A minimal model of collective behaviour based on non-reciprocal interactions
The collective behaviour of individuals is widely observed in many natural and social systems. In these systems, Newton’s third law, or the law of action–reaction, is often violated. Hence, interaction between individuals is often nonreciprocal. Several previous studies focused on and partially elucidated the mechanism of the aforementioned systems. In this study, we aim to further deepen the understanding from a general perspective by proposing and analysing a simple mathematical model. The model is proposed by drawing inspiration from friendship formation in human society. It is demonstrated via simulations that various patterns emerge by changing the parameters. Further, these patterns are characterized by two macroscopic variables, based on which they are classified into six categories. Through this classification, we found that lifelike complex patterns emerge at the boundary between the parameter spaces where relative position among particles are fixed and where some particles move infinite distance from the others. Although this study is still in a rudimentary stage, we believe that our finding in which macroscopic patterns are related to local rules could move one step forward in understanding the core principle of collective behaviour.