Analysis of mono- and multi-cluster flocking for a nonlinear Cucker–Smale model with external force

The Cucker–Smale model describes the dynamics of groups of interacting self-propelled agents. This model is developed to study the collective behavior of groups of living things, such as bird flocks, fish schools, and herds of bacteria colonies. Since this model was introduced, it has been studied from a variety of perspectives, such as consensus, mono-cluster flocking, multi-cluster flocking, pattern formation, collision avoidance, and so on. In this study, we introduce the nonlinear Cucker–Smale model via nonlinear external force and discuss the collective motions of agents in this model from the perspective of consensus, mono-cluster flocking, and multi-cluster flocking. In particular, we present some conditions to derive consensus, mono-cluster flocking, and multi-cluster flocking. These conditions are simpler and more tractable to induce consensus, mono-cluster flocking, and multi-cluster flocking than the conditions presented for the (traditional) Cucker–Smale model. More precisely, this study shows that if p∈(1,3) or q∈(1,3), then the mono-cluster flocking is induced, and DMtCF does not occur in our model. In the case where p≥3 and q≥3, this study proves that only the consensus multi-cluster flocking is induced. Finally, the results of this paper are numerically verified.