Table 1.
Predator Agent Parameter Values.
Table 2.
Prey Agent Parameter Values.
Fig 1.
A simplified example of fuzzy reasoning.
Section A shows the current state of the artificial world. The observed prey agent is depicted in black and the nearest predator in red. In this simplified example the observed prey agent performs fuzzy reasoning solely based on the nearest predator’s relative bearing (in this case -126°). The left part of section B presents the evaluation of the degree of truth of the antecedents of individual if-then rules that are listed at the bottom of this section. We assume no uncertainty in the data and model all inputs as singleton fuzzy values. For example, the degree of truth of the antecedent “nearest predator relative bearing is left” is therefore computed as 〈−180, −90,0〉 (−126) = 0.6. The right part of section B presents fuzzy inference or the evaluation of the consequent part of individual rules. Since we use the product t-norm (x ∘ y = xy) for implication this translates to scaling the triangular fuzzy number that is used to define the corresponding output linguistic variable’s value (shaded areas). For example, in the case of consequent “heading change is right,” this means 0.6 ∘ 〈0, 90, 180〉. Section C presents the aggregation of individual consequent parts and based on that the computation of the final, crisp output, the conclusion (desired change in heading). We aggregate rules via the probabilistic sum s-norm (x ◇ y = x + y − xy), and compute the crisp output by means of the centre-of-gravity defuzzification method. This means that the shaded area in section C (aggregated shaded areas from section B) gets translated into 54.375, the desired change in heading for the observed prey agent in this simplified example. For further details on fuzzy reasoning in general and its application to the modelling of collective behaviour consult [30–33, 70].
Table 3.
Prey Agent Fuzzy Data Base.
Table 4.
Evolutionary Process Parameter Values.
Table 5.
Validation Process Parameter Values.
Fig 2.
Prey agents that can influence each other either directly or indirectly via others are considered as being part of the same group.
Presented are two groups (one in a milling state, green shading, and one in a polar state, blue shading) and one straggler (a prey agent that can neither influence nor be influenced by any other prey individual, grey shading).
Fig 3.
Local density, number of groups, and time spent in a specific collective state for each evolutionary run.
The left graph displays the mean and bootstrapped 95% CI of the local density at update step 0, as well as, mean and bootstrap 95% CI of the average local density during update steps 900–1800. The middle graph displays the mean and bootstrapped 95% CI of the number of groups at update step 0 and the mean and bootstrapped 95% CI of the average number of groups during update steps 900–1800. The right graph shows the cumulative proportion of time spent in a specific collective state. The number of replicates was 20. Evolution ids were sorted based on the average local density during update steps 900–1800. Ids marked in bold correspond to representatives of the four classes of evolved behaviour (see main text for details).
Fig 4.
Time series of global polarization and rotation for each of the four classes of evolved behaviour.
Evolutions no. 12, 6, 4 and 10 (marked as bold in Fig 3) were selected as representatives of polarized, milling, swarming and dynamic behaviour based on the proportion of time spent in a specific state, high mean local density and low mean number of groups. For video sequences of the representative evolved behaviours see S2–S6 Videos.
Fig 5.
Density plot of global polarization versus rotation for various group sizes in the case of evolution no. 10.
The density plots visualize the relationship between group size and behaviour stability. Increasing the number of agents leads global behaviour to change from predominantly polarized to predominantly milling.
Fig 6.
Box plots of the distributions of the six parameters through which the evolved rule sets were summarized.
Grouping was based on the evolved behaviour class. Statistical significance of differences was obtained by means of a Benjamini-Yekutieli adjusted Dunn test. Symbols ns, * and ** indicate p ≥ 0.05, p < 0.05, and p < 0.01, respectively. Unmarked cases denote significance at p < 0.001.