Advertisement

Journal of Mathematical Biology

, Volume 70, Issue 4, pp 709–743 | Cite as

Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations

  • Anotida Madzvamuse
  • Hussaini S. Ndakwo
  • Raquel Barreira
Article

Abstract

By introducing linear cross-diffusion for a two-component reaction-diffusion system with activator-depleted reaction kinetics (Gierer and Meinhardt, Kybernetik 12:30–39, 1972; Prigogine and Lefever, J Chem Phys 48:1695–1700, 1968; Schnakenberg, J Theor Biol 81:389–400, 1979), we derive cross-diffusion-driven instability conditions and show that they are a generalisation of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Our most revealing result is that, in contrast to the classical reaction-diffusion systems without cross-diffusion, it is no longer necessary to enforce that one of the species diffuse much faster than the other. Furthermore, it is no longer necessary to have an activator–inhibitor mechanism as premises for pattern formation, activator–activator, inhibitor–inhibitor reaction kinetics as well as short-range inhibition and long-range activation all have the potential of giving rise to cross-diffusion-driven instability. To support our theoretical findings, we compute cross-diffusion induced parameter spaces and demonstrate similarities and differences to those obtained using standard reaction-diffusion theory. Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space; outside, these are chosen to belong to cross-diffusively-driven instability parameter spaces. Our numerical experiments validate our theoretical predictions that parameter spaces induced by cross-diffusion in both the \(u\) and \(v\) components of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spaces of the cross-diffusion induced parameter spaces. Our results allow experimentalists to have a wider range of parameter spaces from which to select reaction kinetic parameter values that will give rise to spatial patterning in the presence of cross-diffusion.

Keywords

Cross-diffusion reaction systems Cross-diffusion driven instability Parameter space identification Pattern formation Planary domains Finite element method 

Mathematics Subject Classification

35K57 92Bxx 37D99 65M60 

Notes

Acknowledgments

This work (AM) is supported by the following grants: EPSRC research grant (EP/J016780/1): Modeling, analysis and simulation of spatial patterning on evolving surfaces.

