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Local Hyperbolic/Kinetic Systems in 1D

  • Raluca Eftimie
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2232)

Abstract

Local hyperbolic systems have been first introduced to describe the movement of a population formed of left-moving and right-moving individuals, in response to the local density of their neighbours. These types of models (also called discrete-speed kinetic models, since they incorporate individual-level information regarding the movement direction of cell/bacteria/individuals into macroscopic models for population dynamics) are applied to describe biological phenomena characterised by sharp turning behaviours (as observed, for example, in bacteria or cells). In this Chapter we discuss these hyperbolic systems in a step-by-step manner: we start with conservative systems with density-dependent turning rates, then we discuss systems with density-dependent speeds, and we conclude by discussing systems that include population dynamics (described by death and birth terms). We also present in more detail an analytical investigation of the stability of spatially-homogeneous steady states and spatially-heterogeneous travelling waves.

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Authors and Affiliations

  • Raluca Eftimie
    • 1
  1. 1.Division of MathematicsUniversity of DundeeDundeeUK

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