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September  2015, 10(3): 443-475. doi: 10.3934/nhm.2015.10.443

A model of riots dynamics: Shocks, diffusion and thresholds

1. 

Ecole des Hautes Etudes en Sciences Sociales and CNRS, Centre d'Analyse et de Mathématique Sociales (CAMS, UMR8557), 190-198, avenue de France - 75013 Paris, France

2. 

Ecole des Hautes Etudes en Sciences Sociales and CNRS, Centre d'Analyse et de Mathématique Sociales (CAMS, UMR8557), 190-198 avenue de France - 75013 Paris, France

3. 

UNC Chapel Hill, Department of Mathematics, Phillips Hall, CB # 3250, Chapel Hill, NC 27599-3250, United States

Received  November 2014 Revised  February 2015 Published  July 2015

We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.
Citation: Henri Berestycki, Jean-Pierre Nadal, Nancy Rodíguez. A model of riots dynamics: Shocks, diffusion and thresholds. Networks & Heterogeneous Media, 2015, 10 (3) : 443-475. doi: 10.3934/nhm.2015.10.443
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show all references

References:
[1]

S. Alcaide, Movimiento 15-M: Los Ciudadanos Exigen Reconstruir La Política,, 2011., ().   Google Scholar

[2]

Houghton Mifflin Harcourt, 1972. Google Scholar

[3]

Policing, 7 (2012), 21-31. doi: 10.1093/police/pas049.  Google Scholar

[4]

Journal of Statistical Mechanics: Theory and Experiment, (2014), P03010, 15 pp.  Google Scholar

[5]

Physica D, 239 (2010), 1048-1060. doi: 10.1016/j.physd.2010.02.016.  Google Scholar

[6]

European Journal of Applied Mathematics, 21 (2010), 371-399. doi: 10.1017/S0956792510000185.  Google Scholar

[7]

to appear in the European Journal of Applied Mathematics, (2015), 28 pp. Google Scholar

[8]

The Crowd: A Study of the Popular Mind, Editions Felix Alcan, 1895 (9th ed. 1905), Viking Press, New York, 1960. Google Scholar

[9]

Journal of Statistical Physics, 151 (2013), 567-606. doi: 10.1007/s10955-012-0687-3.  Google Scholar

[10]

PloS one, 7 (2012), e48596, 1-9. doi: 10.1371/journal.pone.0048596.  Google Scholar

[11]

R. Clarke and C. Lett, What happened when Michael Brown met Officer Darren Wilson,, 2014., ().   Google Scholar

[12]

Proceedings of the National Academy of Sciences of the United States of America, 105 (2008), 15649-15653. doi: 10.1073/pnas.0803685105.  Google Scholar

[13]

ETC Publications, 1995. Google Scholar

[14]

Scientific Reports, 3 (2013), 1-9. doi: 10.1038/srep01303.  Google Scholar

[15]

C. Fizgerald, The Final Report: The L.A. Riots,, 2006., ().   Google Scholar

[16]

Columbia University Press, 2005. Google Scholar

[17]

Scientific reports, 1 (2011), 1-7. Google Scholar

[18]

Mathematical Models and Methods in Applied Sciences (M3AS), 19 (2009), 1441-1481. doi: 10.1142/S0218202509003887.  Google Scholar

[19]

American Journal of Sociology, 83 (1978), 1420-1443. doi: 10.1086/226707.  Google Scholar

[20]

Biometrika, 58 (1971), 83-90. doi: 10.1093/biomet/58.1.83.  Google Scholar

[21]

Mathematical Social Sciences, 69 (2014), 12-21. doi: 10.1016/j.mathsocsci.2014.01.004.  Google Scholar

[22]

Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, (2014), 731-740. doi: 10.1145/2623330.2623679.  Google Scholar

[23]

W. Lowery, C. D. Leonnig and M. Berman, Even before Michael Brown's slaying in Ferguson, racial questions hung over police,, 2014., ().   Google Scholar

[24]

Public Affairs, New York, first edition, 2005. Google Scholar

[25]

White Plains, New York, 1978. Google Scholar

[26]

Journal of the American Statistical Association, 106 (2011), 100-108. doi: 10.1198/jasa.2011.ap09546.  Google Scholar

[27]

Journal of Ethnic and Migration Studies, 35 (2009), 731-751. doi: 10.1080/13691830902826137.  Google Scholar

[28]

The American Journal of Sociology, 106 (2000), 173-208. doi: 10.1086/303110.  Google Scholar

[29]

T. Newburn, The Ferguson riots may seem similar to those in UK in 2011 - but there are stark contrasts,, 2014., ().   Google Scholar

[30]

Annals of the Institute of Statistical Mathematics, 50 (1998), 379-402. doi: 10.1023/A:1003403601725.  Google Scholar

[31]

HAL archives ouvertes, https://hal.archives-ouvertes.fr/halshs-0048422, 2010. Google Scholar

[32]

J. Salter, Police shooting of black teenager in St. Louis reignites anger,, 2014., ().   Google Scholar

[33]

Journal of Conflict Resolution, 17 (1973), 381-428. doi: 10.1177/002200277301700302.  Google Scholar

[34]

Journal of Quantitative Criminology, 25 (2009), 325-339. doi: 10.1007/s10940-009-9068-8.  Google Scholar

[35]

Math. Models Methods Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029.  Google Scholar

[36]

Social Forces, 86 (2007), 385-415. doi: 10.1093/sf/86.2.385.  Google Scholar

[37]

The Guardian, December 2011. Google Scholar

[38]

Neural computation, 10 (1998), 821-835. doi: 10.1162/089976698300017502.  Google Scholar

[39]

American Mathematical Society, Providence, translatio edition, 1994.  Google Scholar

[40]

Biophysics, 12 (1972), 1-24. Google Scholar

[41]

Wiley-Blackwell, 2008. doi: 10.1002/9780470712962.  Google Scholar

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