Advertisement

A user’s guide to PDE models for chemotaxis

  • T. HillenEmail author
  • K. J. Painter
Article

Abstract

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

Mathematics Subject Classification (2000)

92C17 

References

  1. 1.
    Allegretto W., Xie H., Yang S.: Properties of solutions for a chemotaxis system. J. Math. Biol. 35, 949–966 (1997)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Alt W.: Biased random walk model for chemotaxis and related diffusion approximation. J. Math. Biol. 9, 147–177 (1980)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Alt W., Lauffenburger D.A.: Transient behavior of a chemotaxis system modelling certain types of tissue inflammation. J. Math. Biol. 24(6), 691–722 (1987)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Baker M.D., Wolanin P.M., Stock J.B.: Signal transduction in bacterial chemotaxis. Bioessays 28(1), 9–22 (2006)Google Scholar
  5. 5.
    Balding D., McElwain D.L.: A mathematical model of tumour-induced capillary growth. J. Theor. Biol. 114(1), 53–73 (1985)Google Scholar
  6. 6.
    Biler P.: Local and global solvability of some parabolic systems modelling chemotaxis. Adv. Math. Sci. Appl. 8(2), 715–743 (1998)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Biler P.: Global solutions to some parabolic-elliptic systems of chemotaxis. Adv. Math. Sci. Appl. 9(1), 347–359 (1999)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Boon J.P., Herpigny B.: Model for chemotactic bacterial bands. Bull. Math. Biol. 48(1), 1–19 (1986)zbMATHGoogle Scholar
  9. 9.
    Budd C.J., Carretero-Gonzd́flez R., Russell R.D.: Precise computations of chemotactic collapse using moving mesh methods. J. Comput. Phys. 202(2), 463–487 (2005)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Budick S.A., Dickinson M.H.: Free-flight responses of Drosophila melanogaster to attractive odors. J. Exp. Biol. 209(15), 3001–3017 (2006)Google Scholar
  11. 11.
    Budrene E.O., Berg H.C.: Complex patterns formed by motile cells of Escherichia coli. Nature 349(6310), 630–633 (1991)Google Scholar
  12. 12.
    Budrene E.O., Berg H.C.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376(6535), 49–53 (1995)Google Scholar
  13. 13.
    Burger, M., Di Francesco, M., Dolak-Struss, Y.: The Keller-Segel model for chemotaxis: linear vs. nonlinear diffusion. SIAM J. Math. Anal. (2008) (to appear)Google Scholar
  14. 14.
    Byrne H.M., Cave G., McElwain D.L.: The effect of chemotaxis and chemokinesis on leukocyte locomotion: a new interpretation of experimental results. IMA J. Math. Appl. Med. Biol. 15(3), 235–256 (1998)zbMATHGoogle Scholar
  15. 15.
    Byrne H.M., Owen M.R.: A new interpretation of the Keller–Segel model based on multiphase modelling. J. Math. Biol. 49, 604–626 (2004)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Chaplain M.A.: Mathematical modelling of angiogenesis. J. Neurooncol. 50(1–2), 37–51 (2000)Google Scholar
  17. 17.
    Chaplain M.A.J., Stuart A.M.: A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10, 149–168 (1993)zbMATHGoogle Scholar
  18. 18.
    Childress S., Percus J.K.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Condeelis J., Singer R.H., Segall J.E.: The great escape: when cancer cells hijack the genes for chemotaxis and motility. Annu. Rev. Cell Dev. Biol. 21, 695–718 (2005)Google Scholar
  20. 20.
    Corrias L., Perthame B., Zaag H.: Global solutions of some chemotaxis and angiogenesis systems in high space dimensions. Milan J. Math. 72, 1–28 (2004)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Dahlquist F.W., Lovely P., Koshland D.E.: Quantitative analysis of bacterial migration in chemotaxis. Nat. New Biol. 236, 120–123 (1972)Google Scholar
  22. 22.
