Nuno, Juan Carlos and Herrero, Miguel A. and Primicerio, Mario (2008) A triangle model of criminality. Physica A-statistical mechanics and its applications, 387 (12). pp. 2926-2936. ISSN 0378-4371
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This paper is concerned with a quantitative model describing the interaction of three sociological species, termed as owners, criminals and security guards, and denoted by X, Y and Z respectively. In our model, Y is a predator of the species X, and so is Z with respect to Y. Moreover, Z can also be thought of as a predator of X, since this last population is required to bear the costs of maintaining Z. We propose a system of three ordinary differential equations to account for the time evolution of X(t), Y(t) and Z(t) according to our previous assumptions. Out of the various parameters that appear in that system, we select two of them, denoted by H, and h, which are related with the efficiency of the security forces as a control parameter in our discussion. To begin with, we consider the case of large and constant owners population, which allows us to reduce (3)-(5) to a bidimensional system for Y(t) and Z(t). As a preliminary step, this situation is first discussed under the additional assumption that Y(t) + Z(t) is constant. A bifurcation study is then performed in terms of H and h, which shows the key role played by the rate of casualties in Y and Z, that results particularly in a possible onset of histability. When the previous restriction is dropped, we observe the appearance of oscillatory behaviours in the full two-dimensional system. We finally provide a exploratory study of the complete model (3)-(5), where a number of bifurcations appear as parameter H changes, and the corresponding solutions behaviours are described.
|Uncontrolled Keywords:||Criminality; nonlinear dynamics; sociological systems; dynastic cycle; crime; mathematics; sociology|
|Subjects:||Social sciences > Law > Criminology|
Sciences > Mathematics > Differential equations
Social sciences > Sociology > Social research
Sciences > Mathematics > Operations research
Hammurabi’s code of laws is arguably the oldest penal text. It can be retrieved at the address: http://www.fordham.edu/halsall/ancient/hamcode.html.
E. Durkheim, Le crime phénomène normal 1894, in: Les regles de la méthode sociologique. Paris 14 ed. 1960, 65–72.
S. Kanazawa, M.C. Still, Why men commit crimes (and why they desist), Sociological Theory 18 (2000) 434–447.
P. Ormerod, Crime, economic incentives and social networks, The Institute of Economic Affairs, London (2005).
L.E. Cohen, M. Felson, Social change and crime rate trends: A routine activity approach, American Sociological Review 44 (1979) 588–608.
L.E. Cohen, R. Machalek, A general theory of expropriative crime: An evolutionary ecological approach, American Journal of Sociology 3 (1988) 465–501.
M. Felson, R.V. Clarke, Opportunity makes the thief. Police research series, 98. Home Office (UK), 1998.
L. Real, The kinetics of functional response, American Naturalist 111 (1977) 289–300.
A.A. Berryman, The origins and evolution of predator–prey theory, Ecology 73 (1992) 1530–1535.
R.M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 2001.
O. Bäckman, C. Edling, Mathematics matters: On the absence of mathematical models in quantitative sociology, Acta Sociologica 42 (1999) 69–78.
P. Ball, The physical modelling of human social systems, ComPlexUs 1 (2004) 190–206.
C.R. Edling, Mathematics in sociology, Annual Review of Sociology 28 (2002) 197–220.
P. Ormerod, C. Mounfield, L. Smith, Non-linear Modelling of Burglary and Violent Crime in the UK. Volterra Consulting Ltd, 2001.
M. Campbell, P. Ormerod, Social interaction and the dynamics of crime, Volterra Consulting Ltd, 1997.
H. Zhao, Z. Feng, C. Castillo-Chávez, The Dynamics of Poverty and Crime, preprint MTBI-02-08M 9, 2002.
D. Usher, The dynastic cycle and the stationary state, American Economic Review 79 (1989) 1031–1044.
G. Feichtinger, C.V. Forst, C. Piccardi, A nonlinear dynamical model for the dynastic cycle, Chaos, Solitons and Fractals 7 (1996) 257–271.
E. Hairer, G. Wanner, Analysis by its History, Springer, New York, 1995.
Yu.A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 2004.
S.H. Strogatz, Non-linear Dynamics and Chaos, Perseus book Publishing LLC, USA, 2000.
A. Panfilov, Non-linear Dynamical Systems. 2001. Online publication in: http://www-binf.bio.uu.nl/panfilov/nonlin.html.
J.D. Murray, Mathematical Biology, Springer Verlag, New York, 2002.
S. Camazine, J.-L. Deneubourg, N.R. Franks, J. Sneyd, G. Theraulaz, E. Bonabeau, Self-Organization in Biological Systems, Princeton University Press, New Jersey., 2001.
A. Quetelet, Sur l’homme et le developpement de ses facultes, ou Essai de physique sociale, Bachelier, Paris, 1835.
E.L. Glaeser, B. Sacerdote, J.A. Scheinkman, Crime and Social interactions, The Quarterly Journal of Economics 111 (2) (1996) 507–548.
M. Eigen, J.Mc. Caskill, P. Schuster, The molecular quasispecies, Advances in Chemical Physics 75 (1989) 149–263.
S. Cote (Ed.), Criminological Theories. Bridging the Past to the Future, Sage Publications Inc, 2002.
|Deposited On:||10 Sep 2012 12:52|
|Last Modified:||22 Nov 2013 19:43|
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