# 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 & 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. doi: 10.1017/S0956792508007766. Google Scholar [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. doi: 10.1137/09077357X. Google Scholar [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. doi: 10.1512/iumj.2001.50.1873. Google Scholar [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). doi: 10.1090/memo/0737. Google Scholar [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. doi: 10.1016/S0167-2789(98)00180-8. Google Scholar [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. doi: 10.1137/S1111111102405719. Google Scholar [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. doi: 10.1137/050646883. Google Scholar [8] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile,, Math. Meth. Appl. Sci., 27 (2004), 1783. doi: 10.1002/mma.569. Google Scholar [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. doi: 10.1137/S0036139901393676. Google Scholar [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. doi: 10.1016/S0167-2789(00)00206-2. Google Scholar [11] D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model,, J. Math. Biol., 49 (2004), 358. doi: 10.1007/s00285-003-0258-y. Google Scholar [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. doi: 10.1093/imamat/hxl028. Google Scholar [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. doi: 10.1017/S0956792503005254. Google Scholar [14] T. Kolokolnikov and M. J. Ward, Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model,, DCDS-B, 4 (2004), 1033. doi: 10.3934/dcdsb.2004.4.1033. Google Scholar [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. doi: 10.1111/j.1467-9590.2005.01554. Google Scholar [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. doi: 10.1007/s00332-008-9024-z. Google Scholar [17] T. Kolokolnikov, M. J. Ward, and J. Wei, Self-replication of mesa patterns in reaction-diffusion models,, Physica D, 236 (2007), 104. doi: 10.1016/j.physd.2007.07.014. Google Scholar [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. doi: 10.4171/IFB/140. Google Scholar [19] T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion,, SIAM J. Appl. Math., 71 (2011), 1428. doi: 10.1137/100808381. Google Scholar [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. doi: 10.1103/PhysRevE.51.1899. Google Scholar [21] W. Liu, A. L. Bertozzi, and T. Kolokolnikov, Diffuse interface surface tension models in an expanding flow,, Comm. Math. Sci., 10 (2012), 387. doi: 10.4310/CMS.2012.v10.n1.a16. Google Scholar [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. Google Scholar [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. doi: 10.1137/S0036139901384285. Google Scholar [24] C. B. Muratov and V. V. Osipov, Static spike autosolitons in the Gray-Scott model,, J. Phys. A: Math Gen., 33 (2000), 8893. doi: 10.1088/0305-4470/33/48/321. Google Scholar [25] Y. Nishiura, Far-from Equilibrium Dynamics,, Translated from the 1999 Japanese original by Kunimochi Sakamoto. Translations of Mathematical Monographs, (1999). Google Scholar [26] K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis Model,, Physica D, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011. Google Scholar [27] J. E. Pearson, Complex patterns in a simple system,, Science, 216 (1993), 189. Google Scholar [28] A. Potapov and T. Hillen, Metastability in chemotaxis models,, J. Dynam. Diff. Eq., 17 (2005), 293. doi: 10.1007/s10884-005-2938-3. Google Scholar [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. doi: 10.1142/S0218202508003029. Google Scholar [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. doi: 10.1137/090759069. Google Scholar [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. Google Scholar [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. doi: 10.1137/S0036139902415117. Google Scholar [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. doi: 10.1137/040620990. Google Scholar [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. doi: 10.1512/iumj.2005.54.2792. Google Scholar [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. doi: 10.1007/s00332-002-0531-z. Google Scholar [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. doi: 10.1111/1467-9590.00223. Google Scholar [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. doi: 10.1017/S0956792501004442. Google Scholar [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. doi: 10.1017/S0956792503005278. Google Scholar [39] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415. doi: 10.1007/s00332-001-0380-1. Google Scholar [40] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the strong coupling case,, J. Diff. Eq., 178 (2002), 478. doi: 