    Dallon J.C., Othmer H.G.: A discrete cell model with adaptive signalling for aggregation of dictyostelium discoideum. Philos. Trans. R. Soc. B 352, 391–417 (1997)Google Scholar
  23. 23.
    Dkhil F.: Singular limit of a degenerate chemotaxis-fisher equation. Hiroshima Math. J. 34, 101–115 (2004)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Dolak Y., Hillen T.: Cattaneo models for chemotaxis, numerical solution and pattern formation. J. Math. Biol. 46(2), 153–170 (2003)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Dolak Y., Schmeiser C.: The Keller-Segel model with logistic sensitivity function and small diffusivity. SIAM J. Appl. Math. 66, 286–308 (2005)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Dormann D., Weijer C.J.: Chemotactic cell movement during Dictyostelium development and gastrulation. Curr. Opin. Genet. Dev. 16(4), 367–373 (2006)Google Scholar
  27. 27.
    Eberl H.J., Parker D.F., van Loosdrecht M.C.M.: A new deterministic spatio-temporal continuum model for biofilm development. J. Theor. Med. 3(3), 161–175 (2001)zbMATHGoogle Scholar
  28. 28.
    Eisenbach M.: Chemotaxis. Imperial College Press, London (2004)Google Scholar
  29. 29.
    Ford R.M., Lauffenburger D.A.: Measurement of bacterial random motility and chemotaxis coefficients: II. application of single cell based mathematical model. Biotechnol. Bioeng. 37, 661–672 (1991)Google Scholar
  30. 30.
    Gajewski H., Zacharias K.: Global behavior of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 159, 77–114 (1998)MathSciNetGoogle Scholar
  31. 31.
    Gueron S., Liron N.: A model of herd grazing as a travelling wave, chemotaxis and stability. J. Math. Biol. 27(5), 595–608 (1989)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Henry M., Hilhorst D., Schätzle R.: Convergence to a viscocity solution for an advection- reaction-diffusion equation arising from a chemotaxis-growth model. Hiroshima Math. J. 29, 591–630 (1999)zbMATHMathSciNetGoogle Scholar
  33. 33.
    Hildebrand E., Kaupp U.B.: Sperm chemotaxis: a primer. Ann. N. Y. Acad. Sci. 1061, 221–225 (2005)Google Scholar
  34. 34.
    Hillen T.: A classification of spikes and plateaus. SIAM Rev. 49(1), 35–51 (2007)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Hillen T., Othmer H.G.: The diffusion limit of transport equations derived from velocity jump processes. SIAM J. Appl. Math. 61(3), 751–775 (2000)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Hillen T., Painter K.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. Appl. Math. 26, 280–301 (2001)zbMATHMathSciNetGoogle Scholar
  37. 37.
    Hillen T., Painter K., Schmeiser C.: Global existence for chemotaxis with finite sampling radius. Discr. Cont. Dyn. Syst. B 7(1), 125–144 (2007)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Höfer T., Sherratt J.A., Maini P.K.: Dictyostelium discoideum: cellular self-organisation in an excitable biological medium. Proc. R. Soc. Lond. B. 259, 249–257 (1995)Google Scholar
  39. 39.
    Horstmann D.: Lyapunov functions and L p-estimates for a class of reaction-diffusion systems. Coll. Math. 87, 113–127 (2001)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Horstmann D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences I. Jahresberichte DMV 105(3), 103–165 (2003)zbMATHMathSciNetGoogle Scholar
  41. 41.
    Horstmann D., Stevens A.: A constructive approach to traveling waves in chemotaxis. J. Nonlin. Sci. 14(1), 1–25 (2004)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Jabbarzadeh E., Abrams C.F.: Chemotaxis and random motility in unsteady chemoattractant fields: a computational study. J. Theor. Biol. 235(2), 221–232 (2005)MathSciNetGoogle Scholar
  43. 43.
    Kareiva P., Odell G.: Swarms of predators exhibit ’prey-taxis’ if individual predators use area-restricted search. Am. Nat. 130(2), 233–270 (1987)Google Scholar
