}},i}-{{\bf{r}}}_{{\rm{c}},j}{| }^{2}}-{C}_{{\rm{f}}}^{\prime\prime\prime }\displaystyle \frac{{{\bf{v}}}_{{\rm{e}},i}-{{\bf{v}}}_{{\rm{c}},j}}{| {{\bf{r}}}_{{\rm{e}},i}-{{\bf{r}}}_{{\rm{c}},j}{| }^{2}}\right){\rm{\Theta }}({r}_{\mathrm{sens}}-| {{\bf{r}}}_{i}-{{\bf{r}}}_{j}| )\right],\end{eqnarray} \tag{ 15 }$$

where ${C}_{{\rm{f}}}^{\prime\prime\prime }$ is the coefficient of the velocity alignment term.

Erratic escaping. In nature many species tend to use certain kinds of erratic escaping strategies in which they combine direct escaping with some more advanced patterns of motion. Here we implemented the most basic one called zigzagging [29, 31]. During zigzagging, the escaper tries to trick its chaser(s) with a set of sudden and unexpected changes in direction. For this we introduced a so-called panic parameter (${p}_{\mathrm{panic}}$), which is an exponential function of the distance between the escaper and the nearest chaser $({d}_{\min })$, with the value of 0 if the chaser is outside the escaper's range of sensitivity and 1 when the escaper gets caught:

$$\begin{eqnarray}&&{p}_{\mathrm{panic}}=\displaystyle \frac{1}{e-1}\ [{{\rm{e}}}^{-{d}_{\min }/{r}_{\mathrm{sens}}+1}-1].\end{eqnarray} \tag{ 16 }$$

The escaper starts zigzagging when the panic parameter reaches a certain threshold (${p}_{\mathrm{thres}}$) and it is not closer to the wall than a certain distance (${r}_{\mathrm{zigzag}}$):

$$\begin{eqnarray}&&{p}_{\mathrm{panic}}\lt {p}_{\mathrm{thres}},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{r}_{{\rm{e}}}\lt {r}_{{\rm{a}}}-{r}_{\mathrm{zigzag}},\end{eqnarray} \tag{ 18 }$$

where ${r}_{{\rm{e}}}$ is the absolute value of ${{\bf{r}}}_{{\rm{e}}}$. If these conditions are true, the escaper starts moving in a certain direction for a given amount of time. Afterwards, the prey zigzags in another direction for a period of time if the zigzag conditions are still true, otherwise it continues to escape directly. The zigzagging gets interrupted when the distance between the chaser and the escaper decreases below ${r}_{\mathrm{zigzag}}$. We determine the direction of a zigzag segment in the following way:

$$\begin{eqnarray}&&{{\bf{v}}}_{{\rm{e}},i}^{\mathrm{zigzag}}={v}_{\max ,{\rm{e}}}\ { \mathcal N }\left\{{ \mathcal R }\left[\displaystyle \sum _{j}\left(\displaystyle \frac{{{\bf{r}}}_{{\rm{e}},i}-{{\bf{r}}}_{{\rm{c}},j}}{| {{\bf{r}}}_{{\rm{e}},i}-{{\bf{r}}}_{{\rm{c}},j}{| }^{2}}\right)\ {\rm{\Theta }}({r}_{\mathrm{sens}}-| {{\bf{r}}}_{i}-{{\bf{r}}}_{j}| )\right]\right\},\end{eqnarray} \tag{ 19 }$$

where ${ \mathcal R }$ is a rotating operator that rotates its argument vector by an angle in two dimensions randomly chosen from the interval $[-\pi ,\pi ]$ and by two random angles (one azimuth and one polar) in three dimensions from the intervals $[0,2\pi ]$ and $[0,\pi ]$. The duration of a zigzag segment is also a random parameter, which we chose from a power-law distribution with an exponent of −2, a lower limit of ${r}_{\mathrm{zigzgag}}/{v}_{\max ,{\rm{e}}}$, and an upper limit of ${r}_{{\rm{a}}}/{v}_{\max ,{\rm{e}}}$.

