Skip to main content
Log in

The Stochastic Gierer–Meinhardt System

  • Published:
Applied Mathematics & Optimization Aims and scope Submit manuscript

A Correction to this article was published on 06 July 2022

This article has been updated

Abstract

The Gierer–Meinhardt system occurs in morphogenesis, where the development of an organism from a single cell is modelled. One of the steps in the development is the formation of spatial patterns of the cell structure, starting from an almost homogeneous cell distribution. Turing proposed different activator–inhibitor systems with varying diffusion rates in his pioneering work, which could trigger the emergence of such cell structures. Mathematically, one describes these activator–inhibitor systems as coupled systems of reaction-diffusion equations with different diffusion coefficients and highly nonlinear interaction. One famous example of these systems is the Gierer–Meinhardt system. These systems usually are not of monotone type, such that one has to apply other techniques. The purpose of this article is to study the stochastic reaction-diffusion Gierer–Meinhardt system with homogeneous Neumann boundary conditions on a one or two-dimensional bounded spatial domain. To be more precise, we perturb the original Gierer–Meinhardt system by an infinite-dimensional Wiener process and show under which conditions on the Wiener process and the initial conditions, a solution exists. In dimension one, we even show the pathwise uniqueness. In dimension two, uniqueness is still an open question.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

Change history

References

  1. Barbu, V., Da Prato, G., Röckner, M.: Existence and uniqueness of non-negative solutions to the stochastic porous media equation. Indiana Univ. Math. J. 57(1), 187–211 (2008)

    Article  MathSciNet  Google Scholar 

  2. Barbu, V., Da Prato, G., Röckner, M.: Stochastic Porous Media Equations. Lecture Notes in Mathematics, vol. 2163. Springer, Berlin (2016)

  3. Bashkirtseva, I., L. Ryashko, L., Ryazanova, T.: Analysis of noise-induced bifurcations in the stochastic tritrophic population system. Int. J. Bifurc. Chaos Appl. Sci. Eng. 27(13), Article ID 1750208 (2017)

  4. Bergh, J., Löfström, J.: An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer, Berlin (1976)

  5. Biancalani, T., Jafarpour, F., Goldenfeld, N.: Giant Amplification of noise in fluctuation-induced pattern formation. Phys. Rev. Lett. 118(1), 018101 (2017)

    Article  Google Scholar 

  6. Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations, vol. XIV. Springer, New York (2011)

  7. Britton, N.F.: Essential Mathematical Biology. Springer, London (2003)

    Book  Google Scholar 

  8. Chen, L., Shao, Y., Wu, R., Zhou, Y.: Turing and Hopf bifurcation of Gierer–Meinhardt activator-substrate model. Electron. J. Differ. Equ. 2017(173), 1–19 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Cherny, A.S.: On the strong and weak solutions of stochastic differential equations governing Bessel processes. Stoch. Stoch. Rep. 70, 213–219 (2000)

    Article  MathSciNet  Google Scholar 

  10. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, 2nd edn. Encyclopedia of Mathematics and its Applications, vol. 152. Cambridge University Press, Cambridge (2014)

  11. Duan, J., Wang, W.: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, Amsterdam (2014)

    MATH  Google Scholar 

  12. Engelbert, H.: On the theorem of T. Yamada and S. Watanabe. Stoch. Stoch. Rep. 36, 205–216 (1991)

  13. Ethier, S., Kurtz, T.: Markov Processes, Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1986)

  14. Ghergu, M., Ruadulescu, V.: Nonlinear PDEs. Springer Monographs in Mathematics. Springer, Heidelberg (2012). Mathematical Models in Biology, Chemistry and Population Genetics, With a foreword by Viorel Barbu

  15. Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik 12, 30–39 (1972)

    Article  Google Scholar 

  16. Gierer, A., Meinhardt, H.: Generation and regeneration of sequence of structures during morphogenesis. J. Theor. Biol. 85(3), 429–450 (1980)

    Article  MathSciNet  Google Scholar 

  17. Gong, P., Gu, L., Wang, H.: Hopf bifurcation and turing instability analysis for the Gierer–Meinhardt model of the depletion type. Discret. Dyn. Nat. Soc. 2020, 1–10 (2020)

    MathSciNet  MATH  Google Scholar 

  18. Gonpot, P., Collet, J., Sookia, N.: Gierer–Meinhardt model: bifurcation analysis and pattern formation. Trends Appl. Sci. Res. 3(2), 115–128 (2008)

    Article  Google Scholar 

  19. Hausenblas, E., Panda, A.A.: The stochastic Gierer–Meinhardt system. Online resource. Application. Math. Optim. (2022). https://doi.org/10.1007/s00245-022-09835-6

  20. Hausenblas, E., Seidler, J.: A note on maximal inequality for stochastic convolutions. Czech. Math. J. 51(4), 785–790 (2001)

    Article  MathSciNet  Google Scholar 

  21. Jiang, H.: Global existence of solutions of an activator–inhibitor system. Discret. Contin. Dyn. Syst. 14(4), 737–751 (2006)

    Article  MathSciNet  Google Scholar 

  22. Jacod, J.: Weak and strong solutions of stochastic differential equations. Stochastics 3, 171–191 (1980)

    Article  MathSciNet  Google Scholar 

  23. Karig, D., Martini, K., Lu, T., DeLateur, N., Goldenfeld, N., Weiss, R.: Stochastic Turing patterns in a synthetic bacterial population. Proc. Natl Acad. Sci. U.S.A. 115(26), 6572–6577 (2018)