  44. 44.
    Keller E.F., Segel L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)Google Scholar
  45. 45.
    Keller E.F., Segel L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)Google Scholar
  46. 46.
    Keller E.F., Segel L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 377–380 (1971)Google Scholar
  47. 47.
    Kennedy J.S., Marsh D.: Pheromone-regulated anemotaxis in flying moths. Science 184, 999–1001 (1974)Google Scholar
  48. 48.
    Kim I.C.: Limits of chemotaxis growth model. Nonlinear Anal. 46, 817–834 (2001)zbMATHMathSciNetGoogle Scholar
  49. 49.
    Kolokolnikov T., Erneux T., Wei J.: Mesa-type patterns in the one-dimensional Brusselator and their stability. Physica D 214, 63–77 (2006)zbMATHMathSciNetGoogle Scholar
  50. 50.
    Kowalczyk R.: Preventing blow-up in a chemotaxis model. J. Math. Anal. Appl. 305, 566–588 (2005)zbMATHMathSciNetGoogle Scholar
  51. 51.
    Kuiper H.: A priori bounds and global existence for a strongly coupled quasilinear parabolic system modelling chemotaxis. Electron. J. Differ. Equ. 52, 1–18 (2001)MathSciNetGoogle Scholar
  52. 52.
    Kuiper H., Dung L.: Global attractors for cross-diffusion systems on domains of arbitrary dimensions. Rocky Mountain J. Math. 37(5), 1645–1668 (2007)zbMATHMathSciNetGoogle Scholar
  53. 53.
    Landman K.A., Pettet G.J., Newgreen D.F.: Chemotactic cellular migration: smooth and discontinuous travelling wave solutions. SIAM J. Appl. Math. 63(5), 1666–1681 (2003)zbMATHMathSciNetGoogle Scholar
  54. 54.
    Landman K.A., Pettet G.J., Newgreen D.F.: Mathematical models of cell colonization of uniformly growing domains. Bull. Math. Biol. 65(2), 235–262 (2003)Google Scholar
  55. 55.
    Lapidus I.R., Schiller R.: Model for the chemotactic response of a bacterial population. Biophys. J 16(7), 779–789 (1976)Google Scholar
  56. 56.
    Larrivee B., Karsan A.: Signaling pathways induced by vascular endothelial growth factor (review). Int. J. Mol. Med. 5(5), 447–456 (2000)Google Scholar
  57. 57.
    Lauffenburger D.A., Kennedy C.R.: Localized bacterial infection in a distributed model for tissue inflammation. J. Math. Biol. 16(2), 141–163 (1983)zbMATHGoogle Scholar
  58. 58.
    Lee, J.M., Hillen, T., Lewis, M.A.: Continuous travelling waves for prey-taxis. Bull. Math. Biol. (2007) (in review)Google Scholar
  59. 59.
    Levine H.A., Sleeman B.D.: A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J. Appl. Math. 57, 683–730 (1997)zbMATHMathSciNetGoogle Scholar
  60. 60.
    Logan J.A., White B.J., Bentz P., Powell J.A.: Model analysis of spatial patterns in Mountain Pine Beetle outbreaks. Theor. Popul. Biol. 53(3), 236–255 (1998)zbMATHGoogle Scholar
  61. 61.
    Luca M., Chavez-Ross A., Edelstein-Keshet L., Mogilner A.: Chemotactic signaling, microglia, and Alzheimer’s disease senile plaques: is there a connection? Bull. Math. Biol. 65(4), 693–730 (2003)Google Scholar
  62. 62.
    Maini P.K., Myerscough M.R., Winters K.H., Murray J.D.: Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53(5), 701–719 (1991)zbMATHGoogle Scholar
  63. 63.
    Mantzaris N.V., Webb S., Othmer H.G.: Mathematical modeling of tumor-induced angiogenesis. J. Math. Biol. 49(2), 111–187 (2004)zbMATHMathSciNetGoogle Scholar
  64. 64.
    Maree A.F., Hogeweg P.: How amoeboids self-organize into a fruiting body: multicellular coordination in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 98(7), 3879–3883 (2001)Google Scholar
  65. 65.
    Mimura M., Tsujikawa T.: Aggregation pattern dynamics in a chemotaxis model including growth. Physica A 230, 499–543 (1996)Google Scholar
  66. 66.
    Mittal N., Budrene E.O., Brenner M.P., Van Oudenaarden A.: Motility of Escherichia coli cells in clusters formed by chemotactic aggregation. Proc. Natl. Acad. Sci. USA 100(23), 13259–13263 (2003)Google Scholar
