Critical coupling strength of the Cucker-Smale model for flocking

We present a non-trivial positive lower bound for the critical coupling strength of the Cucker-Smale model with a short-range communication weight from the viewpoint of mono-cluster (global) flocking. For a long-range communication weight, it is well known [F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007) 852-862; S.-Y. Ha and J.-G. Liu, A proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297-325] that as long as the coupling strength is positive, mono-cluster flocking occurs asymptotically for any initial configuration. Hence, the critical coupling strength is simply zero. However, for a short-range communication weight, numerical simulations indicate that for a given initial configuration, mono-cluster flocking is possible only in a large-coupling-strength regime depending on the initial configuration. This suggests the positivity of the critical coupling strength in the sense that if the coupling strength is above the critical value, mono-cluster flocking emerges, whereas if it is below the critical value, mono-cluster flocking does not occur. Thus, it is interesting to determine the exact critical value for the coupling strength depending on the initial configuration or at least to estimate the possible range of the coupling strength. In this paper, we show that the critical coupling strength exists and is positive by providing a positive lower bound. We also present the results of several numerical simulations for the upper and lower bounds of the critical coupling strength depending on the initial configurations, and we compare them with analytical results.