Aggregating pattern dynamics in a chemotaxis model including growth

We study a parabolic-elliptic chemotactic system describing the evolution of a population's density "u" and a chemoattractant's concentration "v". The system considers a nonconstant chemotactic sensitivity given by "/(JV -u)", for JV > 0, and a source term of logistic type "Xu(\ -u)". The existence of global bounded classical solutions is proved for any / > 0, JV > 0 and X > 0. By using a comparison argument we analyze the stability of the constant steady state u = \,v = 1, for a range of parameters.-For JV > 1 and NX > 2/, any positive and bounded solution converges to the steady state. -For JV < 1 the steady state is locally asymptotically stable and for /JV < X, the steady state is globally asymptotically stable.

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“…n > 1. Similar result for the parabolic-elliptic problem can be found in Mimura and Tsujikawa [10] (see also [11] and [15]). …”

Section: Introductionsupporting

confidence: 69%

On a parabolic–elliptic chemotactic system with non-constant chemotactic sensitivity

Negreanu

Tello

2013

Nonlinear Analysis: Theory, Methods & Applications

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“…The case for a cell kinetics term describing an Allee effect ( f (u) = u(1−u)(u −α)) has been studied in a series of papers [23,32,48,65], where it is assumed that the kinetics term acts on a faster time scale compared to that of cell movement. Depending on the choice of scaling, solutions in the form of viscous shock solutions can be found.…”

Section: (M8) the Cell Kinetics Modelmentioning

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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“…The system (E) was presented by Mimura and Tsujikawa [6] for the pattern dynamics of aggregating regions of bacteria. The unknown functions uðx; tÞ and vðx; tÞ are the population density of bacteria and the concentration of a chemical substance at the position x and time t, respectively.…”

Section: Introductionmentioning

confidence: 99%