Interplay of time-delay effect on the Cucker–Smale model with fractional time derivative

We study the interaction between the time-delay effect and the memory effect within the Cucker–Smale model. We use the Caputo fractional derivative, rather than the standard time derivative, to incorporate the memory effect into the Cucker–Smale model. For the proposed model, formulated as a fractional delay differential equation, the main goal of this paper is to develop sufficient conditions to exhibit its anomalous emergent dynamics. Additionally, we observe an algebraic decay in the flocking dynamics of the proposed model, which arises from several unique features of the fractional derivative, in contrast to the usual first-order time derivative.