Skip to main content
SpringerLink
Log in

A review on reaction–diffusion approximation

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

Abstract

Recently reaction–diffusion approximation has been intensively studied. This paper reviews the studies in reaction–diffusion approximation and is based on Iida and Ninomiya (Sugaku 66:225–248, 2014) added recent studies to. This paper explains reaction–diffusion approximation of the Stefan problem, nonlinear diffusion, nonlocal dispersal and wave equations. Moreover, some instabilities of equations mentioned above will be explained by the Turing instability through reaction–diffusion approximations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: H. Schmeisser and H. Triebel (eds.) Function spaces, differential operators and nonlinear analysis, vol. 133, pp. 9–126 . Teubner-Texte Math (1993)

  2. Bothe, D.: The instantaneous limit of a reaction-diffusion system. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math. vol. 215, pp. 215–224 Dekker, New York (2001)

  3. Bothe, D., Hilhorst, D.: A reaction-diffusion system with fast reversible reaction. J. Math. Anal. Appl. 286, 125–135 (2003)

    Article  MathSciNet  Google Scholar 

  4. Choi, Y.S., Lui, R., Yamada, Y.: Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion. Discret. Contin. Dynam. Systems 9, 1193–1200 (2003)

    Article  MathSciNet  Google Scholar 

  5. Crooks, E.C.M., Dancer, E.N., Hilhorst, D., Mimura, M., Ninomiya, H.: Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions. Nonlinear Anal. Real World Appl. 5, 645–665 (2004)

    Article  MathSciNet  Google Scholar 

  6. Dancer, E.N., Hilhorst, D., Mimura, M., Peletier, L.A.: Spatial segregation limit of a competition-diffusion system. Eur. J. Appl. Math. 10, 97–115 (1999)

    Article  MathSciNet  Google Scholar 

  7. Eymard, R., Hilhorst, D., van der Hout, R., Peletier, L. A.: A reaction-diffusion system approximation of a one-phase Stefan problem. In: J. L. Menaldi, E. Rofman and A. Sulem (eds.) Optimal Control and Partial Differential Equations, pp. 156–170 . IOS Press Amsterdam, Berlin, Oxford, Tokyo and Washington DC (2001)

  8. Hilhorst, D., van der Hout, R., Peletier, L.A.: The fast reaction limit for a reaction-diffusion system. J. Math. Anal. Appl. 199, 349–373 (1996)

    Article  MathSciNet  Google Scholar 

  9. Hilhorst, D., van der Hout, R., Peletier, L.A.: Diffusion in the presence of fast reaction: the case of a general monotone reaction term. J. Math. Sci. Univ. Tokyo 4, 469–517 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Hilhorst, D., van der Hout, R., Peletier, L.A.: Nonlinear diffusion in the presence of fast reaction. Nonlinear Anal. TMA 41, 803–823 (2000)

    Article  MathSciNet  Google Scholar 

  11. Hilhorst, D., Iida, M., Mimura, M., Ninomiya, H.: A competition-diffusion system approximation to the classical two-phase Stefan problem. Japan J. Ind. Appl. Math. 18, 161–180 (2001)

    Article  MathSciNet  Google Scholar 

  12. Hilhorst, D., Iida, M., Mimura, M., Ninomiya, H.: Relative compactness in \(L^p\) of solutions of some \(2m\) components competition-diffusion system. Discret. Contin. Dyn. Syst. 21, 233–244 (2008)

    Article  Google Scholar 

  13. Hilhorst, D., Mimura, M., Ninomiya, H.: Fast Reaction Limit of Competition-Diffusion Systems, Evolutionary Equations, Vol 5, Handbook of Differential Equations (edited by C.M. Dafermos and Milan Pokorny), pp. 105–168. Hungary: North-Holland (2009)

  14. Huang, Y.: How do cross-migration models arise? Math. Biosci. 195, 127–140 (2005)

    Article  MathSciNet  Google Scholar 

  15. Iida, M., Mimura, M., Ninomiya, H.: Diffusion, cross-diffusion and competitive interaction. J. Math. Biol. 53(4), 617–641 (2006)

    Article  MathSciNet  Google Scholar 

  16. Iida, M., Monobe, H., Murakawa, H., Ninomiya, H.: Vanishing, moving and immovable interfaces in fast reaction limits. J. Differ. Equ. 263, 2715–2735 (2017)

    Article  MathSciNet  Google Scholar 

  17. Iida, M., Nakashima, K., Yanagida, E.: On certain one-dimensional elliptic systems under different growth conditions at respective infinities. Adv. Stud. Pure Math. 47, 565–572 (2007)

    MathSciNet  MATH  Google Scholar 

  18. Iida, M., Ninomiya, H.: A reaction-diffusion approximation to a cross-diffusion system. In: M. Chipot and H. Ninomiya (eds.) Recent Advances on Elliptic and Parabolic Issues, pp. 145–164. World Scientific (2006)

  19. Iida, M., Ninomiya, H.: Reaction-diffusion approximation and related topics. Sugaku (in Japanese) 66, 225–248 (2014)

    MathSciNet  Google Scholar 

  20. Izuhara, H., Mimura, M.: Reaction-diffusion system approximation to the cross-diffusion competition system. Hiroshima Math. J. 38, 315–347 (2008)

    MathSciNet  MATH  Google Scholar 

  21. Kan-on, Y.: Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics. Hiroshima Math. J. 23, 509–536 (1993)

    MathSciNet  MATH  Google Scholar 

  22. Kan-on, Y., Mimura, M.: Segregation structures of competing species mediated by a diffusive predator. Mathematical topics in population biology, morphogenesis and neurosciences, vol. 71, pp. 123–133, Lecture Notes in Biomath. (1985)

