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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 10

Special issue on Chemotaxis

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Preface
Thomas Hillen and Michael Winkler
2013, 18(10): i-ii doi: 10.3934/dcdsb.2013.18.10i +[Abstract](282) +[PDF](94.8KB)
Abstract:
Self-organization of micro organisms through the oriented movement of individuals along chemical gradients has caught researchers' imagination and interest for a long time. In fact, the process of aggregation of cells is a first step in the transition from individuals to a collective. Chemotaxis has been identified to play an important role in areas as diverse as ecological species (e.g. slime molds) and bacteria (E. coli), embryogenesis, immune response, wound healing, and cancer development. The first mathematical model for chemotaxis was introduced by Patlak (1953) and became later known as the Keller and Segel (1970) system of equations. The model has become a hot topic not only for the description of biological phenomena, but also mathematically. Sophisticated mathematical analysis has developed and it is the purpose of this special issue of DCDS-B to showcase some of the interesting and challenging mathematical questions that currently arise in the analysis of chemotaxis models.

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Trudinger-Moser type inequality for radially symmetric functions in a ring and applications to Keller-Segel in a ring
Tomasz Cieślak
2013, 18(10): 2505-2512 doi: 10.3934/dcdsb.2013.18.2505 +[Abstract](385) +[PDF](337.3KB)
Abstract:
We prove that for radially symmetric functions in a ring $\Omega = ${$ x \in \mathbb{R}^n, n \geq 2 : r \leq |x| \leq R $} a special type of Trudinger-Moser-like inequality holds. Next we show how to infer from it a lack of blowup of radially symmetric solutions to a Keller-Segel system in $\Omega$.
Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model
Thomas Hillen, Jeffery Zielinski and Kevin J. Painter
2013, 18(10): 2513-2536 doi: 10.3934/dcdsb.2013.18.2513 +[Abstract](350) +[PDF](7301.1KB)
Abstract:
In a recent study (K.J. Painter and T. Hillen, Spatio-temporal chaos in a chemotaxis model, Physica D, 240 (4), 363-375, 2011) a model for chemotaxis incorporating logistic growth was investigated for its pattern formation properties. In particular, a variety of complex spatio-temporal patterning was found, including stationary, periodic and chaotic. Complicated dynamics appear to arise through a sequence of ``merging and emerging'' events: the merging of two neighbouring aggregates or the emergence of a new aggregate in an open space. In this paper we focus on a time-discrete dynamical system motivated by these dynamics, which we call the merging-emerging system (MES). We introduce this new class of set-valued dynamical systems and analyse its capacity to generate similar ``pattern formation'' dynamics. The MES shows remarkably close correspondence with patterning in the logistic chemotaxis model, strengthening our assertion that the characteristic length scales of merging and emerging are responsible for the observed dynamics. Furthermore, the MES describes a novel class of pattern-forming discrete dynamical systems worthy of study in its own right.
Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$
Sachiko Ishida, Yusuke Maeda and Tomomi Yokota
2013, 18(10): 2537-2568 doi: 10.3934/dcdsb.2013.18.2537 +[Abstract](393) +[PDF](582.9KB)
Abstract:
This paper gives the gradient estimate for solutions to the quasilinear non-degenerate parabolic-parabolic Keller-Segel system (KS) on the whole space $\mathbb{R}^N$. The gradient estimate for (KS) on bounded domains is known as an application of Amann's existence theory in [1]. However, in the whole space case it seems necessary to derive the gradient estimate directly. The key to the proof is a modified Bernstein's method. The result is useful to obtain the whole space version of the global existence result by Tao-Winkler [13] except for the boundedness.
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Sachiko Ishida and Tomomi Yokota
2013, 18(10): 2569-2596 doi: 10.3934/dcdsb.2013.18.2569 +[Abstract](484) +[PDF](646.1KB)
Abstract:
This paper gives a blow-up result for the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. It is known that the system has a global solvability in the case where $q < m + \frac{2}{N}$ ($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity) without any restriction on the size of initial data, and where $q \geq m + \frac{2}{N}$ and the initial data are ``small''. However, there is no result when $q \geq m + \frac{2}{N}$ and the initial data are ``large''. This paper discusses such case and shows that there exist blow-up energy solutions from initial data having large negative energy.
Pattern formation of the attraction-repulsion Keller-Segel system
