The Diffusion Limit of Transport Equations II: Chemotaxis Equations

Othmer, Hans G.; Hillen, Thomas

doi:10.1137/s0036139900382772

Cited by 328 publications

(368 citation statements)

References 42 publications

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“…In either case, the signal detected by the cell is intrinsically non-local and it may therefore be appropriate to consider movement based on a non-local gradient by the integration of the signal by the cell over some region. A model incorporating this was postulated in [77] and a detailed derivation, analytical and numerical study has been carried out by [37].…”

Section: (M4) Non-local Samplingmentioning

confidence: 99%

A user’s guide to PDE models for chemotaxis

2008

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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.

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Section: (M4) Non-local Samplingmentioning

confidence: 99%

A user’s guide to PDE models for chemotaxis

2008

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“…Simplified cases can be obtained suppressing one or more of the three terms that characterize the model. For instance models in absence of evolution of biological functions have been studied by Othmer and Hillen [25], [27], who have proposed the stochastic jumping dynamics described by the operator L . Similarly, it is possible to study the early stage of the evolution characterized by mutations, but not yet by proliferative events.…”

Section: Remark 23 the Above Modelling Approach Is Based On The Assumentioning

confidence: 99%

“…Various contributions are given by papers [3], [4], [5], [17], [18], [20], [25], [27], while the book [11] and the survey [8] report about applications of the kinetic theory to model complex systems in biology. Moreover different models in life sciences are presented in the book [2] and the survey [7].…”

Section: Introductionmentioning

confidence: 99%

From kinetic models of multicellular growing systems to macroscopic biological tissue models

Bellouquid

Angelis

2011

Nonlinear Analysis: Real World Applications

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This paper deals with the derivation of macroscopic equations for a class of equations modelling complex multicellular systems. Cellular interactions generate both modification of biological functions and proliferative/destructive events. The asymptotic analysis refers to the derivation of hyperbolic models focused on the influence of existence of a global equilibrium solution. The asymptotic analysis shows how the macroscopic tissue behavior can be described from the underlying cellular description.

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“…For those bacteria that tumble at r, the second and the third terms in (1) represent such contributions of bacterial tumbling out of and into e, respectively. The small circle in the integral symbol denotes that the integration is performed over the enclosed spherical surface constructed by the swimming vector e. Equation (1) was characterized as the "velocity-jump" process for chemotactic bacteria by Othmer and coworkers [29,30]; but it is in fact derived from the more complicated Boltzmann equation [13] after certain simplifications. The constant swimming speed is the so-called one-speed approximation in the Boltzmann theory.…”

Section: Cell Balance Equationmentioning

confidence: 99%

Digital Object Identifier (DOI)

Celko¹

2010

Joe Celko's Data, Measurements and Standards in SQL

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The Diffusion Limit of Transport Equations II: Chemotaxis Equations

Cited by 328 publications

References 42 publications

A user’s guide to PDE models for chemotaxis

A user’s guide to PDE models for chemotaxis

From kinetic models of multicellular growing systems to macroscopic biological tissue models

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