Advertisement

Journal of Mathematical Biology

, Volume 61, Issue 3, pp 377–399 | Cite as

Wavespeed in reaction–diffusion systems, with applications to chemotaxis and population pressure

  • Sanjeeva Balasuriya
  • Georg A. GottwaldEmail author
Article

Abstract

We present a method based on the Melnikov function used in dynamical systems theory to determine the wavespeed of travelling waves in perturbed reaction–diffusion systems. We study reaction–diffusion systems which are subject to weak nontrivial perturbations in the reaction kinetics, in the diffusion coefficient, or with weak active advection. We find explicit formulæ for the wavespeed and illustrate our theory with two examples; one in which chemotaxis gives rise to nonlinear advection and a second example in which a positive population pressure results in both a density-dependent diffusion coefficient and a nonlinear advection. Based on our theoretical results we suggest an experiment to distinguish between chemotactic and population pressure in bacterial colonies.

Mathematics Subject Classification (2000)

37Mxx 92Bxx 92D25 35C07 34C37 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adler J (1966) Chemotaxis in bacteria. Science 153: 708–716CrossRefGoogle Scholar
  2. Allee WC (1938) The social life of animals. Norton, New YorkGoogle Scholar
  3. Arrowsmith DK, Place CM (1990) An introduction to dynamical systems. Cambridge University Press, CambridgezbMATHGoogle Scholar
  4. Balasuriya S, Gottwald GA, Hornibrook J, Lafortune S (2007) High Lewis number combustion wavefronts: a perturbative Melnikov analysis. SIAM J Appl Math 67: 464–486zbMATHCrossRefMathSciNetGoogle Scholar
  5. Balasuriya S, Volpert VA (2008) Wavespeed analysis: approximating Arrhenius kinetics with step-function kinetics. Combust Theor Model 12: 643–670zbMATHCrossRefMathSciNetGoogle Scholar
  6. Bazazi S, Buhl J, Hale JJ, Anstey ML, Sword GA, Simpson SJ, Couzin ID (2008) Collective motion and cannibalism in locust migratory bands. Curr Biol 18: 1–5CrossRefGoogle Scholar
  7. Benguria RD, Depassier MC, Méndez V (2004) Minimal speed of fronts of reaction–convection–diffusion equations. Phys Rev E 69: 031106CrossRefMathSciNetGoogle Scholar
  8. Berg HC, Turner L (1990) Chemotaxis of bacteria in glass capillary arrays. Biophys J 58: 919–930CrossRefGoogle Scholar
  9. Bonner JT (1967) The cellular slime moulds. Princeton University Press, PrincetonGoogle Scholar
  10. Brenner MP, Levitov LS, Budrene EO (1998) Physical mechanisms for chemotactic pattern formation in bacteria. Biophys J 74: 1677–1693CrossRefGoogle Scholar
  11. Budrene E, Berg H (1991) Complex patterns formed by motile cells of Escherichia coli. Nature 349: 630–633CrossRefGoogle Scholar
  12. Buhl J, Sumpter DJT, Couzin ID, Despland EM, Hale JJ, Miller E, Simpson SJ (2006) From disorder to order in marching locusts. Science 312: 1402–1406CrossRefGoogle Scholar
  13. Carl EA (1971) Population control in Arctic ground squirrels. Ecology 52: 395–413CrossRefGoogle Scholar
  14. Cox SM, Gottwald GA (2006) A bistable reaction–diffusion system in a stretching flow. Phys D 216: 307–318zbMATHCrossRefMathSciNetGoogle Scholar
  15. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Indiana Univ Math J 21: 193–226zbMATHCrossRefMathSciNetGoogle Scholar
  16. Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugenics 7: 335–369Google Scholar
  17. Gilding BH, Kersner R (2004) Travelling waves in nonlinear diffusion–convection reaction. Birkhauser, BaselzbMATHGoogle Scholar
  18. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, New YorkzbMATHGoogle Scholar
  19. Gueron S, Levin SA, Rubenstein DI (1996) The dynamics of herds: from individuals to aggregations. J Theor Biol 182: 85–98CrossRefGoogle Scholar
  20. Gurney WSC, Nisbet RM (1975) The regulation of inhomogeneous population. J Theors Biol 52: 441–457CrossRefGoogle Scholar
  21. Holmes PJ (1980) Averaging and chaotic motions in forced oscillations. SIAM J Appl Math 38: 65–80zbMATHCrossRefMathSciNetGoogle Scholar
  22. Keller E, Segel L (1970) Initiation of slime mold aggregation viewed as an instability. J Theor Biol 26: 399–415CrossRefGoogle Scholar
  23. King JR, McCabe PM (2003) On the Fisher-KPP equation with fast nonlinear diffusion. Proc R Soc Lond A 459: 2529–2546zbMATHCrossRefMathSciNetGoogle Scholar
  24. Kobayashi R, Tero A, Nakagaki T (2006) Mathematical model for rhythmic amoeboid movement in the true slime mold. J Math Biol 53: 273–286zbMATHCrossRefMathSciNetGoogle Scholar
  25. Kolmogorov A, Petrovsky I, Piscounoff N (1937) Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bulletin de l’Université d’Etat à Moscou, Série Internationale 1: 1Google Scholar
