TY - JOUR

T1 - Aggregation, blowup, and collapse

T2 - The ABC'S of taxis in reinforced random walks

AU - Othmer, Hans G.

AU - Stevens, Angela

PY - 1997/8

Y1 - 1997/8

N2 - 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 affected by the local modulator concentration. We then show that a variety of dynamics is possible, which we classify as aggregation, blowup, or collapse, depending on whether the dynamics admit stable bounded peaks, whether solutions blow up in finite time, or whether a suitable spatial norm of the density function is asymptotically less than its initial value.

AB - 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 affected by the local modulator concentration. We then show that a variety of dynamics is possible, which we classify as aggregation, blowup, or collapse, depending on whether the dynamics admit stable bounded peaks, whether solutions blow up in finite time, or whether a suitable spatial norm of the density function is asymptotically less than its initial value.

KW - Aggregation

KW - Blowup

KW - Chemotaxis equations

KW - Collapse

KW - Diffusion approximation

KW - Reinforced random walk

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U2 - 10.1137/s0036139995288976

DO - 10.1137/s0036139995288976

M3 - Article

AN - SCOPUS:0031207030

SN - 0036-1399

VL - 57

SP - 1044

EP - 1081

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

IS - 4

ER -