Quantification

To describe the emergent patterns in a spatial-explicit agent-based model, specific quantifications have been developed in many previous works (Parrish et al., 2002). For example, the degree of coordination in a fish aggregate is usually measured based on the addition of all individuals’ directions (Huth & Wissal, 1994; Couzin et al., 2002). According to the length of this sum vector, a moving school can be distinguished from a disordered swarm or a milling herd due to the large value. However, these metrics can only measure a single aggregate, and lose their effectiveness when fish flock into multiple groups. A simple instance is when two fish schools moving in opposite directions, in which case, the sum of their directions is mediated to a small vector, as the situation of a swarm.

In the proposed model, fish agents are given the ability to leave their neighbours, as free as to herd together. Therefore, a state of multiple groups is a common situation during an evolutionary process. Instead of clustering these groups from a global view, the bottom-up metric, RPFC, is originally designed by the research work to quantify collective patterns simply and effectively, as drawn in Figure 3.3. The metric takes advantage of each agent’s sensory perceptions, by which an agent’s positional status, at each time step, can be categorised into one of the following six types: R, P, Fm, Fc, Cm and Cc (Figure 3.3). According to the average frequencies of these six positional types

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in a population, three collective patterns can be identified, as the schooling pattern (composed of P, Fm and Fc), the swarming pattern (composed of Cm and Cc) and the dispersal pattern (composed of R).

Figure 3.3: Categorisation of Positions

Each fish agent at each step can be categorised into exactly one position: ranger (without neighbours in the sensory range), pioneer (schooling without leading neighbours), marginal follower (schooling with leading neighbours and without neighbours at some other side), central follower (schooling and surrounded by neighbours), marginal coward (not schooling, and not surrounded by neighbours) and central coward (not schooling, and surrounded by neighbours). This categorisation is based on an agent’s sensory perceptions, which ensures that an agent can recognise these positional differences.

The implementation of the RPFC metric is as follows. If an agent’s five visual sensors are all zero, which means there is no neighbour in its sensory range, this agent reports its position as ‘ranger (R)’. For those agents with neighbours, an agent is considered ‘schooling’ if

𝑃𝐿_{(0°) > 𝜔(𝑃}𝐿_{(120°) + 𝑃}𝐿_{(240°)), (3.9)}

where 𝜔 = 2 based on the experiment in Figure 3.4. This condition means an agent’s neighbours are on average in a direction similar to that of itself, and if the condition is not satisfied, the agent is considered ‘swarming’ at this step.

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Figure 3.4: RPFC Metric vs. Group-level Index

The model in Couzin et al. (2002) showed that the increase of ℓ𝑂, as a parameter of the

tendency of parallel orientation, can lead the collective pattern to transit from a disordered swarm, via a milling group and a relatively disordered school, to a highly coordinated school, which outcome is duplicated as the visualisation. To quantify the degree of coordination, the length of the sum vector of all individuals’ directions was used in the referenced work, which measurement is duplicated as the black line. As a comparison, the proposed RPFC metric displays that the swarming pattern and the schooling pattern can also be recognised clearly given 𝜔 ≥ 2. Specifically, a disordered

swarm and a milling herd cannot be separated by both of these two metrics, which was measured by another group-level metric in the referenced work. Besides, the relatively disordered school moves through the repulsion behaviours by overlapped agents, which is not a potential state in the proposed overlap-free model.

Subsequently, for further analyses in the latter chapter, a schooling agent is subdivided into one of the three positions according to the information from its visual sensors: viz., ‘pioneer (P)’, if 𝑃𝐸(0°) = 0, which means there is no neighbour in the front sector of 144°; ‘central follower (Fc)’, if all of its five visual sensors are greater than zero, which means it is surrounded by neighbours; and ‘marginal follower (Fm)’ if none of the both. It should be noted that the ‘straggler’ of a moving school is not separated from followers, in consideration of the existence of a blind zone at the rear. Lastly, a swarming agent is subdivided into one of the two positions: ‘central coward (Cc)’, if all of its visual sensors are greater than zero, and ‘marginal coward (Cm)’, otherwise. The term ‘coward’ is chosen to describe swarming individuals according to the appearance that these agents prefer to hide in the crowd rather than being ‘pioneers’ or ‘rangers’.

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Since the proposed categorisation is based on an agent’s two perceptions, it is guaranteed that these positional differences can be recognised by the fish agents.

Although an agent often changes its position with time, the distribution of these positions in the population is highly stable, except when the population size is too small. Specific exceptions are when a ranger is followed by other agents in its blind zone, a follower is actually in a huge milling aggregate, and two parallel pioneers are not followed by other agents. These errors, however, are insignificant to the measured results in the model. Hence, more complicated categorisations are prevented.

Apart from the proposed RPFC metric, the nearest neighbour’s distance (NND), as the distance between one and its nearest neighbour, is introduced to quantify the flocking degree of agents and the level of crowd density in a group. This metric has been widely used as a measurement of the crowding degree in animals (Parrish et al., 2002). There are k-NND metrics (Ballerini, 2008), as the kth nearest neighbour’s distance, to reduce the influence of exceptional situations, like pairs. Since there is no survival benefit to pairwise couples in the model, NND is adopted for its simplicity.