Journal of Mathematical Biology
, Volume 61, Issue 4, pp 545579
First online:
An investigation of a nonlocal hyperbolic model for selforganization of biological groups
 Razvan C. FetecauAffiliated withDepartment of Mathematics, Simon Fraser University Email author
 , Raluca EftimieAffiliated withDepartment of Mathematics and Statistics, McMaster University
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In this article, we introduce and study a new nonlocal hyperbolic model for the formation and movement of animal aggregations. We assume that the nonlocal attractive, repulsive, and alignment interactions between individuals can influence both the speed and the turning rates of group members. We use analytical and numerical techniques to investigate the effect of these nonlocal interactions on the longtime behavior of the patterns exhibited by the model. We establish the local existence and uniqueness and show that the nonlinear hyperbolic system does not develop shock solutions (gradient blowup). Depending on the relative magnitudes of attraction and repulsion, we show that the solutions of the model either exist globally in time or may exhibit finitetime amplitude blowup. We illustrate numerically the various patterns displayed by the model: dispersive aggregations, finitesize groups and blowup patterns, the latter corresponding to aggregations which may collapse to a point. The transition from finitesize to blowup patterns is governed by the magnitude of the social interactions and the random turning rates. The presence of these types of patterns and the absence of shocks are consequences of the biologically relevant assumptions regarding the form of the speed and the turning rate functions, as well as of the kernels describing the social interactions.
Keywords
Biological aggregations Nonlinear hyperbolic systems Nonlocal interactions Alignment BlowupMathematics Subject Classification (2000)
92D25 92D50 35L65 Title
 An investigation of a nonlocal hyperbolic model for selforganization of biological groups
 Journal

Journal of Mathematical Biology
Volume 61, Issue 4 , pp 545579
 Cover Date
 201010
 DOI
 10.1007/s0028500903116
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Biological aggregations
 Nonlinear hyperbolic systems
 Nonlocal interactions
 Alignment
 Blowup
 92D25
 92D50
 35L65
 Authors

 Razvan C. Fetecau ^{(1)}
 Raluca Eftimie ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada
 2. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada