Skip to main content

Local Bifurcation

  • Chapter
  • First Online:
Non-Local Cell Adhesion Models

Part of the book series: CMS/CAIMS Books in Mathematics ((CMS/CAIMS BM,volume 1))

  • 446 Accesses

Abstract

The success of the Armstrong–Painter–Sherratt adhesion model (2.14) is that it can replicate the complicated patterns observed in cell sorting experiments.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. N.J. Armstrong, K.J. Painter, J.A. Sherratt, A continuum approach to modelling cell-cell adhesion. J. Theor. Biol. 243(1), 98–113 (2006)

    Article  MathSciNet  Google Scholar 

  2. P.L. Buono, R. Eftimie, Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation. J. Math. Biol. 71(4), 847–881 (2015)

    Article  MathSciNet  Google Scholar 

  3. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations. International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1955)

    Google Scholar 

  4. M.G. Crandall, P.H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree. J. Math. Mech. 19(12), 1083–1102 (1970)

    MathSciNet  MATH  Google Scholar 

  5. M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    Article  MathSciNet  Google Scholar 

  6. E.N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. Bull. Lond. Math. Soc. 34(5), 533–538 (2002)

    Article  MathSciNet  Google Scholar 

  7. M. Golubitsky, I. Stewart, The Symmetry Perspective. Progress in Mathematics (Birkhäuser, Basel, 2002)

    Google Scholar 

  8. T.J. Healey, Global bifurcations and continuation in the presence of symmetry with an application to solid mechanics. SIAM J. Math. Anal. 19(4), 824–840 (1988)

    Article  MathSciNet  Google Scholar 

  9. T.J. Healey, H.J. Kielhöfer, Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations. Arch. Ration. Mech. Anal. 113(4), 299–311 (1991)

    Article  MathSciNet  Google Scholar 

  10. T.J. Healey, H.J. Kielhöfer, Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry. J. Differ. Equ. 106(1), 70–89 (1993)

    Article  MathSciNet  Google Scholar 

  11. T. Hillen, C. Painter, K.J. Schmeiser, Global existence for chemotaxis with finite sampling radius. Discrete Contin. Dyn. Syst. Ser. B 7(1), 125–144 (2006)

    MathSciNet  MATH  Google Scholar 

  12. T. Hillen, K.J. Painter, A user’s guide to pde models for chemotaxis. J. Math. Biol. 58(1–2), 183–217 (2009)

    Article  MathSciNet  Google Scholar 

  13. J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis. Chapman & Hall/CRC Research Notes in Mathematics Series (Taylor & Francis, Hoboken, 2001)

    Google Scholar 

  14. J. López-Gómez, Global bifurcation for Fredholm operators. Rend. Ist. Mat. Univ. Trieste Int. J. Math. 48, 539–564 (2016)

    MathSciNet  MATH  Google Scholar 

  15. H. Matano, Asymptotic behavior of solutions of semilinear heat equations on S1, in Nonlinear Diffusion Equations and Their Equilibrium States II (Springer, Cham, 1988), pp. 139–162

    Book  Google Scholar 

  16. H.G. Othmer, T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250 (2002)

    Article  MathSciNet  Google Scholar 

  17. P.H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations. Commun. Pure Appl. Math. 23(1), 970, 939–961 (1970)

    Google Scholar 

  18. P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 513, 487–513 (1971)

    Article  MathSciNet  Google Scholar 

  19. J. Shi, X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246(7), 2788–2812 (2009)

    Article  MathSciNet  Google Scholar 

  20. X. Wang, Q. Xu, Spiky and transition layer steady states of chemotaxis systems via global bifurcation and helly’s compactness theorem. J. Math. Biol. 66, 1241–1266 (2013)

    Article  MathSciNet  Google Scholar 

  21. T. Xiang, A study on the positive nonconstant steady states of nonlocal chemotaxis systems. Discrete Contin. Dyn. Syst. Ser. B 18(9), 2457–2485 (2013)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Buttenschön, A., Hillen, T. (2021). Local Bifurcation. In: Non-Local Cell Adhesion Models. CMS/CAIMS Books in Mathematics, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-67111-2_4

Download citation

Publish with us

Policies and ethics