  67. 67.
    Mori I., Ohshima Y.: Molecular neurogenetics of chemotaxis and thermotaxis in the nematode Caenorhabditis elegans. Bioessays 19(12), 1055–1064 (1997)Google Scholar
  68. 68.
    Murray J.D.: Mathematical Biology II: Spatial Models and Biochemical Applications, 3rd edn. Springer, New York (2003)Google Scholar
  69. 69.
    Murray J.D., Myerscough M.R.: Pigmentation pattern formation on snakes. J. Theor. Biol. 149(3), 339–360 (1991)Google Scholar
  70. 70.
    Myerscough M.R., Maini P.K., Painter K.J.: Pattern formation in a generalized chemotactic model. Bull. Math. Biol. 60(1), 1–26 (1998)zbMATHGoogle Scholar
  71. 71.
    Nagai T., Senba T., Yoshida K.: Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997)zbMATHMathSciNetGoogle Scholar
  72. 72.
    Nanjundiah V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)Google Scholar
  73. 73.
    Odell G.M., Keller E.F.: Traveling bands of chemotactic bacteria revisited. J. Theor. Biol. 56(1), 243–247 (1976)Google Scholar
  74. 74.
    Osaki K., Tsujikawa T., Yagi A., Mimura M.: Exponential attractor for a chemotaxis-growth system of equations. Nonlinear Anal. 51, 119–144 (2002)zbMATHMathSciNetGoogle Scholar
  75. 75.
    Osaki K., Yagi A.: Finite dimensional attractor for one-dimensional Keller-Segel equations. Funkcial. Ekvac. 44, 441–469 (2001)zbMATHMathSciNetGoogle Scholar
  76. 76.
    Othmer H.G., Dunbar S.R., Alt W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)zbMATHMathSciNetGoogle Scholar
  77. 77.
    Othmer H.G., Hillen T.: The diffusion limit of transport equations II: chemotaxis equations. SIAM J. Appl. Math. 62(4), 1122–1250 (2002)MathSciNetGoogle Scholar
  78. 78.
    Othmer H.G., Stevens A.: Aggregation, blowup and collapse: The ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 1044–1081 (1997)zbMATHMathSciNetGoogle Scholar
  79. 79.
    Owen M.R., Sherratt J.A.: Pattern formation and spatiotemporal irregularity in a model for macrophage-tumour interactions. J. Theor. Biol. 189(1), 63–80 (1997)Google Scholar
  80. 80.
    Painter K., Hillen T.: Volume-filling and quorum-sensing in models for chemosensitive movement. Can. Appl. Math. Quart. 10(4), 501–543 (2002)zbMATHMathSciNetGoogle Scholar
  81. 81.
    Painter K.J., Maini P.K., Othmer H.G.: Complex spatial patterns in a hybrid chemotaxis reaction-diffusion model. J. Math. Biol. 41(4), 285–314 (2000)zbMATHMathSciNetGoogle Scholar
  82. 82.
    Painter K.J., Maini P.K., Othmer H.G.: Development and applications of a model for cellular response to multiple chemotactic cues. J. Math. Biol. 41(4), 285–314 (2000)zbMATHMathSciNetGoogle Scholar
  83. 83.
    Painter K.J., Othmer H.G., Maini P.K.: Stripe formation in juvenile pomacanthus via chemotactic response to a reaction-diffusion mechanism. Proc. Natl. Acad. Sci. USA 96, 5549–5554 (1999)Google Scholar
  84. 84.
    Palsson E., Othmer H.G.: A model for individual and collective cell movement in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA 97(19), 10448–10453 (2000)Google Scholar
  85. 85.
    Park H.T., Wu J., Rao Y.: Molecular control of neuronal migration. Bioessays 24(9), 821–827 (2002)Google Scholar
  86. 86.
    Patlak C.S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)MathSciNetGoogle Scholar
  87. 87.
    Perthame B.: Transport Equations in Biology. Birkhäuser, Basel (2007)zbMATHGoogle Scholar
  88. 88.
    Perumpanani A.J., Sherratt J.A., Norbury J., Byrne H.M.: Biological inferences from a mathematical model for malignant invasion. Invas. Metastas. 16(4–5), 209–221 (1996)Google Scholar
  89. 89.
    Post. K.: A non-linear parabolic system modeling chemotaxis with sensitivity functions (1999)Google Scholar