Behaviour at the wall. Confronting the wall would significantly decrease the escaper's speed; therefore, the escaper should avoid getting too close to the wall. This is ensured by cutting off the radial component of the escaper's velocity vector depending on how close to the wall the agent is:

$$\begin{eqnarray}&&{{\bf{v}}}_{{\rm{e}}}^{\mathrm{final}}={{\bf{v}}}_{{\rm{e}}}-{C}_{\mathrm{wall}}({r}_{{\rm{e}}},{r}_{{\rm{a}}},{r}_{\mathrm{wall}})\ \displaystyle \frac{{{\bf{r}}}_{{\rm{e}}}}{{r}_{{\rm{e}}}}\left(\displaystyle \frac{{{\bf{r}}}_{{\rm{e}}}^{{\rm{T}}}}{{r}_{{\rm{e}}}}\cdot {{\bf{v}}}_{{\rm{e}}}\right)=\hat{{\bf{W}}}{{\bf{v}}}_{{\rm{e}}},\end{eqnarray} \tag{ 20 }$$

where ${C}_{\mathrm{wall}}$ is defined as

$$\begin{eqnarray}{C}_{\mathrm{wall}}=\left\{\begin{array}{ll}1 & \mathrm{if}\ {r}_{{\rm{e}}}\gt {r}_{{\rm{a}}}-2{r}_{\mathrm{wall}},\\ \displaystyle \frac{1}{2}\left[1-\sin \left(\pi \ \displaystyle \frac{{r}_{{\rm{e}}}-{r}_{{\rm{a}}}+2{r}_{\mathrm{wall}}}{{r}_{\mathrm{zigzag}}}-\displaystyle \frac{\pi }{2}\right)\right] & \mathrm{if}\ {r}_{{\rm{e}}}\gt {r}_{{\rm{a}}}-2{r}_{\mathrm{wall}}-{r}_{\mathrm{zigzag}},\\ 0 & \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 21 }$$

Applying this method exclusively would mean that once the escaper reaches the wall it is stuck there forever, which would be unrealistic. Thus, we extended this rule: if the escaper is close to the wall (${C}_{\mathrm{wall}}\gt 0.5$) and it can slip through the gap between the two closest chasers, it does, and it returns to the central part of the arena. In order to determine whether or not jumping back to the arena means a successful escape, the prey routinely calculates how long it would take for itself and the two nearest chasers to reach certain points. These points are the ones we get when we project the difference vectors pointing from the prey to the chasers onto the line defined by the sum of these two vectors (equation (22), figure 5):

$$\begin{eqnarray}&&{\bf{e}}=\displaystyle \frac{({{\bf{r}}}_{{\rm{c}},1}-{{\bf{r}}}_{{\rm{e}}})+({{\bf{r}}}_{{\rm{c}},2}-{{\bf{r}}}_{{\rm{e}}})}{| ({{\bf{r}}}_{{\rm{c}},1}-{{\bf{r}}}_{{\rm{e}}})+({{\bf{r}}}_{{\rm{c}},2}-{{\bf{r}}}_{{\rm{e}}})| },\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{{\bf{r}}}_{{\rm{p}},k}={\bf{e}}[{{\bf{e}}}^{{\rm{T}}}\cdot ({{\bf{r}}}_{{\rm{c}},k}-{{\bf{r}}}_{{\rm{e}}})],\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\tau }_{{\rm{e}},k}=\displaystyle \frac{{r}_{{\rm{p}},k}}{{v}_{\max ,{\rm{e}}}},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{\tau }_{{\rm{c}},k}=\displaystyle \frac{| {{\bf{r}}}_{{\rm{p}},k}-{{\bf{r}}}_{{\rm{c}},k}+{{\bf{r}}}_{{\rm{e}}}| -{r}_{\mathrm{cd}}}{{v}_{\max ,{\rm{c}}}},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{\tau }_{{\rm{e}},k}\mathop{\lt }\limits^{?}{\tau }_{{\rm{c}},k},\end{eqnarray} \tag{ 26 }$$

where $k=1,2$ are the indices of the two nearest chasers (see figure 5). If condition (26) is true, the escaper can run back to the arena without getting caught; otherwise it stays at the wall where it will either be caught or chased.

