Metastability in Chemotaxis Models

We consider pattern formation in a chemotaxis model with a vanishing chemotaxis coefficient at high population densities. This model was developed in Hillen and Painter (2001, Adv. Appli. Math. 26(4), 280–301.) to model volume effects. The solutions show spatio-temporal patterns which allow for ultra-long transients and merging or coarsening. We study the underlying bifurcation structure and show that the existence time for the pseudo- structures exponentially grows with the size of the system. We give approximations for one-step steady state solutions. We show that patterns with two or more steps are metastable and we approximate the two-step interaction using asymptotic expansions. This covers the basic effects of coarsening/merging and dissolving of local maxima. These effects are similar to pattern dynamics in other chemotaxis models, in spinodal decomposition of Cahn–Hilliard models, or to metastable patterns in microwave heating models.

[1]  C. Patlak.  Random walk with persistence and external bias , 1953 .

[2]  L. Segel,et al.  Model for chemotaxis. , 1971, Journal of theoretical biology.

[3]  R. Schaaf.  Stationary solutions of chemotaxis systems , 1985 .

[4]  Crutchfield,et al.  Are attractors relevant to turbulence? , 1988, Physical review letters.

[5]  Peter W. Bates,et al.  Slow motion for the Cahn-Hilliard equation in one space dimension , 1991 .

[6]  Amy Novick-Cohen,et al.  The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor , 1999 .

[7]  Michael J. Ward,et al.  Dynamics and Coarsening of Interfaces for the Viscous Cahn—Hilliard Equation in One Spatial Dimension , 2000 .

[8]  Thomas Hillen,et al.  Global Existence for a Parabolic Chemotaxis Model with Prevention of Overcrowding , 2001, Adv. Appl. Math..

[9]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[10]  T Hillen,et al.  Cattaneo models for chemosensitive movement: numerical solution and pattern formation. , 2003, Journal of mathematical biology.

[11]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[12]  Y. Dolak,et al.  Cattaneo models for chemosensitive movement , 2003, Journal of mathematical biology.

[13]  M. Ward,et al.  THE STABILITY AND DYNAMICS OF HOT-SPOT SOLUTIONS TO TWO ONE-DIMENSIONAL MICROWAVE HEATING MODELS , 2004 .

[14]  C. Schmeiser,et al.  The Keller-Segel model with small diffusivity , 2004 .

[15]  Christian Schmeiser,et al.  The Keller-Segel Model with Logistic Sensitivity Function and Small Diffusivity , 2005, SIAM J. Appl. Math..

[16]  Juncheng Wei,et al.  The Existence and Stability of Spike Patterns in a Chemotaxis Model , 2005, SIAM J. Appl. Math..