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Analysis of a heterogeneous model for riot dynamics: the effect of censorship of information

Published online by Cambridge University Press:  15 July 2015

EHESS, CAMS, 190 - 198 avenue de France, 75013 Paris, France email:
UNC Chapel Hill, Department of Mathematics, Phillips Hall, CB#3250, Chapel Hill, NC 27599-3250, USA email:


This paper is concerned with modelling the dynamics of social outbursts of activity, such as protests or riots. In this sequel to our work in Berestycki et al. (Networks and Heterogeneous Media, vol. 10, no. 3, 1–34), written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore its impact on the existence of upheaval waves. The system involves the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in ${\mathbb R}^d$ as well as the existence of traveling wave solutions in certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which lead to pulsating waves in certain cases. We analyse the effects of periodic domains as well as the barrier problem with the help of numerical simulations. The barrier problem refers to the potential blockage of a wavefront due to a spatial heterogeneity in the system which leads to an area of low excitability (referred to as the barrier). We conclude with a variety of open problems.

Copyright © Cambridge University Press 2015 

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[1]Attia, A. M., Aziz, N., Friedman, B. & Elhusseiny, M. F. (2011) Commentary: The impact of social networking tools on political change in Egypt's “Revolution 2.0”. Electron. Commer. Res. Appl. 10 (4), 369374.Google Scholar
[2]Berestycki, H. & Hamel, F. (2002) Front propagation in periodic excitable media. Commun. Pure Appl. Math. 55 (8), 9491032.Google Scholar
[3]Berestycki, H., Hamel, F. & Roques, L. (2004) Analysis of the periodically fragmented environment model: I-Influence of periodic heterogeneous environment on species persistence. J. Math. Biol. 51 (1), 75113.CrossRefGoogle Scholar
[4]Berestycki, H., Larrouturou, B. & Lions, P. L. (1990) Multi-dimensional travelling-wave solutions of a flame propagation model. Arch. Ration. Mech. Anal. 111 (1), 3349.Google Scholar
[5]Berestycki, H. & Nadal, J-P. (2010) Self-organised critical hot spots of criminal activity. Eur. J. Appl. Math. 21 (4–5), 371399.Google Scholar
[6]Berestycki, H., Nadal, J-P. & Rodríguez, N.A model of riot dynamics: Shocks, diffusion, and thresholds. Networks and Heterogeneous Media 10 (3), 134.Google Scholar
[7]Berestycki, H., Rodríguez, N. & Ryzhik, L. (2013) Traveling wave solutions in a reaction-diffusion model for criminal activity. Multiscale Model. Simul. 11 (4), 10971126.CrossRefGoogle Scholar
[8]Braha, D. (2012) Global civil unrest: Contagion, self-organization, and prediction. PloS One 7 (10), 111.CrossRefGoogle ScholarPubMed
[9]Clarke, R. & Lett, C. (2014) What happened when Michael Brown met Officer Darren Wilson, CNN, Nov. 11, 2014.Google Scholar
[10]Davey, M. & Bosman, J. (November 2, 2014) Protests flare after Ferguson police officer is not indicted. The New York Times.Google Scholar
[11]Davies, T. P., Fry, H. M., Wilson, A. G. & Bishop, S. R. (2013) A mathematical model of the London riots and their policing. Sci. Rep. 3 (1303), 19.Google Scholar
[12]Eversley, M. & James, M. (December 4, 2014) No charges in NYC chokehold death; federal inquiry launched. USA Today.Google Scholar
[13]Faye, G. (2013) Existence and stability of traveling pulses in a neural field equation with synaptic depression. SIAM J. Appl. Dyn. Syst. 12 (4), 20322067.Google Scholar
[14]Fizgerald, C. (2006) The Final Report: The L.A. Riots (documentary). National Geographic Chanel.Google Scholar
[15]Ghannam, J. (2011) Social media in the Arab World: Leading up to the uprisings of 2011. A Report to the Center for International Media Assistance, 32 pgs.Google Scholar
[16]Goodman, D. & Baker, A. (December 3, 2014) Wave of protests after grand jury doesn't indict officer in Eric Garner chokehold case. The New York Times.Google Scholar
[17]Hickey, J. (December 1, 2014) 76 arrested in London protest over NY chokehold death. NewsMax.Google Scholar
[18]Hwa Ang, P. & Nadarajan, B. (1996) Censorship and the internet: A Singapore perspective. Commun. ACM 39 (6), 7278.Google Scholar
[19]Kanel', Y. I. (1961) Certain problems of burning-theory equations. Sov. Math. Dokl. 2, 4851.Google Scholar
[20]Kanel', Y. I. (1962) Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory. Matematicheskii Sb. 59 (101), 245288.Google Scholar
[21]Lam, K.-Y. & Lou, Y. (2014) Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal. Bull. Math. Biol. 76 (2), 261–91.Google Scholar
[22]Lang, J. C. & De Sterck, H. (2014) The Arab Spring: A simple compartmental model for the dynamics of a revolution. Math. Soc. Sci. 69, 1221.Google Scholar
[23]Lewis, T. & Keener, J. (2011) Wave-block in excitable media due to regions of depressed excitability. SIAM J. Appl. Math. 61 (1), 293316.Google Scholar
[24]Maciel, G. A. & Lutscher, F. (2013) How individual movement response to habitat edges affects population persistence and spatial spread. Amer. Naturalist 182 (1), 4252.Google Scholar
[25]Nirenberg, L. & Berestycki, H. (1992) Travelling fronts in cylinders. Ann. lI. H. P., Sect. C 9 (5), 497572.Google Scholar
[26]Rodriguez, M., Jaworski, J. & Gorner, J. (December 5, 2014) More protests vowed in Chicago over no charges in police chokehold death. Chicago Tribune.Google Scholar
[27]Rodríguez, N. (2014) On an integro-differential model for pest control in a heterogeneous environment. J. Math. Biol 1177–206. doi: 10.1007/s00285-014-0793-8.Google Scholar
[28]Rodríguez, N. & Ryzhik, L. (2015) Exploring the effects of social preference, economic disparity, and heterogeneous environments on segregation. To appear in Commun. Math. Sci., 1–25.Google Scholar
[29]Schweitzer, F. & Holyst, J. A. (2000) Modelling collective opinion formation by means of active Brownian particles. Eur. Phys. J. B 15, 723732.Google Scholar
[30]Sherratt, J. (1998) On the transition from initial data to travelling waves in the Fisher-KPP equation. Dyn. Stab. Syst. 13 (2), 167174.Google Scholar
[31]Short, M. B., D'Orsogna, M. R., Brantingham, P. J. & Tita, G. E. (2009) Measuring and modeling repeat and near-repeat burglary effects. J. Quant. Criminology 25 (3), 325339.Google Scholar
[32]Short, M. B., D'Orsogna, M. R., Pasour, V. B., Tita, G. E., Brantingham, P. J., Bertozzi, A. L. & Chayes, L. B. (2008) A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 18 (Suppl.), 12491267.Google Scholar
[33]Tufekci, Z. & Wilson, C. (2012) Social media and the decision to participate in political protest: Observations from Tahrir Square. J. Commun. 62 (2), 363379.Google Scholar
[34]Volpert, A. I., Volpert, V. A. & Volpert, V. A. (1994) Traveling Wave Solutions of Parabolic Systems, 140th ed., American Mathematical Society, Providence, RI.Google Scholar