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The driving force of the ith escaper will be the sum of the interaction with the wall, collision-avoiding repulsion and either the direct or the zigzagging escaping term, finally transformed by the $\hat{{\bf{W}}}$ operator (20):

$$\begin{eqnarray}&&{{\bf{f}}}_{{\rm{e}},i}=\hat{{\bf{W}}}({{\bf{v}}}_{i}^{{\rm{a}}}+{{\bf{v}}}_{{\rm{e}},i}^{\mathrm{coll}}+({{\bf{v}}}_{{\rm{e}},i}^{\mathrm{esc}}\vee {{\bf{v}}}_{{\rm{e}},i}^{\mathrm{zigzag}})).\end{eqnarray} \tag{ 27 }$$

Although in our model the number of expressions and parameters is significantly larger than in the simplest statistical mechanical models, these expressions and parameters are far from arbitrary; almost all of them are in a direct relation with the observed systems and originate from such trivial facts as the existence of time delays (e.g. reaction time). The parameters are 'freely' tunable from the point of the model, but most of them are experimentally measurable for an actual predator–prey system.

These parameters can be separated into two sets: environmental and tactical parameters (table 2). Except for ${t}_{\mathrm{del}}$, all the environmental parameters had fixed values during the measurements: for instance, ${a}_{\max }=6$ m s^–2, ${r}_{{\rm{a}}}\,=$ 150 m and ${r}_{\mathrm{wall}}=5$ m. On the other hand, all the tactical parameters except the fixed velocities (${v}_{\max ,{\rm{c}}}=6$ m s^–1, ${v}_{\max ,{\rm{e}}}\,=8$ m s^–1) can be adjusted at the beginning of each run to create and study different chasing and escaping tactics. We studied most cases in both 2D and 3D.

Table 2.  The environmental and tactical parameters of the model.

Environmental parameters	Typical range	Dimension	Definition
${a}_{\max }$	6	m s–2	Maximum acceleration of the agent
σ	0.0–1.0	m2 s–3	Standard deviation of the Gaussian noise
${\tau }_{\mathrm{CTRL}}$	0.06	s	Characteristic time of the acceleration
${t}_{\mathrm{del}}$	0–6	s	Delay time
${C}_{{\rm{f}}}^{\prime }$	1.1	ms	Friction coefficient in the direct chasing term
$C{^{\prime\prime} }_{{\rm{f}}}$	1.1	m2 s	Friction coefficient in the velocity alignment between chasers
${C}_{{\rm{f}}}^{\prime\prime\prime }$	1.1	s	Friction coefficient in the direct escaping term
${r}_{{\rm{a}}}$	150	m	Radius of the arena
${r}_{\mathrm{cd}}$	1	m	Catching distance
${r}_{\mathrm{wall}}$	5	m	Width of the wall
Parameters of chasers' tactics	Typical range	Dimension	Definition
${v}_{\max ,{\rm{c}}}$	6	m s–1	Maximum velocity of the chasers
${\tau }_{\mathrm{pred}}$	0–6	s	Upper limit of the chaser's prediction
${C}_{\mathrm{inter}}$	0–1	1	Strength of interaction between chasers
${r}_{\mathrm{inter}}$	0–300	m	Interaction distance between chasers
Parameters of escapers' tactics	Typical range	Dimension	Definition
${v}_{\max ,{\rm{e}}}$	8	m/s	Maximum velocity of the escapers
${p}_{\mathrm{thresh}}$	0–1	1	Escaper's panic threshold
${r}_{\mathrm{sens}}$	0–120	m	Escaper's range of sensitivity
${r}_{\mathrm{zigzag}}$	0–80	m	Minimal length of a zigzag segment; the distance limit between an escaper and a chaser when the escaper stops zigzagging