  90. 90.
    Potapov A., Hillen T.: Metastability in chemotaxis models. J. Dyn. Diff. Equ. 17, 293–330 (2005)zbMATHMathSciNetGoogle Scholar
  91. 91.
    Rascle M., Ziti C.: Finite time blow up in some models of chemotaxis. J. Math. Biol. 33, 388–414 (1995)zbMATHMathSciNetGoogle Scholar
  92. 92.
    Rivero M.A., Tranquillo R.T., Buettner H.M., Lauffenburger D.A.: Transport models for chemotactic cell populations based on individual cell behavior. Chem. Eng. Sci. 44, 1–17 (1989)Google Scholar
  93. 93.
    Segel L.A.: Incorporation of receptor kinetics into a model for bacterial chemotaxis. J. Theor. Biol. 57(1), 23–42 (1976)MathSciNetGoogle Scholar
  94. 94.
    Segel L.A.: A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32, 653–665 (1977)zbMATHGoogle Scholar
  95. 95.
    Sherratt J.A.: Chemotaxis and chemokinesis in eukaryotic cells: the Keller-Segel equations as an approximation to a detailed model. Bull. Math. Biol. 56(1), 129–146 (1994)zbMATHGoogle Scholar
  96. 96.
    Sherratt J.A., Sage E.H., Murray J.D.: Chemical control of eukaryotic cell movement: a new model. J. Theor. Biol. 162(1), 23–40 (1993)Google Scholar
  97. 97.
    Stevens A.: The derivation of chemotaxis-equations as limit dynamics of moderately interacting stochastic many particle systems. SIAM J. Appl. Math. 61(1), 183–212 (2000)zbMATHMathSciNetGoogle Scholar
  98. 98.
    Suzuki T.: Free Energy and Self-Interacting Particles. Birkhäuser, Boston (2005)zbMATHGoogle Scholar
  99. 99.
    Tranquillo R.T., Lauffenburger D.A., Zigmond S.H.: A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations. J. Cell Biol. 106(2), 303–309 (1988)Google Scholar
  100. 100.
    Tyson R., Lubkin S.R., Murray J.D.: A minimal mechanism for bacterial pattern formation. Proc. R. Soc. Lond. B 266, 299–304 (1999)Google Scholar
  101. 101.
    Tyson R., Lubkin S.R., Murray J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Math. Biol. 38(4), 359–375 (1999)zbMATHMathSciNetGoogle Scholar
  102. 102.
    Velazquez J.J.L.: Point dynamics for a singular limit of the Keller-Segel model. I. motion of the concentration regions. SIAM J. Appl. Math. 64(4), 1198–1223 (2004)zbMATHMathSciNetGoogle Scholar
  103. 103.
    Velazquez J.J.L.: Point dynamics for a singular limit of the Keller-Segel model. II. formation of the concentration regions. SIAM J. Appl. Math. 64(4), 1198–1223 (2004)zbMATHMathSciNetGoogle Scholar
  104. 104.
    Wang X.: Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics. SIAM J. Math. Ana. 31, 535–560 (2000)zbMATHGoogle Scholar
  105. 105.
    Wang, Z.A., Hillen, T.: Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos, 17(037108) (2007), 13 ppGoogle Scholar
  106. 106.
    Wang Z.A., Hillen T.: Shock formation in a chemotaxis model. Math. Methods Appl. Sci. 31(1), 45–70 (2008)zbMATHMathSciNetGoogle Scholar
  107. 107.
    Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal dependent sensitivityGoogle Scholar
  108. 108.
    Woodward D.E., Tyson R., Myerscough M.R., Murray J.D., Budrene E.O., Berg H.C.: Spatio-temporal patterns generated by Salmonella typhimurium. Biophys. J. 68(5), 2181–2189 (1995)Google Scholar
  109. 109.
    Wrzosek D.: Long time behaviour of solutions to a chemotaxis model with volume filling effect. Proc. Roy. Soc. Edinb. Sect. A 136, 431–444 (2006)zbMATHMathSciNetGoogle Scholar
  110. 110.
    Wrzosek, D.: Global attractor for a chemotaxis model with prevention of overcrowding. Nonlin. Ana. 59, 1293–1310, P2004Google Scholar
  111. 111.
    Wu D.: Signaling mechanisms for regulation of chemotaxis. Cell Res. 15(1), 52–56 (2005)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of Mathematics and Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburghUK

Personalised recommendations