Effect of time delay on flocking dynamics

We propose a time-delayed Cucker-Smale type model(CS model), which can be applied to modeling (1) collective dynamics of self-propelling agents and (2) the dynamical system of stock return volatility in a financial market. For both models, we assume that it takes a certain amount of time to collect/process information about the current position/return configuration until velocity/volatility adjustment is made. We provide a sufficient condition under which flocking phenomena occur. We also identify the initial configuration for a two-agent case, in which collective behaviors are accelerated by changes in the delay parameter. Numerical illustrations and financial simulations are carried out to verify the validity of the model.