Journal of Mathematical Biology

, Volume 65, Issue 1, pp 35–75 | Cite as

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

Article

Abstract

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.

Keywords

Hyperbolic models Kinetic models Self-organized movement Aggregation 

Mathematics Subject Classification (2000)

45K05 35L40 92C17 92D50 

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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