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Journal of Mathematical Biology

, Volume 72, Issue 6, pp 1441–1465 | Cite as

Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model

  • Mostafa Bendahmane
  • Ricardo Ruiz-Baier
  • Canrong Tian
Article

Abstract

In this paper we analyze the effects of introducing the fractional-in-space operator into a Lotka-Volterra competitive model describing population super-diffusion. First, we study how cross super-diffusion influences the formation of spatial patterns: a linear stability analysis is carried out, showing that cross super-diffusion triggers Turing instabilities, whereas classical (self) super-diffusion does not. In addition we perform a weakly nonlinear analysis yielding a system of amplitude equations, whose study shows the stability of Turing steady states. A second goal of this contribution is to propose a fully adaptive multiresolution finite volume method that employs shifted Grünwald gradient approximations, and which is tailored for a larger class of systems involving fractional diffusion operators. The scheme is aimed at efficient dynamic mesh adaptation and substantial savings in computational burden. A numerical simulation of the model was performed near the instability boundaries, confirming the behavior predicted by our analysis.

Keywords

Turing instability Pattern formation Amplitude equations  Super-diffusion Cross-diffusion Linear stability Lévy flights  Finite volume approximation Fully adaptive multiresolution 

Mathematics Subject Classification

Primary 35B35 Secondary 35B40 47D20 

Notes

Acknowledgments

RRB gratefully acknowledges the support by the Swiss National Science Foundation through the research Grant PP00P2_144922; and CT acknowledges partial support by the PRC Grant NSFC 11201406 and by the Qinglan Project. Finally, we thank the helpful remarks by an anonymous referee, which resulted in substantial improvements to the initial version of this manuscript.

