Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

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

Hans G. Othmer and Angela Stevens
SIAM Journal on Applied Mathematics
Vol. 57, No. 4 (Aug., 1997), pp. 1044-1081
Stable URL: http://www.jstor.org/stable/2951915
Page Count: 38
  • Subscribe ($19.50)
  • Cite this Item
Aggregation, Blowup, and Collapse: The ABC's of Taxis in Reinforced Random Walks
Preview not available

Abstract

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.

Page Thumbnails

  • Thumbnail: Page 
1044
    1044
  • Thumbnail: Page 
1045
    1045
  • Thumbnail: Page 
1046
    1046
  • Thumbnail: Page 
1047
    1047
  • Thumbnail: Page 
1048
    1048
  • Thumbnail: Page 
1049
    1049
  • Thumbnail: Page 
1050
    1050
  • Thumbnail: Page 
1051
    1051
  • Thumbnail: Page 
1052
    1052
  • Thumbnail: Page 
1053
    1053
  • Thumbnail: Page 
1054
    1054
  • Thumbnail: Page 
1055
    1055
  • Thumbnail: Page 
1056
    1056
  • Thumbnail: Page 
1057
    1057
  • Thumbnail: Page 
1058
    1058
  • Thumbnail: Page 
1059
    1059
  • Thumbnail: Page 
1060
    1060
  • Thumbnail: Page 
1061
    1061
  • Thumbnail: Page 
1062
    1062
  • Thumbnail: Page 
1063
    1063
  • Thumbnail: Page 
1064
    1064
  • Thumbnail: Page 
1065
    1065
  • Thumbnail: Page 
1066
    1066
  • Thumbnail: Page 
1067
    1067
  • Thumbnail: Page 
1068
    1068
  • Thumbnail: Page 
1069
    1069
  • Thumbnail: Page 
1070
    1070
  • Thumbnail: Page 
1071
    1071
  • Thumbnail: Page 
1072
    1072
  • Thumbnail: Page 
1073
    1073
  • Thumbnail: Page 
1074
    1074
  • Thumbnail: Page 
1075
    1075
  • Thumbnail: Page 
1076
    1076
  • Thumbnail: Page 
1077
    1077
  • Thumbnail: Page 
1078
    1078
  • Thumbnail: Page 
1079
    1079
  • Thumbnail: Page 
1080
    1080
  • Thumbnail: Page 
1081
    1081