On the discrete kinetic theory for active particles. Mathematical tools

Chauvière, Arnaud; Brazzoli, Ilaria

doi:10.1016/j.mcm.2005.10.001

Cited by 15 publications

(13 citation statements)

References 21 publications

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“…a m ) denotes m "activity states". Note that v can be a continuous variable (if m = 1, as in Bellomo and Delitala 2008) or a discrete variable (if m > 1, as in Chauviere and Brazzoli 2006). The term H (t)·∇ v u describes the change in the density u over the activity space.…”

Section: Kinetic Models For Active Particlesmentioning

confidence: 99%

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

Eftimie

2011

J. Math. Biol.

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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.

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Section: Kinetic Models For Active Particlesmentioning

confidence: 99%

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

Eftimie

2011

J. Math. Biol.

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“…A mathematical discrete kinetic theory for describing the collective behavior of large systems of interacting active particles has been developed in [1], where the concept of active particles refers to those interacting entities whose microscopic state includes, in addition to mechanical variables, also additional variables called activities, related to their peculiar functions. Interactions among the active particles modify their individual state and generate proliferating and destructive events.…”

Section: Introductionmentioning

confidence: 99%

“…Paper [1] deals with a kinetic methodological approach where a class of evolution equations are proposed for modelling large systems of interacting individuals such that the microscopic state is a discrete variable; see also [2,3]. The aim for analyzing such systems is induced by modelling requirements rather than by the intention of reducing computational complexity.…”

Section: Introductionmentioning

confidence: 99%

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From the discrete kinetic theory to modelling open systems of active particles

Brazzoli

2008

Applied Mathematics Letters

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This work deals with a methodological development of the kinetic theory for open systems of active particles with discrete states. It essentially refers to the derivation of mathematical tools which provide the guidelines for modelling open systems in different fields of applied sciences. After a description of closed systems, mathematical frameworks suitable for depicting the evolution of open systems are proposed. Finally, some research perspectives towards modelling are outlined.

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“…Following this line, a systematic analysis of socio-political systems was developed in three recent papers by Bertotti and Delitala (2004, 2007, within the framework of the kinetic theory for particles with discrete states. These papers, together with Chauviere and Brazzoli (2006), offer the guiding lines of the analysis to investigate the ability of the model proposed in Bertotti and Delitala (2007) in describing how a social politics and the overall wealth disposable may have a relevant influence towards the trend of the wealth distribution. This paper develops a critical analysis which shows that the model provides, despite its simplicity, several interesting indications.…”

Section: Introductionmentioning

confidence: 99%