Dynamical Behaviours Of A Class Of Delayed Cucker-Smale Model With Regional Influence

Flocking is a kind of ordered collective behaviour,which appears very naturally in nature and human society,such as the collective migration of birds,the foraging of fish in groups,the formation of common language in human society and so on.As a kind of classical flocking model,Cucker-Smale(C-S)model has been widely studied and developed in recent years.One of the important generalizations is the C-S model with time delay,and to deal such models,people usually assume that the interactions between agents are globally symmetric or asymmetric.But in reality,when the distance between two agents exceeds a certain range,the interaction will decrease rapidly or even can be negligible.That is to say,agents in a flock will only be effectively affected by others in a certain area.Such influence is known as regional influence.In this paper,we study a class of symmetric delayed C-S model with regional influence:where xi,vi∈Rn denotes the position and velocity of ith partical respectively,λ is a positive constant,|·| is Euclidean norm,the interaction strength χr is a cut-off function which is defined by r is a positive constant,I is a strictly positive monotonically non increasing continuous function.Firstly,we consider the special case of I(s)≡1.Combined with the theory of functional differential equation and connected stochastic matrix,we get some sufficient conditions for the system to achieve flocking under the case of non-critical and general situations respectively.Furthermore,the asymptotic velocity is obtained.Then the same results under the case of general situation are obtained for general I(s).