Traveling wave solutions in a reaction-diffusion model for criminal activity

Multiscale Modeling and Simulation

Volumn 11, Issue 4, 2013, Pages 1097-1126

  Berestycki, H ^a   Rodriguez N ^b   Ryzhik, L ^b  

^a ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES   (France)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Criminal activity modeling; Gap problem; Partial differential equations; Traveling wave solutions

Indexed keywords

CRIMINAL ACTIVITIES; ENTIRE SOLUTIONS; EXCITABLE MEDIA; GAP PROBLEM; LONG TIME BEHAVIOR; REACTION DIFFUSION SYSTEMS; REACTION-DIFFUSION MODELS; TRAVELING WAVE SOLUTION;

PARTIAL DIFFERENTIAL EQUATIONS; WAVE PROPAGATION;

CRIME;

EID: 84890901806     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/12089884X     Document Type: Article

References (21)

1. M. Al-Refai, Existence, uniqueness and bounds for a problem in combustion theory, J. Comput. Appl. Math., 167 (2004), pp. 255-269.
2. H. Berestycki, F. Hamel, and R. Monneau, One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math J., 103 (2000), pp. 375-396.
3. H. Berestycki, F. Hamel, and H. Matano, Bistable traveling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), pp. 729-788.
4. H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity, European J. Appl. Math., 21 (2010), pp. 371-399.
5. V. Booth and T. Erneux, Understanding propagation failure as a slow capture near a limit point, SIAM J. Appl. Math., 55 (1995), pp. 1372-1389.
6. L. E. Cohen and M. Felson, Social change and crime rate trends: A routine activity approach, American Sociological Review, 44 (1979), pp. 588-608.
7. P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), pp. 335-361.
8. P. C. Fife and L. A. Peletier, Clines induced by variable migration, in BiologicalGrowth and Spread (Proc. Conf., Heidelberg, 1979), Lecture Notes in Biomath. 38, Springer, Berlin, 1980, pp. 276-278.
9. R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), pp. 353-369.
10. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.
11. J.-S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), pp. 193-212.
12. S. Johnson, K. Bowers, and Hirschfield, New insights into the spatial and temporal distribution of repeat victimisation, Br. J. Criminology, 37 (1997), pp. 224-244.
13. T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), pp. 293-316.
14. G. O. Mohler and M. B. Short, Geographic profiling from kinetic models of criminal behavior, SIAM J. Appl. Math., 72 (2012), pp. 163-180.
15. G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. Schoenberg, and G. E. Tita, Selfexciting point process modeling of crime, J. Amer. Statist. Assoc., 106 (2011), pp. 100-108.
16. J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), pp. 2061-2070.
17. F. Ramón, R. W. Joyner, and J. W. Moore, Propagation of action potentials in inhomogeneous axon regions, Federation Proceedings, 34 (1975), pp. 1357-1363.
18. D. H. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Math. 309, Springer-Verlag, Berlin, New York, 1973.
19. M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), pp. 1249-1267.
20. A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. 140, translated from the Russian manuscript by J. F. Heyda, American Mathematical Society, Providence, RI, 1994.
21. J. Q. Wilson and G. L. Kelling, Broken windows and police and neighborhood safety, Atlantic Mon., 249 (1998), pp. 29-38.