References

  1. Bard J, Lauder I (1974) How well does Turing’s Theory of morphogenesis work? J Theoret Biol 45:501–531CrossRefGoogle Scholar
  2. Barreira R, Elliott CM, Madzvamuse A (2011) The surface finite element method for pattern formation on evolving biological surfaces. J Math Biol 63:1095–1119 Google Scholar
  3. Capasso V, Liddo D (1994) Asymptotic behaviour of reaction-diffusion systems in population and epidemic models. The role of cross-diffusion. J Math Biol 32:453–463CrossRefMATHMathSciNetGoogle Scholar
  4. Capasso V, Liddo D (1993) Global attractivity for reaction-diffusion systems. The case of nondiagonal diffusion matrices. J Math Anal Appl 177:510–529CrossRefMATHMathSciNetGoogle Scholar
  5. Crampin EJ, Hackborn WW, Maini PK (2002) Pattern formation in reaction-diffusion models with nonuniform domain growth. Bull Math Biol 64:746–769CrossRefGoogle Scholar
  6. Dillion R, Maini PK, Othmer HG (1994) Pattern formation in generalized Turing systems. J Math Biol 32:345–393CrossRefMathSciNetGoogle Scholar
  7. Epstein IR, Pojman JA (1998) An introduction to nonlinear chemical dynamics (Topics in Physical Chemistry). Oxford University Press, New YorkGoogle Scholar
  8. Gambino G, Lombardo MC, Sammartino M (2012) Turing instability and traveling fronts for nonlinear reaction-diffusion system with cross-diffusion. Math Comp Sim 82:1112–1132CrossRefMATHMathSciNetGoogle Scholar
  9. Gambino G, Lombardo MC, Sammartino M (2013) Pattern formation driven by cross-diffusion in 2-D domain. Nonlinear Anal Real World Appl 14:1755–1779CrossRefMATHMathSciNetGoogle Scholar
  10. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12:30–39CrossRefGoogle Scholar
  11. Gray P, Scott SK (1990) Chemical oscillations and instabilities. Oxford University Press, New York, Nonlinear chemical kineticsGoogle Scholar
  12. Hillen T, Painter KJ (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58:183–217CrossRefMATHMathSciNetGoogle Scholar
  13. Iida M, Mimura M (2006) Diffusion, cross-diffusion an competitive interaction. J Math Biol 53:617–641CrossRefMATHMathSciNetGoogle Scholar
  14. Kovács S (2004) Turing bifurcation in a system with cross-diffusion. Nonlinear Anal 59:567–581CrossRefMATHMathSciNetGoogle Scholar
  15. Li L, Jin Z, Sun G (2008) Spatial pattern of an epidemic model with cross-diffusion. Chin Phys Lett 25:3500CrossRefGoogle Scholar
  16. Madzvamuse A, Gaffney EA, Maini PK (2010) Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains. J Math Biol 61(1):133–164CrossRefMATHMathSciNetGoogle Scholar
  17. Madzvamuse A, Maini PK, Wathen AJ (2003) A moving grid finite element method applied to a model biological pattern generator. J Comp Phys 190:478–500CrossRefMATHMathSciNetGoogle Scholar
  18. Madzvamuse A, Thomas Roger K, Maini Philip K, Wathen Andrew J (2002) A Numerical Approach to the Study of Spatial Pattern Formation in the Ligaments of Arcoid Bivalves. Bull Math Biol 64:501–530CrossRefGoogle Scholar
  19. McAfree MS, Annunziata O (2013) Cross-diffusion in a colloid-polymer aqueous system. Fluid Phase Equilibr 356:46–55. Bull Math Biol 64:501–530Google Scholar
  20. Murray JD (1982) Parameter space for Turing instabiity in reaction diffusion mechanisms: a comparison of models. J Theor Biol 98:143–163CrossRefGoogle Scholar
  21. Murray JD (2003) Mathematical Biology II: spatial models and biomedical applications, 3rd edn. Springer, BerlinGoogle Scholar
  22. Painter KJ, Othmer HG, Maini PK (1999) Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis. Proc Natl Acad Sci 96:5549CrossRefGoogle Scholar
  23. Prigogine I, Lefever R (1968) Symmetry breaking instabilities in dissipative systems, II. J Chem Phys 48:1695–1700CrossRefGoogle Scholar
  24. Rossi F, Vanag VK, Tiezzi E, Epstein IR (2010) Quaternary cross-diffusion in water-in-oil microemulsions loaded with a component of the Belousov-Zhabotinsky reaction. J Phys Chem B 114:8140–8146Google Scholar
  25. Ruiz-Baier R, Tian C (2013) Mathematical analysis and numerical simulation of pattern formation under cross-diffusion. Nonlinear Anal Real World Appl 14:601–612CrossRefMATHMathSciNetGoogle Scholar
  26. Schnakenberg J (1979) Simple chemical reaction systems with limit cycle behaviour. J Theor Biol 81:389–400CrossRefMathSciNetGoogle Scholar
  27. Shi J, Xie Z, Little K (2011) Cross-diffusion induced instability and stability in reaction-diffusion systems. J Appl Anal Comp 1(1):95–119MathSciNetGoogle Scholar
  28. Thomas D (1975) Artificial enzyme membrane, transport, memory and oscillatory phenomena. In: Thomas D, Kervenez J-P (eds) Analysis and control of immobilised enzyme systems. Springer, Heidelberg, pp 115–150Google Scholar
  29. Tian Lin Z, Pedersen M (2010) Instability induced by cross-diffusion in reaction-diffusion systems. Nonlinear Anal Real World Appl 11:1036–1045CrossRefMATHMathSciNetGoogle Scholar
  30. Turing A (1952) On the chemical basis of morphogenesis. Phil Trans R Soc B 237:37–72CrossRefGoogle Scholar
  31. Tyrrell HJV, Harris KR (1984) Diffusion in liquids. Butterworths, LondonGoogle Scholar
  32. Vanag VK, Epstein IR (2009) Cross-diffusion and pattern formation in reaction diffusion systems. Phys Chem Chem Phys 11:897–912CrossRefGoogle Scholar
  33. Vergara A, Capuano F, Paduano L, Sartorio R (2006) Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol). Macromolecules 39:4500–4506CrossRefGoogle Scholar
  34. Xie Z (2012) Cross-diffusion induced Turing instability for a three species food chain model. J Math Anal Appl 388:539–547CrossRefMATHMathSciNetGoogle Scholar
  35. Zhabotinsky AM (1991) A history of chemical oscillations and waves. Chaos 1:379–86CrossRefGoogle Scholar
  36. Zhang J-F, Li W-T, Wang Y-X (2011) Turing patterns of a strongly coupled predator-prey system with diffusion effects. Nonlinear Anal 74:847–858CrossRefMATHMathSciNetGoogle Scholar
  37. Zemskov EP, Vanag VK, Epstein IR (2011) Amplitude equations for reaction-diffusion systems with cross-diffusion. Phys Rev E 84:036216CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Anotida Madzvamuse
    • 1
  • Hussaini S. Ndakwo
    • 2
  • Raquel Barreira
    • 3
  1. 1.Department of Mathematics, School of Mathematical and Physical SciencesUniversity of SussexBrightonUK
  2. 2.Department of Mathematics, School of Mathematical and Physical SciencesUniversity of SussexBrightonUK
  3. 3.Rua Américo da Silva Marinho-LavradioEscola Superior de Tecnologia do Barreiro/IPSBarreiroPortugal

Personalised recommendations