    Article  Google Scholar 

  24. Kavallaris, I.N., Suzuki, T.: Gierer–Meinhardt System. Non-Local Partial Differential Equations for Engineering and Biology. Mathematics for Industry, vol. 31, pp. 163–193. Springer, Berlin (2018)

  25. Kelkel, J., Surulescu, C.: A weak solution approach to a reaction-diffusion system modeling pattern formation on seashells. Math. Methods Appl. Sci. 32(17), 2267–2286 (2009)

    Article  MathSciNet  Google Scholar 

  26. Kelkel, J., Surulescu, C.: On a stochastic reaction-diffusion system modelling pattern formation on seashells. J. Math. Biol. 60, 765–796 (2010)

    Article  MathSciNet  Google Scholar 

  27. Kolinichenko, A., Ryashko, L.: Multistability and stochastic phenomena in the distributed Brusselator model. J. Comput. Nonlinear Dyn. 15(1), 011007 (2020)

    Article  Google Scholar 

  28. Kolinichenko, A., Pisarchik, A.N., Ryashko, L.: Stochastic phenomena in pattern formation for distributed nonlinear systems. Philos. Trans. R. Soc. (2020). https://doi.org/10.1098/rsta.2019.0252

    Article  MATH  Google Scholar 

  29. Krylov, N.V.: Itô’s formula for the \(L_p\)-norm of stochastic \(W^1_p\)-valued processes. Probab. Theory Relat. Fields 147(3), 583–605 (2010)

    Article  Google Scholar 

  30. Li, F., Xu, L.: Finite time blowup of the stochastic shadow Gierer–Meinhardt system. Electron. Commun. Probab. 20(65), 1–13 (2015)

    MathSciNet  MATH  Google Scholar 

  31. Masuda, K., Takahashi, K.: Reaction-diffusion systems in the Gierer–Meinhardt theory of biological pattern formation. Jpn. J. Appl. Math. 4(1), 47–58 (1987)

    Article  MathSciNet  Google Scholar 

  32. Meinhardt, H.: Models of Biological Pattern Formation. Academic Press, New York (1982)

    Google Scholar 

  33. Murray, J.D.: Mathematical Biology: I: An Introduction. Interdisciplinary Applied Mathematics, vol. 17, 3rd edn. Springer, New York (2002)

  34. Murray, J.D.: Mathematical Biology II: Spatial Models and Biomedical Applications (Interdisciplinary Applied Mathematics, vol. 18), 3rd edn. Springer, New York (2003)

  35. Ondreját, M.: Uniqueness for stochastic evolution equations in Banach spaces. Diss. Math. 426, 1–63 (2004)

    MathSciNet  MATH  Google Scholar 

  36. Perthame, B.: Parabolic Equations in Biology. Lecture Notes on Mathematical Modelling in the Life Sciences. Springer, Berlin (2015)

  37. Qiao, H.: A theorem dual to Yamada–Watanabe theorem for stochastic evolution equations. Stoch. Dyn. 10, 367–374 (2010)

    Article  MathSciNet  Google Scholar 

  38. Stratonovich, R.L.: Topics in the Theory of Random Noise, vol. 1: General Theory of Random Processes Nonlinear Transformations of Signals and Noise. Gordon and Breach Science Publishers, New York (1963)

  39. Stratonovich, R.L.: Topics in the Theory of Random Noise: Volume II: Peaks of Random Functions and the Effect of Noise on Relays Nonlinear Self-excited. Gordon and Breach Science Publishers, New York (1967)

  40. Tappe, S.: The Yamada–Watanabe theorem for mild solutions to stochastic partial differential equations. Electron. Commun. Probab. 18, 1–13 (2013)

    Article  MathSciNet  Google Scholar 

  41. Tessitore, G., Zabczyk, J.: Strict positivity for stochastic heat equations. Stoch. Processes Appl. 77, 83–98 (1998)

    Article  MathSciNet  Google Scholar 

  42. Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237(641), 37–72 (1952)

    Article  MathSciNet  Google Scholar 

  43. Van Neerven, J., Veraar, M., Weis, L.: Maximal \(L^p\)-regularity for stochastic evolution equations. SIAM J. Math. Anal. 44(3), 1372–1414 (2012)

    Article  MathSciNet  Google Scholar 

  44. Wei, J., Winter, M.: Mathematical Aspects of Pattern Formation in Biological Systems, vol. 189. Springer, London (2014)

  45. Wei, L., Röckner, M.: Stochastic Partial Differential Equations: An Introduction. Universitext, Springer, Cham (2015)

  46. Winter, M., Xu, L., Zhai, J., Zhang, T.: The dynamics of the stochastic shadow Gierer–Meinhardt system. J. Differ. Equ. 260, 84–114 (2016)

    Article  MathSciNet  Google Scholar 

  47. Yamada, T., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11, 155–167 (1971)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by the Austrian Science Foundation (FWF) project number P28010 and P28819.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Akash Ashirbad Panda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file 1 (tex 65 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hausenblas, E., Panda, A.A. The Stochastic Gierer–Meinhardt System. Appl Math Optim 85, 24 (2022). https://doi.org/10.1007/s00245-022-09835-6

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00245-022-09835-6

Keywords

Mathematics Subject Classification

Navigation