10.1006/jdeq.2001.4019. Google Scholar [41] J. Wei and M. Winter, Existence and stability of multiple spot solutions for the Gray-Scott model in $\mathbbR^2$,, Physica D., 176 (2003), 147. doi: 10.1016/S0167-2789(02)00743-1. Google Scholar [42] J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems,, J. Math. Biol., 57 (2008), 53. doi: 10.1007/s00285-007-0146-y. Google Scholar [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. doi: 10.1016/j.matpur.2003.09.006. Google Scholar [44] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbbR^2$,, Studies in Appl. Math., 110 (2003), 63. doi: 10.1111/1467-9590.00231. Google Scholar [45] J. Wei and L. Zhang, On a nonlocal eigenvalue problem,, Ann. Sc. Norm. Sup. Pisa C1. Sci., 30 (2001), 41. Google Scholar [46] J. Wei(2008), Existence and stability of spikes for the Gierer-Meinhardt system,, book chapter in Handbook of Differential Equations, (): 487. doi: 10.1016/S1874-5733(08)80013-7. Google Scholar

show all references

##### 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. doi: 10.1017/S0956792508007766. Google Scholar [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. doi: 10.1137/09077357X. Google Scholar [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. doi: 10.1512/iumj.2001.50.1873. Google Scholar [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). doi: 10.1090/memo/0737. Google Scholar [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. doi: 10.1016/S0167-2789(98)00180-8. Google Scholar [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. doi: 10.1137/S1111111102405719. Google Scholar [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. doi: 10.1137/050646883. Google Scholar [8] T. Hillen and A. Potapov, The one-dimensional chemotaxis model: Global existence and asymptotic profile,, Math. Meth. Appl. Sci., 27 (2004), 1783. doi: 10.1002/mma.569. Google Scholar [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. doi: 10.1137/S0036139901393676. Google Scholar [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. doi: 10.1016/S0167-2789(00)00206-2. Google Scholar [11] D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model,, J. Math. Biol., 49 (2004), 358. doi: 10.1007/s00285-003-0258-y. Google Scholar [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. doi: 10.1093/imamat/hxl028. Google Scholar [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. doi: 10.1017/S0956792503005254. Google Scholar [14] T. Kolokolnikov and M. J. Ward, Bifurcation of spike equilibria in the near-shadow Gierer-Meinhardt model,, DCDS-B, 4 (2004), 1033. doi: 10.3934/dcdsb.2004.4.1033. Google Scholar [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. doi: 10.1111/j.1467-9590.2005.01554. Google Scholar [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. doi: 10.1007/s00332-008-9024-z. Google Scholar [17] T. Kolokolnikov, M. J. Ward, and J. Wei, Self-replication of mesa patterns in reaction-diffusion models,, Physica D, 236 (2007), 104. doi: 10.1016/j.physd.2007.07.014. Google Scholar [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. doi: 10.4171/IFB/140. Google Scholar [19] T. Kolokolnikov and J. Wei, Stability of spiky solutions in a competition model with cross-diffusion,, SIAM J. Appl. Math., 71 (2011), 1428. doi: 10.1137/100808381. Google Scholar [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. doi: 10.1103/PhysRevE.51.1899. Google Scholar [21] W. Liu, A. L. Bertozzi, and T. Kolokolnikov, Diffuse interface surface tension models in an expanding flow,, Comm. Math. Sci., 10 (2012), 387. doi: 10.4310/CMS.2012.v10.n1.a16. Google Scholar [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. Google Scholar [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. doi: 10.1137/S0036139901384285. Google Scholar [24] C. B. Muratov and V. V. Osipov, Static spike autosolitons in the Gray-Scott model,, J. Phys. A: Math Gen., 33 (2000), 8893. doi: 10.1088/0305-4470/33/48/321. Google Scholar [25] Y. Nishiura, Far-from Equilibrium Dynamics,, Translated from the 1999 Japanese original by Kunimochi Sakamoto. Translations of Mathematical Monographs, (1999). Google Scholar [26] K. