# American Institute of Mathematical Sciences

July  2014, 19(5): 1373-1410. doi: 10.3934/dcdsb.2014.19.1373

## The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime

 1 Dalhousie University, Department of Mathematics and Statistics, Halifax, Nova Scotia, B3H 3J5, Canada 2 Mathematics Department, University of British Columbia, Vancouver, B.C. V6T 1Z2, Canada 3 Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  January 2012 Revised  April 2012 Published  April 2014

The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${\mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.
Citation: Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373
##### References:
 [1] W. Chen and M. J. Ward, Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model, Europ. J. Appl. Math, 20 (2009), 187-214. doi: 10.1017/S0956792508007766. [2] W. Chen and M. J. Ward, The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model, SIAM J. Appl. Dyn. Sys., 10 (2011), 582-666. doi: 10.1137/09077357X. [3] A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana U. Math. J., 50 (2001), 443-507. doi: 10.1512/iumj.2001.50.1873. [4] A. Doelman, R. A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray Scott model, Memoirs of the AMS, 155 (2002), xii+64 pp. doi: 10.1090/memo/0737. [5] A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotic approach, Physica D, 122 (1998), 1-36. doi: 10.1016/S0167-2789(98)00180-8. [6] A. Doelman and T. J. Kaper, Semistrong pulse interactions in a class of coupled reaction-diffusion systems, SIAM J. Appl. Dyn. Sys., 2 (2003), 53-96. doi: 10.1137/S1111111102405719. [7] A. Doelman, T. J. Kaper and K. Promislow, Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM J. Math. Anal., 38 (2007), 1760-1787. doi: 10.1137/050646883. [8] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Meth. Appl. Sci., 27 (2004), 1783-1801. doi: 10.1002/mma.569. [9] D. Iron and M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model, SIAM J. Appl. Math., 62 (2002), 1924-1951. doi: 10.1137/S0036139901393676. [10] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62. doi: 10.1016/S0167-2789(00)00206-2. [11] D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390. doi: 10.1007/s00285-003-0258-y. [12] K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in a one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162. doi: 10.1093/imamat/hxl028. [13] T. Kolokolnikov and M. J. Ward, Reduced-wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, Europ. J. Appl. Math., 14 (2003), 513-545. doi: 10.1017/S0956792503005254. [14] T. Kolokolnikov and M. J. Ward, Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model, DCDS-B, 4 (2004), 1033-1064. doi: 10.3934/dcdsb.2004.4.1033. [15] T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Studies in Appl. Math., 115 (2005), 21-71. doi: 10.1111/j.1467-9590.2005.01554. [16] T. Kolokolnikov, M. J. Ward and J. Wei, Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Sci., 19 (2009), 1-56. doi: 10.1007/s00332-008-9024-z. [17] T. Kolokolnikov, M. J. Ward, and J. Wei, Self-replication of mesa patterns in reaction-diffusion models, Physica D, 236 (2007), 104-122. doi: 10.1016/j.physd.2007.07.014. [18] T. Koloklonikov, M. J. Ward and J. Wei, Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model, Interfaces and Free Boundaries, 8 (2006), 185-222. doi: 10.4171/IFB/140. [19] T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., 71 (2011), 1428-1457. doi: 10.1137/100808381. [20] K. J. Lee and H. L. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion systems, Phys. Rev. E., 51 (1995), 1899-1915. doi: 10.1103/PhysRevE.51.1899. [21] W. Liu, A. L. Bertozzi, and T. Kolokolnikov, Diffuse interface surface tension models in an expanding flow, Comm. Math. Sci., 10 (2012), 387-418. doi: 10.4310/CMS.2012.v10.n1.a16. [22] R. McKay and T. Kolokolnikov, Theodore Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 191-220. [23] C. B. Muratov and V. V. Osipov, Stability of static spike autosolitons in the Gray-Scott model, SIAM J. Appl. Math., 62 (2002), 1463-1487. doi: 10.1137/S0036139901384285. [24] C. B. Muratov and V. V. Osipov, Static spike autosolitons in the Gray-Scott model, J. Phys. A: Math Gen., 33 (2000), 8893-8916. doi: 10.1088/0305-4470/33/48/321. [25] Y. Nishiura, Far-from Equilibrium Dynamics, Translated from the 1999 Japanese original by Kunimochi Sakamoto. Translations of Mathematical Monographs, 209. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2002. [26] K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis Model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011. [27] J. E. Pearson, Complex patterns in a simple system, Science, 216 (1993), 189-192. [28] A. Potapov and T. Hillen, Metastability in chemotaxis models, J. Dynam. Diff. Eq., 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3. [29] 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. Meth. Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029. [30] M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime - hotpsots, bifurcations, and suppression, SIAM J. Appl. Dyn. Sys., 9 (2010), 462-483. doi: 10.1137/090759069. [31] M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotpsots in reaction-diffusion models of crime, Proc. Nat. Acad. Sci., 107 (2010), 3961-3965. [32] B. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117. [33] W. Sun, M. J. Ward and R. Russell, The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities, SIAM J. Appl. Dyn. Syst., 4 (2005), 904-953. doi: 10.1137/040620990. [34] H. Van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations, Indiana Univ. Math. J., 54 (2005), 1219-1301. doi: 10.1512/iumj.2005.54.2792. [35] M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonlinear Sci., 13 (2003), 209-264. doi: 10.1007/s00332-002-0531-z. [36] M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns in the Schnakenburg model, Studies in Appl. Math., 109 (2002), 229-264. doi: 10.1111/1467-9590.00223. [37] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, Europ. J. Appl. Math., 13 (2002), 283-320. doi: 10.1017/S0956792501004442. [38] M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model, Europ. J. Appl. Math., 14 (2003), 677-711. doi: 10.1017/S0956792503005278. [39] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458. doi: 10.1007/s00332-001-0380-1. [40] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the strong coupling case, J. Diff. Eq., 178 (2002), 478-518. doi: 10.1006/jdeq.2001.4019. [41] J. Wei and M. Winter, Existence and stability of multiple spot solutions for the Gray-Scott model in $\mathbb{R}^2$, Physica D., 176 (2003), 147-180. doi: 10.1016/S0167-2789(02)00743-1. [42] J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y. [43] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl. (9), 83 (2004), 433-476. doi: 10.1016/j.matpur.2003.09.006. [44] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbb{R}^2$, Studies in Appl. Math., 110 (2003), 63-102. doi: 10.1111/1467-9590.00231. [45] J. Wei and L. Zhang, On a nonlocal eigenvalue problem, Ann. Sc. Norm. Sup. Pisa C1. Sci., 30 (2001), 41-62. [46] J. Wei(2008), Existence and stability of spikes for the Gierer-Meinhardt system, book chapter in Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 5 (M. Chipot ed.), Elsevier, pp. 487-585. doi: 10.1016/S1874-5733(08)80013-7.