At first we studied the effect of the interaction between the chasers as a function of ${N}_{{\rm{c}}}$ with two different spatial distributions of the agents and the interaction range fixed (${r}_{\mathrm{inter}}$ = 300 m). In the first distribution, all the agents were spaced randomly on the field with no regard to their type. In the second one, the escaper was placed randomly in a circle with a radius of ${r}_{{\rm{a}}}/4$ and a center with coordinates (${r}_{{\rm{a}}}/2$, ${r}_{{\rm{a}}}/2$) while the chasers were put in another circle of the same radius but with a center of (0, $-3{r}_{{\rm{a}}}/4$), where the center of the field is the origin (figure 6). In 3D the method is the same but using spheres. The completely random starting position is widely used in the literature, but in reality it is more common that the chasers are closer to each other when they start the pursuit (it is also not logical for the prey to wander into a place where the predators are scattered all around).

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For the random distribution, the chasers sometimes happened to be initially around the escaper, which leads to an immediate encircling and a very quick (and unrealistic) pursuit. However, with the pack distribution the chasers have to build up their chasing formation to catch the prey, which produces a well-defined regime of the parameter space in which the chasers are successful. Despite this, there is no big difference in the maximal Effectiveness_c values in these two cases, as figure 7 demonstrates. For the case of the pack distribution, the chasers' effectiveness can be seen in figure 7 as a function of the number of chasers and the interaction strength between them. It is important to point out that these results are the solid proof of the fact that in these scenarios (i.e. where the escaper is faster) a single chaser is not able to catch the prey and multiple agents only have a chance to do so if there is a strong tactical interaction between them. As an example, we took a closer look on certain cross sections around the optimum of figure 7, which are presented in figure 8. figure 8 illustrates that chasing in three dimensions is a lot harder than in two dimensions, and while three chasers can form an optimal group for 2D, five is more advantageous for 3D.

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On one hand, we can see that the effectiveness decreases monotonically above ${N}_{{\rm{c}}}=6$, which happens because of the division by ${N}_{{\rm{c}}}$ in (28); on the other hand, one and two chasers have no chance to ever catch the faster prey. Between these two regimes the tendency is not monotonic. This happens for geometric reasons (symmetric arrangements), which will be averaged out for larger ${N}_{{\rm{c}}}$ values: an odd number of chasers have a much bigger chance to block all the escape directions, but it is much easier for a gap to remain when the number of chasers is even (see figure 5).

In this subsection we study a chaser's cognitive prediction regarding the escaper's path, how the time delay can affect the outcome of the pursuit and how the prediction can compensate for the delay. As it was introduced in section 2, the prediction has an upper time limit, the maximum time of prediction (${\tau }_{\mathrm{pred}}$), which is studied in this subsection. This quantity is proportional to how far ahead the chasers forecast their prey's position. If we add the prediction (figure 9) to the previously optimized chasers, their effectiveness increases a little (with a maximum at ${\tau }_{\mathrm{pred}}\,\approx$ 0.38 s), but after that it starts to decrease. This happens because the optimized chasers are already hunting with a 100% success rate ($n={n}_{\mathrm{tot}}$ in (29)), while a long prediction time extends the pursuit's duration because the chasers do not rush immediately towards to the escaper but extrapolate and chase the escaper's future position, which is always further away, and this longer duration leads to lower efficiency, while in 3D even the efficiency of an optimized (with regard to the interaction between the chasers) group can be significantly increased by the prediction. However, if we take a look at chasing packs with non-optimized interaction parameters (in 2D), we can see that prediction improves their efficiency to near the level of the optimized group.

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We tested the tolerance of the Effectiveness_c against delay with different prediction times for different groups of chasers (figure 10). These experiments led to the conclusion that if the prediction time is large enough in comparison to the delay (but still of the same order of magnitude), prediction will compensate for the delay and the pursuit remains successful. However, a very large delay (e.g. 5 s) with a small prediction can completely prevent the success of the pursuit. Based on these results, the optimal values of the chasers' tactical parameters can be found in table 3 if the system has a delay of 1 s.

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Table 3.  The optimal chasing parameters in two and three dimensions with prediction and delay (1 s).

