Skip to main content

2021 | OriginalPaper | Buchkapitel

7. Discussion and Future Directions

verfasst von : Andreas Buttenschön, Thomas Hillen

Erschienen in: Non-Local Cell Adhesion Models

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

The central building block to include adhesive interactions between cells in reaction-advection-diffusion models of tissues is to use a non-local term.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
10.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
36.
Zurück zum Zitat J.A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure (2014). Preprint, arXiv:1402.4252 J.A. Carrillo, A. Chertock, Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure (2014). Preprint, arXiv:1402.4252
37.
Zurück zum Zitat J.A. Carrillo, R.S. Gvalani, G.A. Pavliotis, A. Schlichting, Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus. Arch. Ration. Mech. Anal. 235(1), 635–690 (2020)MathSciNetMATH J.A. Carrillo, R.S. Gvalani, G.A. Pavliotis, A. Schlichting, Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus. Arch. Ration. Mech. Anal. 235(1), 635–690 (2020)MathSciNetMATH
39.
Zurück zum Zitat J.A. Carrillo, H. Murakawa, M. Sato, H. Togashi, O. Trush, A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation (2019). Preprint, arXiv:1901.02919 J.A. Carrillo, H. Murakawa, M. Sato, H. Togashi, O. Trush, A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation (2019). Preprint, arXiv:1901.02919
49.
Zurück zum Zitat F.A. Davidson, N. Dodds, Spectral properties of non-local differential operators. Appl. Anal. 85(6–7), 717–734 (2006)MathSciNetMATH F.A. Davidson, N. Dodds, Spectral properties of non-local differential operators. Appl. Anal. 85(6–7), 717–734 (2006)MathSciNetMATH
56.
Zurück zum Zitat P. Domschke, D. Trucu, A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. J. Theor. Biol. 361C, 41–60 (2014)MathSciNetMATH P. Domschke, D. Trucu, A. Gerisch, M.A.J. Chaplain, Mathematical modelling of cancer invasion: implications of cell adhesion variability for tumour infiltrative growth patterns. J. Theor. Biol. 361C, 41–60 (2014)MathSciNetMATH
68.
Zurück zum Zitat P. Freitas, Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations. J. Dyn. Differ. Equ. 6(4), 613–629 (1994)MathSciNetMATH P. Freitas, Bifurcation and stability of stationary solutions of nonlocal scalar reaction-diffusion equations. J. Dyn. Differ. Equ. 6(4), 613–629 (1994)MathSciNetMATH
69.
Zurück zum Zitat P. Freitas, A nonlocal Sturm–Liouville eigenvalue problem. Proc. R. Soc. Edinb. Math. 124(01), 169–188 (1994)MathSciNetMATH P. Freitas, A nonlocal Sturm–Liouville eigenvalue problem. Proc. R. Soc. Edinb. Math. 124(01), 169–188 (1994)MathSciNetMATH
70.
Zurück zum Zitat P. Freitas, M. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space. Differ. Integral Equ. 13(1–3), 265–288 (2000)MathSciNetMATH P. Freitas, M. Vishnevskii, Stability of stationary solutions of nonlocal reaction-diffusion equations in m-dimensional space. Differ. Integral Equ. 13(1–3), 265–288 (2000)MathSciNetMATH
73.
Zurück zum Zitat H. Fujii, Y. Nishiura, Global bifurcation diagram in nonlinear diffusion systems, in Northholland Mathematics Studies (Elsevier, Amsterdam, 1983), pp. 17–35 H. Fujii, Y. Nishiura, Global bifurcation diagram in nonlinear diffusion systems, in Northholland Mathematics Studies (Elsevier, Amsterdam, 1983), pp. 17–35
88.
Zurück zum Zitat J.K. Hale, Asymptotic Behavior of Dissipative Systems (American Mathematical Soc., Providence, 1988) J.K. Hale, Asymptotic Behavior of Dissipative Systems (American Mathematical Soc., Providence, 1988)
96.
Zurück zum Zitat D.B. Henry, Some infinite-dimensional morse-smale systems defined by parabolic partial differential equations. J. Differ. Equ. 59(2), 165–205 (1985)MathSciNetCrossRef D.B. Henry, Some infinite-dimensional morse-smale systems defined by parabolic partial differential equations. J. Differ. Equ. 59(2), 165–205 (1985)MathSciNetCrossRef
125.
Zurück zum Zitat H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo 1A 29, 401–441 (1982)MathSciNetMATH H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo 1A 29, 401–441 (1982)MathSciNetMATH
126.
Zurück zum Zitat 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. 139–162CrossRef 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. 139–162CrossRef
133.
Zurück zum Zitat H. Murakawa, H. Togashi, Continuous models for cell-cell adhesion. J. Theor. Biol. 374, 1–12 (2015)MATH H. Murakawa, H. Togashi, Continuous models for cell-cell adhesion. J. Theor. Biol. 374, 1–12 (2015)MATH
140.
Zurück zum Zitat Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13(4), 555–593 (1982)MathSciNetCrossRef Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13(4), 555–593 (1982)MathSciNetCrossRef
156.
Zurück zum Zitat P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 513, 487–513 (1971)MathSciNetMATH P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 513, 487–513 (1971)MathSciNetMATH
159.
Zurück zum Zitat J.B. Raquepas, J.D. Dockery, Dynamics of a reaction–diffusion equation with nonlocal inhibition. Physica D 134(1), 94–110 (1999)MathSciNetCrossRef J.B. Raquepas, J.D. Dockery, Dynamics of a reaction–diffusion equation with nonlocal inhibition. Physica D 134(1), 94–110 (1999)MathSciNetCrossRef
Metadaten
Titel
Discussion and Future Directions
verfasst von
Andreas Buttenschön
Thomas Hillen
Copyright-Jahr
2021
DOI
https://doi.org/10.1007/978-3-030-67111-2_7