### Mathematical Biosciences and Engineering

2020, Issue 5: 5651-5671. doi: 10.3934/mbe.2020304
Research article Special Issues

# Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays

• Received: 13 May 2020 Accepted: 13 August 2020 Published: 25 August 2020
• We study a variant of the Cucker-Smale system with distributed reaction delays. Using backward-forward and stability estimates on the quadratic velocity fluctuations we derive sufficient conditions for asymptotic flocking of the solutions. The conditions are formulated in terms of moments of the delay distribution and they guarantee exponential decay of velocity fluctuations towards zero for large times. We demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking condition can be resolved analytically, leading to an explicit formula. For the other two distributions, the satisfiability of the assumptions is investigated numerically.

Citation: Jan Haskovec, Ioannis Markou. Exponential asymptotic flocking in the Cucker-Smale model with distributed reaction delays[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5651-5671. doi: 10.3934/mbe.2020304

### Related Papers:

• We study a variant of the Cucker-Smale system with distributed reaction delays. Using backward-forward and stability estimates on the quadratic velocity fluctuations we derive sufficient conditions for asymptotic flocking of the solutions. The conditions are formulated in terms of moments of the delay distribution and they guarantee exponential decay of velocity fluctuations towards zero for large times. We demonstrate the applicability of our theory to particular delay distributions - exponential, uniform and linear. For the exponential distribution, the flocking condition can be resolved analytically, leading to an explicit formula. For the other two distributions, the satisfiability of the assumptions is investigated numerically.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.285 1.3

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