151 research outputs found
A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis
We consider two models which were both designed to describe the movement of
eukaryotic cells responding to chemical signals. Besides a common standard
parabolic equation for the diffusion of a chemoattractant, like chemokines or
growth factors, the two models differ for the equations describing the movement
of cells. The first model is based on a quasilinear hyperbolic system with
damping, the other one on a degenerate parabolic equation. The two models have
the same stationary solutions, which may contain some regions with vacuum. We
first explain in details how to discretize the quasilinear hyperbolic system
through an upwinding technique, which uses an adapted reconstruction, which is
able to deal with the transitions to vacuum. Then we concentrate on the
analysis of asymptotic preserving properties of the scheme towards a
discretization of the parabolic equation, obtained in the large time and large
damping limit, in order to present a numerical comparison between the
asymptotic behavior of these two models. Finally we perform an accurate
numerical comparison of the two models in the time asymptotic regime, which
shows that the respective solutions have a quite different behavior for large
times.Comment: One sentence modified at the end of Section 4, p. 1
Nonlinear Instability for a Volume-Filling Chemotaxis Model with Logistic Growth
This paper deals with a Neumann boundary value problem for a volume-filling chemotaxis model with logistic growth in a d-dimensional box Td=(0,Ï)dââ(d=1,2,3). It is proved that given any general perturbation of magnitude ÎŽ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order lnâĄ(1/ÎŽ). Each initial perturbation certainly can behave drastically different from another, which gives rise to the richness of patterns
Turing Instabilities are Not Enough to Ensure Pattern Formation
Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turingâs reactionâdiffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation
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