    Google Scholar 

  23. Kareiva, P., Odell, G.: Swarms of predators exhibit “preytaxis” if individual predator use area-restricted search. Am. Nat. 130, 233–270 (1987)

    Article  Google Scholar 

  24. Keller, J., Sternberg, P., Rubinstein, J.: Fast reaction, slow diffusion and curve shortening. SIAM J. Appl. Math. 49, 116–133 (1989)

    Article  MathSciNet  Google Scholar 

  25. Lou, Y., Ni, W.-M.: Diffusion vs cross-diffusion: an elliptic approach. J. Differ. Equ. 154, 157–190 (1999)

    Article  MathSciNet  Google Scholar 

  26. Lou, Y., Ni, W.-M., Wu, Y.: On the global existence of a cross-diffusion system. Discret. Contin. Dynam. Syst. 4, 193–203 (1998)

    Article  MathSciNet  Google Scholar 

  27. Lou, Y., Ni, W.-M., Yotsutani, S.: On a limiting system in the Lotka-Volterra competition with cross-diffusion. Discret. Contin. Dynam. Syst. 10, 435–458 (2004)

    Article  MathSciNet  Google Scholar 

  28. Matano, H., Mimura, M.: Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. Kyoto Univ. 19, 1049–1079 (1983)

    Article  MathSciNet  Google Scholar 

  29. Mimura, M., Kawasaki, K.: Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9, 49–64 (1980)

    Article  MathSciNet  Google Scholar 

  30. Mimura, M., Nishiura, Y., Tesei, A., Tsujikawa, T.: Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima Math. J. 14, 425–449 (1984)

    MathSciNet  MATH  Google Scholar 

  31. Mimura, M., Yamada, Y., Yotsutani, S.: A free boundary problem in ecology. Japan J. Appl. Math. 2, 151–186 (1985)

    Article  MathSciNet  Google Scholar 

  32. Mimura, M., Yamada, Y., Yotsutani, S.: Stability analysis for free boundary problems in ecology. Hiroshima Math. J. 16, 477–498 (1986)

    MathSciNet  MATH  Google Scholar 

  33. Moussa, A., Perthame, B., Salort, D.: Backward Parabolicity, Cross-Diffusion and Turing Instability, to appear in Journal of Nonlinear Science

  34. Monneau, R., Weiss, G.S.: Self-propagating High temperature Synthesis (SHS) in the high activation energy regime Acta Math. Univ. Comenianae LXXV I, 99–109 (2007)

    MATH  Google Scholar 

  35. Murakawa, H.: Reaction-diffusion system approximation to degenerate parabolic systems. Nonlinearity 20, 2319–2332 (2007)

    Article  MathSciNet  Google Scholar 

  36. Murakawa, H.: A regularization of a reaction-diffusion system approximation to the two phase Stefan problem. Nonlinear Anal. TMA 69, 3512–3524 (2008)

    Article  MathSciNet  Google Scholar 

  37. Murakawa, H., Ninomiya, H.: Fast reaction limit of a three-component reaction-diffusion system. J. Math. Anal. Appl. 379, 150–170 (2011)

    Article  MathSciNet  Google Scholar 

  38. Murray, J. D.: Mathematical biology. II Spatial models and biomedical applications, Interdisciplinary Applied Mathematics, vol. 18. New York: Springer-New York Incorporated (2001)

  39. Nakaki, T., Murakawa, H.: A numerical method to Stefan problems and its application to the flow through porous media, European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004, 495.pdf (2004), 1–12

  40. Ninomiya, H., Tanaka, Y., Yamamoto, H.: Reaction, diffusion and non-local interaction. J. Math. Biol. 75, 1203–1233 (2017)

    Article  MathSciNet  Google Scholar 

  41. Ninomiya, H., Tanaka, Y., Yamamoto, H.: Reaction-diffusion approximation of nonlocal interactions using Jacobi polynomials. Jpn. J. Ind. Appl. Math. 35, 613–651 (2018)

    Article  MathSciNet  Google Scholar 

  42. Ninomiya, H., Yamamoto, H.: A reaction-diffusion approximation of a semilinear wave equation, p. 14 (preprint)

  43. Nishiura, Y.: Far-from equilibrium dynamics, translations of mathematical monographs, vol 209. American Mathematical Society (2002)

  44. Okubo, A.: Diffusion and ecological problems: mathematical models. vol. 10. Springer, New York (1980)

    MATH  Google Scholar 

  45. Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. Theoret. Biol. 79, 83–99 (1979)

    Article  MathSciNet  Google Scholar 

  46. Turchin, P.: Quantitative analysis of movement: measuring and modeling population redistribution in animals and plants. Sinauer (1998)

  47. Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. Roy. Soc. Lond. B237, 37–72 (1952)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Science and Engineering Education, University of Miyazaki, Miyazaki, 889-2192, Japan

    M. Iida

  2. School of Interdisciplinary Mathematical Sciences, Meiji University, Nakano-ku, Tokyo, 164-8525, Japan

    H. Ninomiya

  3. Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    H. Yamamoto

Authors
  1. M. Iida
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Ninomiya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to H. Ninomiya.

Additional information

This paper was partially supported by JSPS KAKENHI Grant numbers JP2628702401, JP15K04963, JP16K13778, JP16KT0022.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iida, M., Ninomiya, H. & Yamamoto, H. A review on reaction–diffusion approximation. J Elliptic Parabol Equ 4, 565–600 (2018). https://doi.org/10.1007/s41808-018-0029-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-018-0029-y

Keywords

Mathematics Subject Classification

Search

Navigation