  26. Kramer L, Gottwald GA, Krinsky V, Pumir A, Barelko V (2000) Persistence of zero velocity fronts in reaction diffusion systems. Chaos 10: 731zbMATHCrossRefMathSciNetGoogle Scholar
  27. Lega J, Passot T (2007) Hydrodynamics of bacterial colonies. Nonlinearity 20: C1–C16zbMATHCrossRefMathSciNetGoogle Scholar
  28. Lewis MA, Kareiva P (1993) Theor Popul Biol 43:141–158zbMATHCrossRefGoogle Scholar
  29. Lika K, Hallam TG (1999) Traveling wave solutions of a reaction–advection equation. J Math Bio 38: 346–358zbMATHCrossRefMathSciNetGoogle Scholar
  30. Malaguti L, Emilia R, Marcelli C (2002) Travelling wavefronts in reaction–diffusion equations with convection effects and non-regular terms. Math Nachr 242: 148–164zbMATHCrossRefMathSciNetGoogle Scholar
  31. Malaguti L, Marcelli C, Matucci S (2004) Front propagation in bistable reaction–diffusion-advection equations. Adv Differ Equ 9: 1143–1166zbMATHMathSciNetGoogle Scholar
  32. Melnikov VK (1963) On the stability of the centre for time-periodic perturbations. Trans Moscow Math Soc 12: 1–56Google Scholar
  33. Montroll EW, West BJ (1979) On an enriched collection of stochastic processes. In: Montroll EW, Lebowitz JL (eds) Fluctuation phenomena. North Holland, AmsterdamGoogle Scholar
  34. Morisita M (1971) Measuring of habitat value by “environmental density” method. In: Patil GP, Pielou EC, Waters WE (eds) Statistical ecology 1. Spatial patterns and statistical distributions, vol 1, 1 edn. Pennsylvania State University Press, University Park, p 379Google Scholar
  35. Murray JD (1993) Mathematical biology. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  36. Myers JH, Krebs CJ (1974) Population cycles in rodents. Sci Am 6: 38–46CrossRefGoogle Scholar
  37. Nakagaki T, Yamada H, Masami I (1999) Reaction–diffusion-advection model for pattern formation of rhythmic contraction in a giant amoeboid cell of the physarum plasmodium. J Theor Biol 197: 497–506CrossRefGoogle Scholar
  38. Nakagaki T, Yamada H, Ueda T (2000) Interaction between cell shape and contraction pattern in the physarum plasmodium. Biophys Chem 84: 194–204CrossRefGoogle Scholar
  39. Odell GM, Bonner JT (1986) How the dictyostelium discoideum grex crawls. Phil Trans R Soc Lond B 312: 487–525CrossRefGoogle Scholar
  40. Press W, Teukolsky S, Vetterling W, Flannery B (1992) Numerical recipes, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  41. Sanchez-Garduño F, Maini PK (1994) Existence and uniqueness of a sharp travelling wave in degenerate non-linear diffusion Fisher-KPP equation. J Math Biol 33:163–192zbMATHCrossRefMathSciNetGoogle Scholar
  42. Segel LA (1972) Lecture notes on mathematics in the life sciences. American Mathematical Society, ProvidenceGoogle Scholar
  43. Shiguesada N, Kawasaki K, Teramoto E (1979) Spatial segregation of interacting species. J Theor Biol 79: 83–99CrossRefGoogle Scholar
  44. Stephens PA, Sutherland WJ, Freckelton RP (1999) What is the Allee effect. Oikos 87: 185–190CrossRefGoogle Scholar
  45. Stock JB, Surette MG (1996) Chemotaxis. In: Neidardt FC, Curtiss R, Ingraham JL, Lin EC, Low KB, Megasanik B, Reznikoff WS, Riley M, Shaechter M, Umbarger HE (eds) Escherichia coli and Salmonella: cellular and molecular biology, vol 1, 2nd edn. American Society for Microbiology, Washington, pp 1103–1129Google Scholar
  46. Taylor CM, Davis HG, Civille JC, Grevstad FS, Hastings A (2004) Consequences of an Allee effect in the invasion of a Pacific estuary by Spartina alterniflora. Ecology 85: 3254–3266CrossRefGoogle Scholar
  47. Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Springer, New YorkzbMATHGoogle Scholar
  48. Wolfram Research Inc. (2005) Mathematica. Wolfram Research, Inc., 5.2 edn. Champaign, IllinoisGoogle Scholar
  49. Yamada H, Nakagaki T, Baker RE, Maini PK (2007) Dispersion relation in oscillatory reaction–diffusion systems with self-consistent flow in true slime mold. J Math Biol 54: 745–760zbMATHCrossRefMathSciNetGoogle Scholar
  50. Yamada H, Nakagaki T, Ito M (1999) Pattern formation of a reaction–diffusion system with self-consistent flow in the amoeboid organism physarum plasmodium. Phys Rev E 59: 1009–1014CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Goodwin-Niering Center for Conservation Biology and Environmental StudiesConnecticut CollegeNew LondonUSA
  2. 2.School of Mathematical SciencesUniversity of AdelaideAdelaideAustralia
  3. 3.School of Mathematics and Statistics, and Centre for Mathematical BiologyUniversity of SydneySydneyAustralia

Personalised recommendations