	${N}_{{\rm{c}}}$	${C}_{\mathrm{inter}}$	${r}_{\mathrm{inter}}$	${\tau }_{\mathrm{pred}}$
2D	3	0.86	155 m	1.75 s
3D	3	0.9	135 m	2 s

We examined the effect of external, Gaussian noise within the range $\sigma =0$–2 m² s^–3, where 0.2 m² s^–3 is equivalent to, e.g., the wind blowing with medium strength. The values of Effectiveness_c proved to be quite robust and stable against this perturbation, which just slightly increased the deviation of the statistical results. The standard deviation of the noise amplitude–Effectiveness_c dataset was about 2% in two dimensions, and 2.4% in three dimensions, and it stayed fairly constant.

Despite our model's simplicity, while studying it in real time we can observe very interesting, life-like patterns of motion emerging. These can be categorized as follows:

• The optimal group of chasers catches the prey very quickly: they rush towards the escaper on the right trajectories to block all its possible escape routes. This can also happen (with a lower frequency) in three dimensions. This sometimes results in line formation, too.
• If the chasing parameters are moderately efficient, 'on-the-field' pursuit often happens: the agents cross the field several times, the escaper moves along the wall, and hops back to the center of the field several times. In these situations the classical encircling (observed, e.g., in lions) can also happen completely emergently (figure 11). With significantly more chasers, this phenomenon (sometimes called as caging) can be observed in three dimensions as well.

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In this section we have demonstrated that those chasers that are unable to catch the prey alone can successfully pursue it if there is a certain, well-parametrized interaction between them, which is their collective chasing tactics. For this we studied and optimized the parameters of the group of chasers by scanning their parameter space. We have found that even in three dimensions three chasers can efficiently catch the prey if the strength and characteristic distance of the interaction are set properly. We have shown that the prediction method enhances the capabilities of the chasers (even if their interaction parameters are sub-optimal), and that prediction can suppress the negative effects of time delay. We pointed out that these optimal chasing tactics are robust against external noise and we have found interesting, emergent behaviour patterns, such as life-like encircling. The data also implies that chasing in three dimensions is a lot more difficult task than in two dimensions, which is consistent with the fact that truly three-dimensional pursuits are very rare in nature. What is more common in three dimensions is to restrict the degrees of freedom in some way. Flying and swimming animals frequently use the natural borders of their open 3D space (water–air or air–ground boundaries) to drive their prey into a corner. It has also been reported for dolphins hunting in groups that certain individuals tend to have different roles during the hunt, for example they create artificial blockades to restrict the possible escaping routes of schools of fish and catch them more efficiently [47, 48]. Finally, certain species, such as fish-hunting cone snails or all kinds of web spiders use various 'sit-and-wait' strategies—they hide, and attack abruptly only when the prey is close enough [23, 46].

The panic parameter of the zigzag motion was defined in (16) as a real number between 0 and 1 that depends on the distance between the escaper and its nearest chaser. If this value reaches the panic threshold, the escaper begins a zigzag motion. At first we examined the Effectiveness_e values as a function of ${p}_{\mathrm{thresh}}$ to study the zigzag's effect on the pursuit's outcome. The results are presented in figure 12 with and without delay, in two and three dimensions. These figures tell us that zigzagging can significantly improve the escaper's efficiency in all the cases studied; moreover, ${p}_{\mathrm{thresh}}$ does have optimal values.

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The zigzagging has another parameter: ${r}_{\mathrm{zigzag}}$, which is the critical distance at which the escaper ceases zigzagging and returns to direct escape to avoid dangerous situations due to the closeness of the chaser(s). figure 13 implies that in 2D ${r}_{\mathrm{zigzag}}=20$ m is a universal optimum, while in 3D, if ${r}_{\mathrm{zigzag}}\gt 20$ m, the chasers cannot catch the escaper. In other words, (i) zigzagging too close to chasers is disadvantageous in both two and three dimensions, (ii) in two dimensions there is an advantageous regime with an optimal parameter for the zigzagging, and (iii) in three dimensions the Effectiveness_e gets saturated at ${r}_{\mathrm{zigzag}}={r}_{\mathrm{sens}}$, which means that the evader successfully escapes even without the zigzag tactics.

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