Stationary solutions of chemotaxis systems

Schaaf, Renate

doi:10.1090/S0002-9947-1985-0808736-1

Stationary solutions of chemotaxis systems
HTML articles powered by AMS MathViewer

by Renate Schaaf PDF

Trans. Amer. Math. Soc. 292 (1985), 531-556 Request permission

Abstract:

The Keller-Segel Model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance (Chemotaxis). This paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns. Examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response.

References

J. C. Alexander and Stuart S. Antman, Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rational Mech. Anal. 76 (1981), no. 4, 339–354. MR 628173, DOI 10.1007/BF00249970
Wolfgang Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol. 9 (1980), no. 2, 147–177. MR 661424, DOI 10.1007/BF00275919
Wolfgang Alt, Orientation of cells migrating in a chemotactic gradient, Biological growth and spread (Proc. Conf., Heidelberg, 1979) Lecture Notes in Biomath., vol. 38, Springer, Berlin-New York, 1980, pp. 353–366. MR 609371
N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal. 4 (1974/75), 17–37. MR 440205, DOI 10.1080/00036817408839081
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), no. 3-4, 217–237. MR 632161, DOI 10.1016/0025-5564(81)90055-9
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal. 52 (1973), 161–180. MR 341212, DOI 10.1007/BF00282325

Taxis and behaviour, elementary sensory systems in biology

Perturbation theory for linear operators

Initiation of slime mold aggragation viewed as an instability

26

Hansjörg Kielhöfer, On the Lyapunov-stability of stationary solutions of semilinear parabolic differential equations, J. Differential Equations 22 (1976), no. 1, 193–208. MR 450762, DOI 10.1016/0022-0396(76)90011-5
I. Richard Lapidus and M. Levandowsky, Modeling chemosensory responses of swimming eukaryotes, Biological growth and spread (Proc. Conf., Heidelberg, 1979) Lecture Notes in Biomath., vol. 38, Springer, Berlin-New York, 1980, pp. 388–396. MR 609375

Chemotaxis, signal relaying and aggregation morphology

42

Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
Paul H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (1971), 487–513. MR 0301587, DOI 10.1016/0022-1236(71)90030-9
Renate Schaaf, Global behaviour of solution branches for some Neumann problems depending on one or several parameters, J. Reine Angew. Math. 346 (1984), 1–31. MR 727393, DOI 10.1515/crll.1984.346.1

Global solution branches via the time maps

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981), no. 2, 269–290. MR 607786, DOI 10.1016/0022-0396(81)90077-2
J. Smoller, A. Tromba, and A. Wasserman, Nondegenerate solutions of boundary value problems, Nonlinear Anal. 4 (1980), no. 2, 207–216. MR 563804, DOI 10.1016/0362-546X(80)90049-8
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
René P. Sperb, On a mathematical model describing the aggregation of amoebae, Bull. Math. Biol. 41 (1979), no. 4, 555–571. MR 631877, DOI 10.1016/S0092-8240(79)80008-7