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Collective Dynamics of Opposing Groups with Stochastic Communication

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Abstract

We propose models describing the collective dynamics of two opposing groups of individuals with stochastic communication. Individuals from the same group are assumed to align in a stochastic manner, while individuals from different groups are assumed to anti-align. Under reasonable assumptions, we prove the large time behavior of separation, in the sense that the variation inside a group is much less than the distance between the two groups. The separation phenomena are verified by numerical simulations.

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Notes

  1. From now on, summations in \(i,i^{\prime }\) always run from 1 to N1, and \(j,j^{\prime }\) from 1 to N2. \({\sum }_{i^{\prime }\ne i}\) means summation over the index \(i^{\prime }\) which is not equal to i.

  2. This notion is similar to the ‘well-separation’ condition (1.3) in [12].

  3. Notice that 0 is the smallest eigenvalue of A with eigenvector \((1,\dots ,1)^{T}\).

  4. In the rest of this proof, different C may denote different constants which are all independent of N.

  5. This means that at time T the two groups are separated by the perpendicular bisector of the segment connecting \(\bar {\mathbf {x}}(T)\) and \(\bar {\mathbf {y}}(T)\).

  6. Without this normalization, xi(t) and yj(t) may exhibit exponential growth in time, which makes it hard to demonstrate the extent of separation.

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Acknowledgements

SJ was partially supported by NSFC grants Nos. 31571071 and 11871297. RS was supported in part by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR grant N00014-1812465.

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Authors and Affiliations

  1. School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shi Jin

  2. University of Maryland, College Park, 4176 Campus Drive, College Park, MD, 20742, USA

    Ruiwen Shu

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  1. Shi Jin
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  2. Ruiwen Shu
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Correspondence to Shi Jin.

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Dedication to Prof. Enrique Zuazua on the occasion of his 60th birthday.

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Jin, S., Shu, R. Collective Dynamics of Opposing Groups with Stochastic Communication. Vietnam J. Math. 49, 619–636 (2021). https://doi.org/10.1007/s10013-020-00430-2

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