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##### References:
 [1] W. Chen and M. J. Ward, Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model, Europ. J. Appl. Math, 20 (2009), 187-214. doi: 10.1017/S0956792508007766. [2] W. Chen and M. J. Ward, The stability and dynamics of localized spot patterns in the two-dimensional Gray-Scott model, SIAM J. Appl. Dyn. Sys., 10 (2011), 582-666. doi: 10.1137/09077357X. [3] A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana U. Math. J., 50 (2001), 443-507. doi: 10.1512/iumj.2001.50.1873. [4] A. Doelman, R. A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray Scott model, Memoirs of the AMS, 155 (2002), xii+64 pp. doi: 10.1090/memo/0737. [5] A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1D Gray-Scott model: A matched asymptotic approach, Physica D, 122 (1998), 1-36. doi: 10.1016/S0167-2789(98)00180-8. [6] A. Doelman and T. J. Kaper, Semistrong pulse interactions in a class of coupled reaction-diffusion systems, SIAM J. Appl. Dyn. Sys., 2 (2003), 53-96. doi: 10.1137/S1111111102405719. [7] A. Doelman, T. J. Kaper and K. Promislow, Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer-Meinhardt model, SIAM J. Math. Anal., 38 (2007), 1760-1787. doi: 10.1137/050646883. [8] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Meth. Appl. Sci., 27 (2004), 1783-1801. doi: 10.1002/mma.569. [9] D. Iron and M. J. Ward, The dynamics of multi-spike solutions to the one-dimensional Gierer-Meinhardt model, SIAM J. Appl. Math., 62 (2002), 1924-1951. doi: 10.1137/S0036139901393676. [10] D. Iron, M. J. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Physica D, 150 (2001), 25-62. doi: 10.1016/S0167-2789(00)00206-2. [11] D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390. doi: 10.1007/s00285-003-0258-y. [12] K. Kang, T. Kolokolnikov and M. J. Ward, The stability and dynamics of a spike in a one-dimensional Keller-Segel model, IMA J. Appl. Math., 72 (2007), 140-162. doi: 10.1093/imamat/hxl028. [13] T. Kolokolnikov and M. J. Ward, Reduced-wave Green's functions and their effect on the dynamics of a spike for the Gierer-Meinhardt model, Europ. J. Appl. Math., 14 (2003), 513-545. doi: 10.1017/S0956792503005254. [14] T. Kolokolnikov and M. J. Ward, Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model, DCDS-B, 4 (2004), 1033-1064. doi: 10.3934/dcdsb.2004.4.1033. [15] T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: The low feed-rate regime, Studies in Appl. Math., 115 (2005), 21-71. doi: 10.1111/j.1467-9590.2005.01554. [16] T. Kolokolnikov, M. J. Ward and J. Wei, Spot self-replication and dynamics for the Schnakenburg model in a two-dimensional domain, J. Nonlinear Sci., 19 (2009), 1-56. doi: 10.1007/s00332-008-9024-z. [17] T. Kolokolnikov, M. J. Ward, and J. Wei, Self-replication of mesa patterns in reaction-diffusion models, Physica D, 236 (2007), 104-122. doi: 10.1016/j.physd.2007.07.014. [18] T. Koloklonikov, M. J. Ward and J. Wei, Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model, Interfaces and Free Boundaries, 8 (2006), 185-222. doi: 10.4171/IFB/140. [19] T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion, SIAM J. Appl. Math., 71 (2011), 1428-1457. doi: 10.1137/100808381. [20] K. J. Lee and H. L. Swinney, Lamellar structures and self-replicating spots in a reaction-diffusion systems, Phys. Rev. E., 51 (1995), 1899-1915. doi: 10.1103/PhysRevE.51.1899. [21] W. Liu, A. L. Bertozzi, and T. Kolokolnikov, Diffuse interface surface tension models in an expanding flow, Comm. Math. Sci., 10 (2012), 387-418. doi: 10.4310/CMS.2012.v10.n1.a16. [22] R. McKay and T. Kolokolnikov, Theodore Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 191-220. [23] C. B. Muratov and V. V. Osipov, Stability of static spike autosolitons in the Gray-Scott model, SIAM J. Appl. Math., 62 (2002), 1463-1487. doi: 10.1137/S0036139901384285. [24] C. B. Muratov and V. V. Osipov, Static spike autosolitons in the Gray-Scott model, J. Phys. A: Math Gen., 33 (2000), 8893-8916. doi: 10.1088/0305-4470/33/48/321. [25] Y. Nishiura, Far-from Equilibrium Dynamics, Translated from the 1999 Japanese original by Kunimochi Sakamoto. Translations of Mathematical Monographs, 209. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2002. [26] K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis Model, Physica D, 240 (2011), 363-375. doi: 10.1016/j.physd.2010.09.011. [27] J. E. Pearson, Complex patterns in a simple system, Science, 216 (1993), 189-192. [28] A. Potapov and T. Hillen, Metastability in chemotaxis models, J. Dynam. Diff. Eq., 17 (2005), 293-330. doi: 10.1007/s10884-005-2938-3. [29] 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. Meth. Appl. Sci., 18 (2008), 1249-1267. doi: 10.1142/S0218202508003029. [30] M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime - hotpsots, bifurcations, and suppression, SIAM J. Appl. Dyn. Sys., 9 (2010), 462-483. doi: 10.1137/090759069. [31] M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotpsots in reaction-diffusion models of crime, Proc. Nat. Acad. Sci., 107 (2010), 3961-3965. [32] B. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117. [33] W. Sun, M. J. Ward and R. Russell, The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities, SIAM J. Appl. Dyn. Syst., 4 (2005), 904-953. doi: 10.1137/040620990. [34] H. Van der Ploeg and A. Doelman, Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations, Indiana Univ. Math. J., 54 (2005), 1219-1301. doi: 10.1512/iumj.2005.54.2792. [35] M. J. Ward and J. Wei, Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model, J. Nonlinear Sci., 13 (2003), 209-264. doi: 10.1007/s00332-002-0531-z. [36] M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns in the Schnakenburg model, Studies in Appl. Math., 109 (2002), 229-264. doi: 10.1111/1467-9590.00223. [37] M. J. Ward and J. Wei, Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, Europ. J. Appl. Math., 13 (2002), 283-320. doi: 10.1017/S0956792501004442. [38] M. J. Ward and J. Wei, Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model, Europ. J. Appl. Math., 14 (2003), 677-711. doi: 10.1017/S0956792503005278. [39] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J. Nonlinear Sci., 11 (2001), 415-458. doi: 10.1007/s00332-001-0380-1. [40] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the strong coupling case, J. Diff. Eq., 178 (2002), 478-518. doi: 10.1006/jdeq.2001.4019. [41] J. Wei and M. Winter, Existence and stability of multiple spot solutions for the Gray-Scott model in $\mathbb{R}^2$, Physica D., 176 (2003), 147-180. doi: 10.1016/S0167-2789(02)00743-1. [42] J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y. [43] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl. (9), 83 (2004), 433-476. doi: 10.1016/j.matpur.2003.09.006. [44] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbb{R}^2$, Studies in Appl. Math., 110 (2003), 63-102. doi: 10.1111/1467-9590.00231. [45] J. Wei and L. Zhang, On a nonlocal eigenvalue problem, Ann. Sc. Norm. Sup. Pisa C1. Sci., 30 (2001), 41-62. [46] J. Wei(2008), Existence and stability of spikes for the Gierer-Meinhardt system, book chapter in Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 5 (M. Chipot ed.), Elsevier, pp. 487-585. doi: 10.1016/S1874-5733(08)80013-7.
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