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A dynamic spatial model of conflict escalation

Published online by Cambridge University Press:  05 November 2015

P. BAUDAINS ,
H.M. FRY ,
T.P. DAVIES ,
A.G. WILSON  and
S.R. BISHOP
P. BAUDAINS
Affiliation:
UCL Department of Security and Crime Science, University College London, 35 Tavistock Square, London, WC1H 9EZ, UK UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
H.M. FRY
Affiliation:
UCL Centre for Advanced Spatial Analysis, University College London, Gower Street, London, WC1E 6BT, UK
T.P. DAVIES
Affiliation:
UCL Department of Security and Crime Science, University College London, 35 Tavistock Square, London, WC1H 9EZ, UK UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK
A.G. WILSON
Affiliation:
UCL Centre for Advanced Spatial Analysis, University College London, Gower Street, London, WC1E 6BT, UK
S.R. BISHOP
Affiliation:
UCL Department of Security and Crime Science, University College London, 35 Tavistock Square, London, WC1H 9EZ, UK
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Abstract

In both historical and modern conflicts, space plays a critical role in how interactions occur over time. Despite its importance, the spatial distribution of adversaries has often been neglected in mathematical models of conflict. In this paper, we propose an entropy-maximising spatial interaction method for disaggregating the impact of space, employing a general notion of ‘threat’ between two adversaries. This approach addresses a number of limitations that are associated with partial differential equation approaches to spatial disaggregation. We use this method to spatially disaggregate the Richardson model of conflict escalation, and then explore the resulting model with both analytical and numerical treatments. A bifurcation is identified that dramatically influences the resulting spatial distribution of conflict and is shown to persist under a range of model specifications. Implications of this finding for real-world conflicts are discussed.

Keywords

ConflictSpatial ModellingDifferential Equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 3: Mathematical Modelling of Crime and Security , June 2016 , pp. 530 - 553
Copyright
Copyright © Cambridge University Press 2015 

