Pattern formation on networks with reactions: A continuous-time random-walk approach

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 87, Issue 3, 2013, Pages

  Angstmann, C N ^a   Donnelly, I C ^a   Henry, B I ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

BIRTH-DEATH PROCESS; CONTINUOUS-TIME RANDOM WALK; GENERALIZED MASTER EQUATIONS; LAPLACIAN OPERATOR; PATTERN FORMATION; REACTION DIFFUSION; REACTION PROCESS; TURING INSTABILITY;

LAPLACE EQUATION; MATHEMATICAL OPERATORS;

RANDOM PROCESSES;

EID: 84875525984     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.87.032804     Document Type: Article

References (61)

1. A.-L. Barabási and Z. N. Oltvai, Nat. Rev. Genet. 1471-0056 10.1038/nrg1272 5, 101 (2004). (Pubitemid 38160277)
2. R. Zallen, The Physics of Amorphous Solids (Wiley Online Library, New York, 1983), Vol. 26.
3. A.-L. Barabási and R. Albert, Science SCIEAS 0036-8075 10.1126/science.286.5439.509 286, 509 (1999).
4. E. López, S. V. Buldyrev, S. Havlin, and H. E. Stanley, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.248701 94, 248701 (2005). (Pubitemid 41636519)
5. I. Simonsen, Physica A PHYADX 0378-4371 10.1016/j.physa.2005.06.032 357, 317 (2005). (Pubitemid 41327889)
6. B. Tadi, G. J. Rodgers, and S. Thurner, Int. J. Bifurcat. Chaos IJBEE4 0218-1274 10.1142/S0218127407018452 17, 2363 (2007). (Pubitemid 47373285)
7. L. K. Gallos and P. Argyrakis, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.74.056107 74, 056107 (2006). (Pubitemid 44746116)
8. H. Nakao and A. S. Mikhailov, Nat. Phys. 1745-2473 10.1038/nphys1651 6, 544 (2010).
9. M. Wolfrum, Physica D PDNPDT 0167-2789 10.1016/j.physd.2012.05.002 241, 1351 (2012).
10. L. Bachelier, Théorie de la spéculation (Gauthier-Villars, Paris, 1900).
11. K. Pearson, Nature (London) NATUAS 0028-0836 10.1038/072342a0 72, 342 (1905).
12. A. Einstein, Ann. Phys. ANPYA2 0003-3804 10.1002/andp.19053220806 322, 549 (1905).
13. S. Havlin and D. Ben-Avraham, Adv. Phys. ADPHAH 0001-8732 10.1080/00018738700101072 36, 695 (1987).
14. J. D. Noh and H. Rieger, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.118701 92, 118701 (2004).
15. S. Condamin, O. Bénichou, V. Tejedor, R. Voituriez, and J. Klafter, Nature (London) NATUAS 0028-0836 10.1038/nature06201 450, 77 (2007). (Pubitemid 350070561)
16. E. Montroll and G. Weiss, J. Math. Phys. JMAPAQ 0022-2488 10.1063/1.1704269 6, 167 (1965).
17. H. Scher and M. Lax, Phys. Rev. B PRBMDO 1098-0121 10.1103/PhysRevB.7. 4491 7, 4491 (1973).
18. J. Klafter and I. M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford University Press, New York, 2011).
19. I. M. Sokolov, M. G. W. Schmidt, and F. Sagués, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.73.031102 73, 031102 (2006). (Pubitemid 43333244)
20. B. I. Henry, T. A. M. Langlands, and S. L. Wearne, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.74.031116 74, 031116 (2006). (Pubitemid 44410802)
21. T. A. M. Langlands, B. I. Henry, and S. L. Wearne, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.77.021111 77, 021111 (2008). (Pubitemid 351366313)
22. M. O. Vlad and J. Ross, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.66.061908 66, 061908 (2002). (Pubitemid 40606023)
23. S. Eule, R. Friedrich, F. Jenko, and I. M. Sokolov, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.78.060102 78, 060102 (2008).
24. V. Méndez, S. Fedotov, and W. Horsthemke, Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities (Springer, Berlin, 2010).
25. S. Fedotov, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.011117 81, 011117 (2010).
26. E. Abad, S. B. Yuste, and K. Lindenberg, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.81.031115 81, 031115 (2010).
27. D. Le Bihan, J. F. Mangin, C. Poupon, C. Clark, S. Pappata, N. Molko, and H. Chabriat, J. Magn. Reson. Imaging 1053-1807 10.1002/jmri.1076 13, 534 (2001). (Pubitemid 32231507)
