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.
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References
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)
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)
A. Coddington, N. Levinson, Theory of Ordinary Differential Equations. International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1955)
M.G. Crandall, P.H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree. J. Math. Mech. 19(12), 1083–1102 (1970)
M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)
E.N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one. Bull. Lond. Math. Soc. 34(5), 533–538 (2002)
M. Golubitsky, I. Stewart, The Symmetry Perspective. Progress in Mathematics (Birkhäuser, Basel, 2002)
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)
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)
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)
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)
T. Hillen, K.J. Painter, A user’s guide to pde models for chemotaxis. J. Math. Biol. 58(1–2), 183–217 (2009)
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis. Chapman & Hall/CRC Research Notes in Mathematics Series (Taylor & Francis, Hoboken, 2001)
J. López-Gómez, Global bifurcation for Fredholm operators. Rend. Ist. Mat. Univ. Trieste Int. J. Math. 48, 539–564 (2016)
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
H.G. Othmer, T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations. SIAM J. Appl. Math. 62, 1222–1250 (2002)
P.H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations. Commun. Pure Appl. Math. 23(1), 970, 939–961 (1970)
P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 513, 487–513 (1971)
J. Shi, X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246(7), 2788–2812 (2009)
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)
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)
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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
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