Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks

In many biological systems, movement of an organism occurs in response to a diffusible or otherwise transported signal, and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by Patlak [Bull. of Math. Biophys., 15 (1953), pp. 311--338]. However, other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages. In these systems, one example of which is the myxobacteria, the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. Davis [Probab. Theory Related Fields, 84 (1990), pp. 203--229] has studied this question for a certain class of random walks, and here we extend this analysis to the continuum limit of such walks. We first derive several general classes of partial differential equations that depend on how the movement rules are...

[1]  S. Tuljapurkar,et al.  Mathematical Aspects of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, Volume 28.Paul C. Fife , S. Levin , 1980 .

[2]  M. A. Herrero,et al.  Chemotactic collapse for the Keller-Segel model , 1996, Journal of mathematical biology.

[3]  V. Nanjundiah,et al.  Chemotaxis, signal relaying and aggregation morphology. , 1973, Journal of theoretical biology.

[4]  M. A. Herrero,et al.  Singularity patterns in a chemotaxis model , 1996 .

[5]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[6]  M. Rascle,et al.  Finite time blow-up in some models of chemotaxis , 1995, Journal of mathematical biology.

[7]  Howard A. Levine,et al.  A System of Reaction Diffusion Equations Arising in the Theory of Reinforced Random Walks , 1997, SIAM J. Appl. Math..

[8]  Wei-Ming Ni,et al.  Large amplitude stationary solutions to a chemotaxis system , 1988 .

[9]  Angela Stevens,et al.  A Model for Gliding and Aggregation of Myxobacteria , 1991 .

[10]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[11]  Reinforced Random Walks.  Reinforced Random Walks , 2022 .

[12]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[13]  A. Stevens.  TRAIL FOLLOWING AND AGGREGATION OF MYXOBACTERIA , 1995 .

[14]  M. Eastham,et al.  The asymptotic solution of linear differential systems , 1985 .

[15]  Tohru Tsujikawa,et al.  Aggregating pattern dynamics in a chemotaxis model including growth , 1996 .

[16]  W. Jäger,et al.  On explosions of solutions to a system of partial differential equations modelling chemotaxis , 1992 .

[17]  M. S. P. Eastham,et al.  The asymptotic solution of linear differential systems : applications of the Levinson theorem , 1989 .

[18]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

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

[20]  J. V. Hurley,et al.  Chemotaxis , 2005, Infection.

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

[22]  Karl Oelschläger,et al.  A fluctuation theorem for moderately interacting diffusion processes , 1987 .

[23]  明 大久保,et al.  Diffusion and ecological problems : mathematical models , 1980 .

[24]  B. Davis,et al.  Reinforced random walk , 1990 .

[25]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[26]  D. G. Aronson.  The Role of Diffusion in Mathematical Population Biology: Skellam Revisited , 1985 .

[27]  Aurel Wintner,et al.  Asymptotic Integrations of Linear Differential Equations , 1955 .

[28]  Christian Remling,et al.  Asymptotic Integration of Linear Differential Equations , 1997 .

[29]  I. Lapidus,et al.  "Pseudochemotaxis" by micro-organisms in an attractant gradient. , 1980, Journal of theoretical biology.