On a chemotaxis model with saturated chemotactic flux

Chertock, Alina; Kurganov, Alexander; Wang, Xuefeng; Wu, Yaqi

doi:10.3934/krm.2012.5.51

Cited by 101 publications

(96 citation statements)

References 28 publications

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“…However, this may affect the positivitypreserving property of the resulting fully discrete scheme. In Chertock & Kurganov (2008) and Chertock et al (2010), we have shown that if one uses a strong-stabilitypreserving (SSP) method (either a Runge-Kutta or a multistep one) or an IMEX-SSP…”

Section: Appendix a Derivation Of The Numerical Methodsmentioning

confidence: 99%

Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach

et al. 2012

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Aquatic bacteria like Bacillus subtilis are heavier than water yet they are able to swim up an oxygen gradient and concentrate in a layer below the water surface, which will undergo Rayleigh-Taylor-type instabilities for sufficiently high concentrations. In the literature, a simplified chemotaxis-fluid system has been proposed as a model for bioconvection in modestly diluted cell suspensions. It couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier-Stokes equations subject to a gravitational force proportional to the relative surplus of the cell density compared to the water density. In this paper, we derive a high-resolution vorticity-based hybrid finite-volume finite-difference scheme, which allows us to investigate the nonlinear dynamics of a two-dimensional chemotaxis-fluid system with boundary conditions matching an experiment of Hillesdon et al. (Bull. Math. Biol., vol. 57, 1995, pp. 299-344). We present selected numerical examples, which illustrate (i) the formation of sinking plumes, (ii) the possible merging of neighbouring plumes and (iii) the convergence towards numerically stable stationary plumes. The examples with stable stationary plumes show how the surface-directed oxytaxis continuously feeds cells into a high-concentration layer near the surface, from where the fluid flow (recurring upwards in the space between the plumes) transports the cells into the plumes, where then gravity makes the cells sink and constitutes the driving force in maintaining the fluid convection and, thus, in shaping the plumes into (numerically) stable stationary states. Our numerical method is fully capable of solving the coupled chemotaxis-fluid system and enabling a full exploration of its dynamics, which cannot be done in a linearised framework.

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Section: Appendix a Derivation Of The Numerical Methodsmentioning

confidence: 99%

Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach

et al. 2012

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“…Motivated by numerical and modeling issues, the question on how blow up of cells can be avoided has been studied intensively in the last years. Modifying the chemotactic sensitivity, for instance, voluming-filling effects [31,6], has been suggested, to allow for degenerate cell diffusion [4], to include growth-death effects [24,29,38,47,52] or to assume saturated chemotactic fluxes [8] or to incorporate a nonlocal sampling radius [14,51]. Another idea is to introduce additional cell diffusion into the equation for chemical concentration [16,7], where further references on preventing blow-ups can be found.…”

Section: Introduction and Outline Of Main Resultsmentioning

confidence: 99%

“…The verifications of items (i) through (iii) are plain calculations. For property (iv) we adapt the arguments in [8,44] for 1-D domain case. To see this, we decompose (4.10) as…”

Section: Nonexistence and Existence Of Pattern Solutionsmentioning

confidence: 99%

On a class of Keller–Segel chemotaxis systems with cross-diffusion

Xiang

2015

Journal of Differential Equations

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In this paper, we study a class of Keller-Segel chemotaxis systems with cross-diffusion. By using the entropy dissipation method and assuming mainly the chemotactic sensitivity separates the cell density and the chemical signal, we first establish the existence of global weak solutions with the effects of cross diffusion included in ≤ 3-D. Then we show there is a critical cross diffusion rate δ c such that no patterns may be expected for δ ≥ δ c , while patterns are formed for δ < δ c and their stability is also derived. In particular, in 1-D, patterns are always formed whenever δ < δ c and the chemotactic coefficient is larger than an expressible bifurcation value, and there is another critical cross diffusion rate δ c < δ c such that cells with cross-diffusion rate δ ∈ (δ c , δ c ) are stable, while, for cells with δ < δ c to be stable, their degradation rate must be less than a threshold value. Hence, in some sense, cross-diffusion is harmful to enable pattern formation, while it is helpful to stabilize the cells once patterns are formed. Finally, we show that the cross diffusion plays a role in regularizing the cell aggregation phenomenon for large chemotactic coefficient. Our results provide global dynamics and insights on how the biological parameters, especially, the cross diffusion, affect pattern formations.

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“…However, we want to add that very few mathematical techniques are available in rigorously characterizing these alternatives. The methodology recently developed by Wang et al in [Chertock et al 2012] and [Xu & Wang, 2012] turns out to be very useful and user friendly in dealing with certain one-dimensional chemotaxis systems with special structures. In loose terms, the ideas there can be summarized as follows.…”

Section: Bifurcation Of Steady Statesmentioning

confidence: 99%

“…Though the boundary of the square is not smooth at the corners, one can interpret the boundary condition in the weak sense via Green's identity-see the discussions in [Chertock, et al, 2012].…”

Section: Numerical Simulations In 2dmentioning

confidence: 99%

Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth

Jin

Wang

Zhang

2016

Int. J. Bifurcation Chaos

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In this paper we investigate pattern formation in Keller-Segel chemotaxis models over a multi-dimensional bounded domain subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as chemoattraction rate χ increases. Then using Crandall-Rabinowitz local theory with χ being the bifurcation parameter, we obtain the existence of nonhomogeneous steady states of the system which bifurcate from this homogeneous steady state. Stability of the bifurcating solutions is also established through rigorous and detailed calculations. Our results provide a selection mechanism of stable wavemode which states that the only stable bifurcation branch must have a wavemode number that minimizes the bifurcation value. Finally we perform extensive numerical simulations on the formation of stable steady states with striking structures such as boundary spikes, interior spikes, stripes, etc. These nontrivial patterns can model cellular aggregation that develop through chemotactic movements in biological systems.

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On a chemotaxis model with saturated chemotactic flux

Cited by 101 publications

References 28 publications

Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach

Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach

On a class of Keller–Segel chemotaxis systems with cross-diffusion

Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth

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