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References

Atkinson, M. P., Gutfraind, A. & Kress, M. (2011) When do armed revolts succeed: Lessons from Lanchester theory. J. Oper. Res. Soc. 63 (10), 1363–1373.CrossRefGoogle Scholar
Blank, L., Enomoto, C. E., Gegax, D., Mcguckin, T. & Simmons, C. (2008) A dynamic model of insurgency: The case of the war in Iraq. Peace Econ. Peace Sci. Public Policy 14 (2), 1–26.CrossRefGoogle Scholar
Brantingham, P. J., Tita, G. E., Short, M. B. & Reid, S. E. (2012) The ecology of gang territorial boundaries. Criminology 50 (3), 851–885.CrossRefGoogle Scholar
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.CrossRefGoogle ScholarPubMed
Deitchman, S. J. (1962) A Lanchester model of guerrilla warfare. Oper. Res. 10 (6), 818–827.CrossRefGoogle Scholar
Dennett, A. & Wilson, A. G. (2013) A multi-level spatial interaction modelling framework for estimating inter-regional migration in Europe. Environ. Plan. A 45 (6), 1491–1507.CrossRefGoogle Scholar
Durrett, R. & Levin, S. (1994) The importance of being discrete (and spatial). Theor. Population Biol. 46 (3), 363–394.CrossRefGoogle Scholar
Epstein, J. M. (1997) Nonlinear Dynamics, Mathematical Biology, and Social Science, pp. 97–99, Addison-Wesley, Reading, MA.Google Scholar
Fry, H. & Wilson, A. G. (2012) A dynamic global trade model with four sectors: Food, natural resources, manufactured goods and labour. CASA Working Paper, 178.Google Scholar
González, E. & Villena, M. (2011) Spatial Lanchester models. Eur. J. Oper. Res. 210 (3), 706–715.CrossRefGoogle Scholar
Harris, B. & Wilson, A. (1978) Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environ. Plan. A 10 (4), 371–388.CrossRefGoogle Scholar
Ilachinski, A. (2004) Artificial War: Multiagent-Based Simulation of Combat, pp. 22–25, World Scientific Publishing Co. Pte. Ltd., Singapore.CrossRefGoogle Scholar
Intriligator, M. D. & Brito, D. L. (1988) A predator-prey model of guerrilla warfare. Synthese 2 (2), 235–244.CrossRefGoogle Scholar
Jackson, S., Russett, B., Snidal, D. & Sylvan, D. (1978) Conflict and coercion in dependent states. J. Conflict Resolution 22 (4), 627–657.CrossRefGoogle Scholar
Karmeshu, M., Jain, V. & Mahajan, A. (1990) A dynamic model of domestic political conflict process. J. Conflict Resolution 34 (2), 252–269.CrossRefGoogle Scholar
Keane, T. (2011) Combat modelling with partial differential equations. Appl. Math. Modelling 35 (6), 2723–2735.CrossRefGoogle Scholar
Kress, M. & MacKay, N. J. (2014) Bits or shots in combat? The generalized Deitchman model of guerrilla warfare. Oper. Res. Lett. 42 (1), 102–108.CrossRefGoogle Scholar
Lanchester, F. W. (1916) Aircraft in Warfare: The Dawn of the Fourth Arm, Constable and Company Limited, London.Google Scholar
Liebovitch, L. S., Naudot, V., Vallacher, R., Nowak, A., Bui-Wrzosinska, L. & Coleman, P. (2008) Dynamics of two-actor cooperation-competition conflict models. Phys. A: Stat. Mech. Appl. 387 (25), 6360–6378.CrossRefGoogle Scholar
Lotka, A. J. (1925) Elements of Physical Biology, Williams & Wilkins Company, Baltimore.Google Scholar
Mandelbrot, B. (1967) How long is the coast of britain? Statistical self-similarity and fractional dimension. Science 156 (3775), 636–638.CrossRefGoogle Scholar
Pitcher, A. B. (2010) Adding police to a mathematical model of burglary. Eur. J. Appl. Math. 21 (4–5), 401–419.CrossRefGoogle Scholar
Protopopescu, V., Santoro, R. & Dockery, J. (1989) Combat modeling with partial differential equations. Eur. J. Oper. Res. 38 (2), 178–183.CrossRefGoogle Scholar
Qubbaj, M. & Muneepeerakul, R. (2012) Two-actor conflict with time delay: A dynamical model. Phys. Rev. E 86 (5), 056101.CrossRefGoogle Scholar
Richardson, L. F. (1952) Contiguity and deadly quarrels: The local pacifying influence. J. R. Stat. Soc. Ser. A (General) 115 (2), 219–231.CrossRefGoogle Scholar
Richardson, L. F. (1960) Arms and Insecurity, The Boxwood Press, Pittsburgh, PA.Google Scholar
Richardson, L. F. (1961) The problem of contiguity; an appendix to statistics of deadly quarrels. Gen. Syst. Yearbook 6, 140–87.Google Scholar
Rojas-Pacheco, A., Obregón-Quintana, B., Liebovitch, L. S. & Guzmán-Vargas, L. (2013) Time-delay effects on dynamics of a two-actor conflict model. Phys. A: Stat. Mech. Appl. 392 (3), 458–467.CrossRefGoogle Scholar
Saperstein, A. M. (2007) Chaos in models of arms races and the initiation of war. Complexity 12 (3), 22–26.CrossRefGoogle Scholar
Senior, M. (1979) From gravity modelling to entropy maximising: A pedagogic guide. Progr. Human Geography 3 (2), 175–210.CrossRefGoogle Scholar
Short, M. B., Bertozzi, A. & Brantingham, P. (2010a) Nonlinear patterns in urban crime: Hotspots, bifurcations and suppression. SIAM J. Appl. Dyn. Syst. 9 (2), 462–483.CrossRefGoogle Scholar
Short, M. B., Brantingham, P. J., Bertozzi, A. L. & Tita, G. E. (2010b) Dissipation and displacement of hotspots in reaction-diffusion models of crime. Proc. Natl. Acad. Sci. 107 (9), 3961–3965.CrossRefGoogle ScholarPubMed
Smith, L. M., Keegan, M. S., Wittman, T., Mohler, G. O. & Bertozzi, A. L. (2010) Improving density estimation by incorporating spatial information. EURASIP J. Adv. Signal Process. 2010, 265631.CrossRefGoogle Scholar
Tobler, W. R. (1970) A computer model simulating urban growth in the Detroit region. Econ. Geography 46, 234–240.CrossRefGoogle Scholar
Wilson, A. G. (1967) A statistical theory of spatial distribution models. Transp. Res. 1 (3), 253–269.CrossRefGoogle Scholar
Wilson, A. G. (1970) Entropy in Urban and Regional Modelling, Pion, London.Google Scholar
Wilson, A. G. (2006) Ecological and urban systems models: Some explorations of similarities in the context of complexity theory. Environ. Plann. A 38 (4), 633–646.CrossRefGoogle Scholar
Wilson, A. G. (2008) Boltzmann, Lotka and Volterra and spatial structural evolution: An integrated methodology for some dynamical systems. J. R. Soc., Interface/the R. Soc. 5 (25), 865–71.CrossRefGoogle ScholarPubMed
Zinnes, D. A. & Muncaster, R. G. (1984) The dynamics of hostile activity and the prediction of war. J. Conflict Resolution 28 (2), 187–229.CrossRefGoogle Scholar