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis Model,, Physica D, 240 (2011), 363. doi: 10.1016/j.physd.2010.09.011. Google Scholar [27] J. E. Pearson, Complex patterns in a simple system,, Science, 216 (1993), 189. Google Scholar [28] A. Potapov and T. Hillen, Metastability in chemotaxis models,, J. Dynam. Diff. Eq., 17 (2005), 293. doi: 10.1007/s10884-005-2938-3. Google Scholar [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. doi: 10.1142/S0218202508003029. Google Scholar [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. doi: 10.1137/090759069. Google Scholar [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. Google Scholar [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. doi: 10.1137/S0036139902415117. Google Scholar [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. doi: 10.1137/040620990. Google Scholar [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. doi: 10.1512/iumj.2005.54.2792. Google Scholar [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. doi: 10.1007/s00332-002-0531-z. Google Scholar [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. doi: 10.1111/1467-9590.00223. Google Scholar [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. doi: 10.1017/S0956792501004442. Google Scholar [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. doi: 10.1017/S0956792503005278. Google Scholar [39] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case,, J. Nonlinear Sci., 11 (2001), 415. doi: 10.1007/s00332-001-0380-1. Google Scholar [40] J. Wei and M. Winter, Spikes for the two-dimensional Gierer-Meinhardt system: the strong coupling case,, J. Diff. Eq., 178 (2002), 478. doi: 10.1006/jdeq.2001.4019. Google Scholar [41] J. Wei and M. Winter, Existence and stability of multiple spot solutions for the Gray-Scott model in $\mathbbR^2$,, Physica D., 176 (2003), 147. doi: 10.1016/S0167-2789(02)00743-1. Google Scholar [42] J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems,, J. Math. Biol., 57 (2008), 53. doi: 10.1007/s00285-007-0146-y. Google Scholar [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. doi: 10.1016/j.matpur.2003.09.006. Google Scholar [44] J. Wei and M. Winter, Asymmetric spotty patterns for the Gray-Scott model in $\mathbbR^2$,, Studies in Appl. Math., 110 (2003), 63. doi: 10.1111/1467-9590.00231. Google Scholar [45] J. Wei and L. Zhang, On a nonlocal eigenvalue problem,, Ann. Sc. Norm. Sup. Pisa C1. Sci., 30 (2001), 41. Google Scholar [46] J. Wei(2008), Existence and stability of spikes for the Gierer-Meinhardt system,, book chapter in Handbook of Differential Equations, (): 487. doi: 10.1016/S1874-5733(08)80013-7. Google Scholar
 [1] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 [2] Yan-Yu Chen, Yoshihito Kohsaka, Hirokazu Ninomiya. Traveling spots and traveling fingers in singular limit problems of reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 697-714. doi: 10.3934/dcdsb.2014.19.697 [3] Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 [4] José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85 [5] Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 [6] S.-I. Ei, M. Mimura, M. Nagayama. Interacting spots in reaction diffusion systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 31-62. doi: 10.3934/dcds.2006.14.31 [7] Keng Deng. On a nonlocal reaction-diffusion population model. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 65-73. doi: 10.3934/dcdsb.2008.9.65 [8] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [9] Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 [10] Toshi Ogawa. Degenerate Hopf instability in oscillatory reaction-diffusion equations. Conference Publications, 2007, 2007 (Special) : 784-793. doi: 10.3934/proc.2007.2007.784 [11] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [12] Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057 [13] Jorge Ferreira, Hermenegildo Borges de Oliveira. Parabolic reaction-diffusion systems with nonlocal coupled diffusivity terms. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2431-2453. doi: 10.3934/dcds.2017105 [14] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [15] Ricardo Enguiça, Andrea Gavioli, Luís Sanchez. A class of singular first order differential equations with applications in reaction-diffusion. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 173-191. doi: 10.3934/dcds.2013.33.173 [16] Xiao-Hui Li, Huo-Jun Ruan. The "hot spots" conjecture on higher dimensional Sierpinski gaskets. Communications on Pure & Applied Analysis, 2016, 15 (1) : 287-297. doi: 10.3934/cpaa.2016.15.287 [17] Georg Hetzer. Global existence for a functional reaction-diffusion problem from climate modeling. Conference Publications, 2011, 2011 (Special) : 660-671. doi: 10.3934/proc.2011.2011.660 [18] Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369 [19] Tatsuki Mori, Kousuke Kuto, Masaharu Nagayama, Tohru Tsujikawa, Shoji Yotsutani. Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization. Conference Publications, 2015, 2015 (special) : 861-877. doi: 10.3934/proc.2015.0861 [20] Yuxiao Guo, Ben Niu. Bautin bifurcation in delayed reaction-diffusion systems with application to the segel-jackson model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2019118

2018 Impact Factor: 1.008