Ping Liu, Junping Shi and Zhi-An Wang
2013, 18(10): 2597-2625 doi: 10.3934/dcdsb.2013.18.2597 +[Abstract](528) +[PDF](873.3KB)
Abstract:
In this paper, the pattern formation of the attraction-repulsion Keller-Segel (ARKS) system is studied analytically and numerically. By the Hopf bifurcation theorem as well as the local and global bifurcation theorem, we rigorously establish the existence of time-periodic patterns and steady state patterns for the ARKS model in the full parameter regimes, which are identified by a linear stability analysis. We also show that when the chemotactic attraction is strong, a spiky steady state pattern can develop. Explicit time-periodic rippling wave patterns and spiky steady state patterns are obtained numerically by carefully selecting parameter values based on our theoretical results. The study in the paper asserts that chemotactic competitive interaction between attraction and repulsion can produce periodic patterns which are impossible for the chemotaxis model with a single chemical (either chemo-attractant or chemo-repellent).
Global solutions and exponential attractors of a parabolic-parabolic system for chemotaxis with subquadratic degradation
Etsushi Nakaguchi and Koichi Osaki
2013, 18(10): 2627-2646 doi: 10.3934/dcdsb.2013.18.2627 +[Abstract](452) +[PDF](447.7KB)
Abstract:
We construct the global bounded solutions and the attractors of a parabolic-parabolic chemotaxis-growth system in two- and three-dimensional smooth bounded domains. We derive new $L_p$ and $H^2$ uniform estimates for these solutions. We then construct the absorbing sets and the global attractors for the dynamical systems generated by the solutions. We also show the existence of exponential attractors by applying the existence theorem of Eden-Foias-Nicolaenko-Temam.
Well-posedness for a model of individual clustering
Elissar Nasreddine
2013, 18(10): 2647-2668 doi: 10.3934/dcdsb.2013.18.2647 +[Abstract](316) +[PDF](453.1KB)
Abstract:
We study the well-posedness of a model of individual clustering. Given $p>N\geq 1$ and an initial condition in $W^{1,p}(\Omega)$, the local existence and uniqueness of a strong solution is proved. We next consider two specific reproduction rates and show global existence if $N=1$, as well as, the convergence to steady states for one of these rates.
On a comparison method to reaction-diffusion systems and its applications to chemotaxis
Mihaela Negreanu and J. Ignacio Tello
2013, 18(10): 2669-2688 doi: 10.3934/dcdsb.2013.18.2669 +[Abstract](361) +[PDF](421.7KB)
Abstract:
In this paper we consider a general system of reaction-diffusion equations and introduce a comparison method to obtain qualitative properties of its solutions. The comparison method is applied to study the stability of homogeneous steady states and the asymptotic behavior of the solutions of different systems with a chemotactic term. The theoretical results obtained are slightly modified to be applied to the problems where the systems are coupled in the differentiated terms and / or contain nonlocal terms. We obtain results concerning the global stability of the steady states by comparison with solutions of Ordinary Differential Equations.
Numerical simulation of chemotaxis models on stationary surfaces
Andriy Sokolov, Robert Strehl and Stefan Turek
2013, 18(10): 2689-2704 doi: 10.3934/dcdsb.2013.18.2689 +[Abstract](345) +[PDF](1824.7KB)
Abstract:
In this paper we present an implicit finite element method for a class of chemotaxis models, where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative. Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations. The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models. The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity
Youshan Tao
2013, 18(10): 2705-2722 doi: 10.3934/dcdsb.2013.18.2705 +[Abstract](460) +[PDF](451.6KB)
Abstract:
This paper deals with the repulsion chemotaxis system $$ \left\{ \begin{array}{ll} u_t=\Delta u +\nabla \cdot (f(u)\nabla v), & x\in\Omega, \ t>0, \\ v_t=\Delta v +u-v, & x\in\Omega, \ t>0, \end{array} \right. $$ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $\Omega\subset\mathbb{R}^n$ with $n\ge 3$, where $f(u)$ is the density-dependent chemotactic sensitivity function satisfying $$ f \in C^2([0, \infty)),      f(0)=0, 0 < f(u) \le K(u+1)^{\alpha}          for     all     u > 0 $$ with some $K>0$ and $\alpha>0$.
    It is proved that under the assumptions that $0\not\equiv u_0\in C^0(\bar{\Omega})$ and $v_0\in C^1(\bar{\Omega})$ are nonnegative and that $\alpha<\frac{4}{n+2}$, the classical solutions to the above system are uniformly-in-time bounded. Moreover, it is shown that for any given $(u_0, v_0)$, the corresponding solution $(u, v)$ converges to $(\bar{u}_0, \bar{u}_0)$ as time goes to infinity, where $\bar{u}_0 :=\frac{1}{\Omega} f_{\Omega} u_0$.

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