Skip to main content Accessibility help
×
Home
Hostname: page-component-8bbf57454-q5g9d Total loading time: 0.445 Render date: 2022-01-25T21:14:51.604Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

Aggregation under local reinforcement: From lattice to continuum

Published online by Cambridge University Press:  04 March 2005

DIRK HORSTMANN
Affiliation:
Mathematisches Institut der Universität zu Köln, Weyertal 86-90, D-50923 Köln, Germany e-mail dhorst@mi.uni-koeln.de
KEVIN J. PAINTER
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland e-mail painter@ma.hw.ac.uk
HANS G. OTHMER
Affiliation:
School of Mathematics, University of Minnesota, Vincent Hall 207a, 206 Church St. S.E., Minneapolis, MN 55455 USA e-mail othmer@math.umn.edu

Abstract

Movement of biological organisms is frequently initiated in response to a diffusible or otherwise transported signal, and in its simplest form this movement can be described by a diffusion equation with an advection term. In systems in which the signal is localized in space the question arises as to whether aggregation of a population of indirectly-interacting organisms, or localization of a single organism, is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. It has been shown [25] that continuum approximations of reinforced random walks show aggregation and even blowup, but the connections between solutions of the continuum equations and of the master equation for the corresponding lattice walk were not studied. Using variational techniques and the existence of a Lyapunov functional, we study these connections here for certain simplified versions of the model studied earlier. This is done by relating knowledge about the shape of the minimizers of a variational problem to the asymptotic spatial structure of the solution.

Type
Papers
Copyright
2004 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)
24
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Aggregation under local reinforcement: From lattice to continuum
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Aggregation under local reinforcement: From lattice to continuum
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Aggregation under local reinforcement: From lattice to continuum
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *