Interplay of a unit-speed constraint and time-delay in the flocking model with internal variables

We studied the dynamics of thermodynamic Cucker–Smale (TCS) particles moving with a constant speed constraint. The TCS model describes the collective dynamics of the population of birds with a time varying internal variable, and it was first introduced as the generalization of the Cucker–Smale (CS) model. In this paper, we considered a modification of the TCS model in which each agent moves at a constant speed, such as the Vicsek model, and we additionally considered the effect of time-delays due to the finiteness of the information propagation speed between agents. Then, we presented several sufficient conditions in terms of initial data and system parameters to exhibit asymptotic flocking. We presented two kinds of results for this purpose. One was an estimate of the diameter of the velocity and temperature configuration, and the other was an estimate of the diameter of the configuration within the time-delay bound τ.