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2021 | OriginalPaper | Chapter

7. Discussion and Future Directions

Authors : Andreas Buttenschön, Thomas Hillen

Published in: Non-Local Cell Adhesion Models

Publisher: Springer International Publishing

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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.

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Literature
10.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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.
go back to reference 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
Metadata
Title
Discussion and Future Directions
Authors
Andreas Buttenschön
Thomas Hillen
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-67111-2_7

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