References

  1. Andreianov B, Bendahmane M, Ruiz-Baier R (2011) Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math Models Methods Appl Sci 21:307–344MathSciNetCrossRefzbMATHGoogle Scholar
  2. Baeumer B, Kovács M, Meerschaert MM (2007) Fractional reproduction-dispersal equations and heavy tail dispersal kernels. Bull Math Biol 69(7):2281–2297MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bendahmane M (2010) Weak and classical solutions to predator-prey system with cross-diffusion. Nonlinear Anal 73(8):2489–2503MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bendahmane M, Bürger R, Ruiz-Baier R, Schneider K (2009) Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems. Appl Numer Math 59:1668–1692MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bendahmane M, Karlsen KH (2006) Analysis of a class of degenerate reaction-diffusion systems and the bidomain model of cardiac tissue. Netw Heterog Media 1(1):185–218MathSciNetCrossRefzbMATHGoogle Scholar
  6. Berres S, Ruiz-Baier R (2011) A fully adaptive numerical approximation for a two-dimensional epidemic model with nonlinear cross-diffusion. Nonlinear Anal Real World Appl 12:2888–2903MathSciNetCrossRefzbMATHGoogle Scholar
  7. Bouchard JP, Georges A (1990) Anomalous diffusion in disordered media: statistical mechanics, model and physical application. Phys Rep 195:127–293MathSciNetCrossRefGoogle Scholar
  8. Brockmann D (2009) Human mobility and spatial disease dynamics. In: Schuster HG (ed) Rev Nonlinear Dyn Complex. Wiley-VCH, New York, pp 1–24Google Scholar
  9. Brockmann D, Hufnagel L, Geisel T (2006) The scaling laws of human travel. Nature 439:462–465CrossRefGoogle Scholar
  10. Buchanan M (2008) Ecological modelling: the mathematical mirror to animal nature. Nature 453:714–716CrossRefGoogle Scholar
  11. Concezzi M, Spigler R (2012) Numerical solution of two-dimensional fractional diffusion equations by a high-order ADI method. Commun Appl Ind Math 3(2):e-421MathSciNetzbMATHGoogle Scholar
  12. De Jager M, Weissing FJ, Herman PM, Nolet BA, Van de Koppel J (2011) Lévy walks evolve through interaction between movement and environmental complexity. Science 332(6037):1551–1553CrossRefGoogle Scholar
  13. Eymard R, Gallouët T, Herbin R (2000) Finite volume methods. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol VII. North-Holland, Amsterdam, pp 713–1020Google Scholar
  14. Gafiychuk VV, Datsko BY (2006) Pattern formation in a fractional reaction-diffusion system. Phys A 365:300–306CrossRefGoogle Scholar
  15. Gambino G, Lombardo MC, Sammartino M, Sciacca V (2013) Turing pattern formation in the Brusselator system with nonlinear diffusion. Phys Rev E 88:042925CrossRefGoogle Scholar
  16. Golovin AA, Matkowsky BJ, Volpert VA (2008) Turing pattern formation in the Brusselator model with super-diffusion. SIAM J Appl Math 69:251–272MathSciNetCrossRefzbMATHGoogle Scholar
  17. Hanert E, Schumacher E, Deleersnijder E (2011) Front dynamics in fractional-order epidemic models. J Theor Biol 279(1):9–16MathSciNetCrossRefGoogle Scholar
  18. Henry BI, Langlands TAM, Wearne SL (2005) Turing pattern formation in fractional activator-inhibitor systems. Phys Rev E 72:026101MathSciNetCrossRefGoogle Scholar
  19. Henry BI, Wearne SL (2002) Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J Appl Math 62:870–887MathSciNetCrossRefzbMATHGoogle Scholar
  20. Horstmann D (2007) Remarks on some Lotka-Volterra type cross-diffusion models. Nonlinear Anal Real World Appl 8:90–117MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hufnagel L, Brockmann D, Geisel T (2004) Forecast and control of epidemics in a globalized world. Proc Natl Acad Sci USA 101:15124–15129CrossRefGoogle Scholar
  22. Jüngel A (2010) Diffusive and nondiffusive population models. Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Birkhäuser, BostonGoogle Scholar
  23. Kruzhkov SN (1969) Results on the nature of the continuity of solutions of parabolic equations and some of their applications. Mat Zametki 6(1):97–108 (English tr. in. Math. Notes 6(1):517–523)Google Scholar
  24. Langlands TAM, Henry BI, Wearne SL (2007) Turing pattern formation with fractional diffusion and fractional reactions. J Phys Condens Matter 19:065115CrossRefGoogle Scholar
  25. Li BW, Wang J (2003) Anomalous heat conduction and anomalous diffusion in one-dimensional systems. Phys Rev Lett 91:044301CrossRefGoogle Scholar
  26. Lou Y, Ni WM (1996) Diffusion, self-diffusion and cross-diffusion. J Diff Eqs 131:79–131MathSciNetCrossRefzbMATHGoogle Scholar
  27. Lou Y, Nagylaki T, Ni WM (2001) On diffusion-induced blowups in a mutualistic model. Nonlinear Anal Theory Meth Appl 45:329–342MathSciNetCrossRefzbMATHGoogle Scholar
  28. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys Rep 339:1–77MathSciNetCrossRefzbMATHGoogle Scholar
  29. Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 5(6):80–90MathSciNetCrossRefzbMATHGoogle Scholar
  30. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77MathSciNetCrossRefzbMATHGoogle Scholar
  31. Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A 37:R161MathSciNetCrossRefzbMATHGoogle Scholar
  32. Nec Y, Nepomnyashchy AA, Golovin AA (2008) Oscillatory instability in super-diffusive reaction-diffusion systems: fractional amplitude and phase diffusion equations. Europhys Lett 82:58003CrossRefGoogle Scholar
  33. Nec Y, Nepomnyashchy AA (2007) Turing instability in sub-diffusive reaction-diffusion systems. J Phys A 40:14687–14702MathSciNetCrossRefzbMATHGoogle Scholar
  34. Nec Y, Nepomnyashchy AA (2008) Turing instability of anomalous reaction-anomalous diffusion systems. Eur J Appl Math 19:329–349MathSciNetCrossRefzbMATHGoogle Scholar
  35. Okubo A, Levin S (2002) Diffusion and Ecological Problems: Modern Perspectives. Springer, New YorkzbMATHGoogle Scholar
  36. Pearson JE (1993) Complex patterns in a simple system. Science 261:189–192CrossRefGoogle Scholar
  37. Ramos-Fernandez G, Mateos JL, Miramontes O, Cocho G, Larralde H, Ayala-Orozco B (2004) Lévy walk patterns in the foraging movements of spider monkeys (ateles geoffroyi). Behav Ecol Sociobiol 55(3):223–230CrossRefGoogle Scholar
  38. Schmitt FG, Seuront L (2001) Multifractal random walk in copepod behavior. Phys A 301:375–396CrossRefzbMATHGoogle Scholar
  39. Sims DW, Southall EJ, Humphries NE, Hays GC, Brad-shaw CJ, Pitchford JW, James A, Ahmed MZ, Brierley AS, Hindell MA et al (2008) Scaling laws of marine predator search behaviour. Nature 451(7182):1098–1102CrossRefGoogle Scholar
  40. Sokolov IM, Klafter J, Blumen A (2002) Fractional kinetics. Phys Today 55:48–54CrossRefGoogle Scholar
  41. Toner J, Tu Y, Ramaswamy S (2005) Hydrodynamics and phases of flocks. Ann Phys 318:170–244MathSciNetCrossRefzbMATHGoogle Scholar
  42. Viswanathan GM, Afanasyevt V, Buldyrev SV, Murphy EJ, Prince PA, Stanley HE (1996) Lévy flight search patterns of wandering albatrosses. Nature 381:413–415CrossRefGoogle Scholar
  43. Weiss M (2003) Stabilizing Turing patterns with subdiffusion in systems with low particle numbers. Phys Rev E 68:036213CrossRefGoogle Scholar
  44. Yadav A, Horsthemke W (2006) Kinetic equations for reaction-subdiffusion systems: derivation and stability analysis. Phys Rev E 74:066118MathSciNetCrossRefGoogle Scholar
  45. Yadav A, Milu SM, Horsthemke W (2008) Turing instability in reaction-subdiffusion systems. Phys Rev E 78:026116MathSciNetCrossRefGoogle Scholar
  46. Yang Q, Liu F, Turner I (2010) Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl Numer Model 34:200–218MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Mostafa Bendahmane
    • 1
  • Ricardo Ruiz-Baier
    • 2
  • Canrong Tian
    • 3
  1. 1.Institut de Mathématiques de BordeauxUniversité VictorBordeaux CedexFrance
  2. 2.Institute of Earth SciencesUniversity of LausanneLausanneSwitzerland
  3. 3.Department of Basic SciencesYancheng Institute of TechnologyYanchengChina

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