Author
Benoît Perthame
Bio: Benoît Perthame is an academic researcher. The author has contributed to research in topics: Population & Euler equations. The author has an hindex of 1, co-authored 1 publications receiving 839 citations.
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05 Sep 2008
TL;DR: In this paper, the renewal equation is used to describe the structure of a population from a point of view of population balance equations, which is an asymptotic view of the population dynamics.
Abstract: From differential equations to structured population dynamics.- Adaptive dynamics an asymptotic point of view.- Population balance equations: the renewal equation.- Population balance equations: size structure.- Cell motion and chemotaxis.- General mathematical tools.
932 citations
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TL;DR: This paper explores in detail a number of variations of the original Keller–Segel model of chemotaxis from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form.
Abstract: Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.
1,532 citations
TL;DR: It is shown that in adult humans new neurons integrate in the striatum, which is adjacent to this neurogenic niche, and this findings demonstrate a unique pattern of neurogenesis in the adult human brain.
821 citations
TL;DR: For the quasilinear parabolic Keller-Segel system with homogeneous Neumann boundary conditions, this article showed that the classical solutions to the problem are uniformly in time bounded, provided that D ( u ) satisfies some technical conditions such as algebraic upper and lower growth estimates as u → ∞.
610 citations
TL;DR: It is demonstrated that cells add a constant volume each generation, irrespective of their newborn sizes, conclusively supporting the so-called constant Δ model, which was introduced for E. coli and recently revisited.
600 citations
TL;DR: In this paper, coupled chemotaxis (Navier and Stokes) systems generalizing the prototype have been proposed to describe the collective effects arising in bacterial suspensions in fluid drops, and they have been applied to the model of collective effects of bacterial suspensions.
Abstract: In the modeling of collective effects arising in bacterial suspensions in fluid drops, coupled chemotaxis-(Navier–)Stokes systems generalizing the prototype have been proposed to describe the spont
523 citations