28. A. Raj, A. Kuceyeski, and M. Weiner, Neuron NERNET 0896-6273 10.1016/j.neuron.2011.12.040 73, 1204 (2012).
29. V. Colizza, A. Barrat, M. Barthélemy, and A. Vespignani, Proc Natl. Acad. Sci. USA PNASA6 0027-8424 10.1073/pnas.0510525103 103, 2015 (2006). (Pubitemid 43271615)
30. M. Kuperman and G. Abramson, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.86.2909 86, 2909 (2001). (Pubitemid 32293415)
31. A. M. Turing, Philos. Trans. R. Soc. B PTRBAE 0962-8436 10.1098/rstb.1952.0012 237, 37 (1952).
32. A. Gierer and H. Meinhardt, Biol. Cybern. 12, 30 (1972).
33. S. Kondo and T. Miura, Science SCIEAS 0036-8075 10.1126/science.1179047 329, 1616 (2010).
34. A. Madzvamuse, E. A. Gaffney, and P. K. Maini, J. Math. Biol. JMBLAJ 0303-6812 10.1007/s00285-009-0293-4 61, 133 (2010).
35. H. Meinhardt, Interface Focus 2042-8898 10.1098/rsfs.2011.0097 2, 407 (2012).
36. P. K. Maini, T. E. Woolley, R. E. Baker, E. A. Gaffney, and S. S. Lee, Interface Focus 2042-8898 10.1098/rsfs.2011.0113 2, 487 (2012).
37. Q. Ouyang and H. L. Swinney, Nature (London) NATUAS 0028-0836 10.1038/352610a0 352, 610 (1991).
38. G. Sun, Z. Jin, Q. X. Liu, and L. Li, J. Stat. Mech.: Theor. Exp. 1742-5468 10.1088/1742-5468/2007/11/P11011 (2007) P11011.
39. O. Stancevic, C. Angstmann, J. M. Murray, and B. I. Henry, arXiv: 1209.2772.
40. M. Mimura and J. D. Murray, J. Theor. Biol. JTBIAP 0022-5193 10.1016/0022-5193(78)90332-6 75, 249 (1978).
41. H. G. Othmer and L. E. Scriven, J. Theor. Biol. JTBIAP 0022-5193 10.1016/0022-5193(71)90154-8 32, 507 (1971).
42. V. Colizza, R. Pastor-Satorras, and A. Vespignani, Nat. Phys. 1745-2473 10.1038/nphys560 3, 276 (2007).
43. L. K. Gallos and P. Argyrakis, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.92.138301 92, 138301 (2004).
44. W. Horsthemke, K. Lam, and P. K. Moore, Phys. Lett. A PYLAAG 0375-9601 10.1016/j.physleta.2004.06.044 328, 444 (2004).
45. L. Diambra and L. da Fontoura Costa, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.73.031917 73, 031917 (2006).
46. S. Hata, H. Nakao, and A. S. Mikhailov, EPL EULEEJ 0295-5075 10.1209/0295-5075/98/64004 98, 64004 (2012).
47. N. E. Kouvaris, H. Kori, and A. S. Mikhailov, PLoS ONE 1932-6203 10.1371/journal.pone.0045029 7, e45029 (2012).
48. M. Wardetzky, S. Mathur, F. Kälberer, and E. Grinspun, in Proceedings of the Fifth Eurographics Symposium on Geometry Processing (Eurographics Association, Aire-la-Ville, Switzerland, 2007), pp. 33-37.
49. B. I. Henry, T. A. M. Langlands, and S. L. Wearne, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.72.026101 72, 026101 (2005). (Pubitemid 41520774)
50. Y. Nec and A. A. Nepomnyashchy, J. Phys. A 1751-8113 10.1088/1751-8113/ 40/49/005 40, 14687 (2007).
51. D. Hernández, C. Varea, and R. A. Barrio, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.79.026109 79, 026109 (2009).
52. D. J. Watts and S. H. Strogatz, Nature (London) NATUAS 0028-0836 10.1038/30918 393, 440 (1998). (Pubitemid 28292183)
53. K. M. Page, P. K. Maini, and N. A. M. Monk, Physica D PDNPDT 0167-2789 10.1016/j.physd.2005.01.022 202, 95 (2005). (Pubitemid 40371044)
54. R. Metzler and J. Klafter, Phys. Rep. PRPLCM 0370-1573 10.1016/S0370-1573(00)00070-3 339, 1 (2000).
55. A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, J. Phys. A 1361-6447 10.1088/0305-4470/38/42/L03 38, L679 (2005). (Pubitemid 41428584)
56. C. Angstmann and B. I. Henry, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.84.061146 84, 061146 (2011).
57. A. N. Samukhin, S. N. Dorogovtsev, and J. F. F. Mendes, Phys. Rev. E PLEEE8 1539-3755 10.1103/PhysRevE.77.036115 77, 036115 (2008). (Pubitemid 351431339)
58. G. Xu, CAGD 21, 767 (2004).
59. R. G. Plaza, F. Sánchez-Garduno, P. Padilla, R. A. Barrio, and P. K. Maini, J. Dyn. Differ. Equations JDDEEH 1040-7294 10.1007/s10884-004-7834-8 16, 1093 (2004). (Pubitemid 40140400)
60. J. D. Murray, Mathematical Biology (Springer, Berlin, 2002), Vol. 2.
61. P. F. Verhulst, Mem. Acad. R. Sci. B.-L. Bruxelles 18, 1 (1845).