Abstract Biological groups exhibit fascinating collective dynamics without centralised control, through only local interactions between individuals. Desirable group behaviours are typically linked to a certain fitness function, which the group robustly performs under different perturbations in, for instance, group structure, group size, noise, or environmental factors. Deriving this fitness function is an important step towards understanding the collective response, yet it easily becomes non-trivial in the context of complex collective dynamics. In particular, understanding the social feedback - how the collective behaviour adapts to changes in the group size - requires dealing with complex models and limited experimental data. In this work, we assume that the collective response is experimentally observed for a chosen, finite set of group sizes. Based on such data, we propose a framework which allows to: (i) predict the collective response for any given group size, and (ii) automatically propose a fitness function. We use Smoothed Model Checking, an approach based on Gaussian Process Classification, to develop a methodology that is scalable, flexible, and data-efficient; We specify the fitness function as a template temporal logic formula with unknown parameters, and we automatically infer the missing quantities from data. We evaluate the framework over a case study of a collective stinging defence mechanism in honeybee colonies.

Honeybees protect their colony against vertebrates by mass stinging and they coordinate their actions during this crucial event thanks to an alarm pheromone carried directly on the stinger, which is therefore released upon stinging. The pheromone then recruits nearby bees so that more and more bees participate in the defence. However, a quantitative understanding of how an individual bee adapts its stinging response during the course of an attack is still a challenge: Typically, only the group behaviour is effectively measurable in experiment; Further, linking the observed group behaviour with individual responses requires a probabilistic model enumerating a combinatorial number of possible group contexts during the defence; Finally, extracting the individual characteristics from group observations requires novel methods for parameter inference. We first experimentally observed the behaviour of groups of bees confronted with a fake predator inside an arena and quantified their defensive reaction by counting the number of stingers embedded in the dummy at the end of a trial. We propose a biologically plausible model of this phenomenon, which transparently links the choice of each individual bee to sting or not, to its group context at the time of the decision. Then, we propose an efficient method for inferring the parameters of the model from the experimental data. Finally, we use this methodology to investigate the effect of group size on stinging initiation and alarm pheromone recruitment. Our findings shed light on how the social context influences stinging behaviour, by quantifying how the alarm pheromone concentration level affects the decision of each bee to sting or not in a given group size. We show that recruitment is curbed as group size grows, thus suggesting that the presence of nestmates is integrated as a negative cue by individual bees. Moreover, the unique integration of exact and statistical methods provides a quantitative characterisation of uncertainty associated to each of the inferred parameters.