Dirk Horstmann

Generalizing the Keller–Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species

In this paper we extend the famous Keller–Segel model for the chemotactic movement of motile species to some multi-species chemotaxis equations. The presented multi-species chemotaxis models are more general than those introduced so far and also include some interaction effects that have not been studied before. For example, we consider multi-species chemotaxis models with attraction and repulsion between interacting motile species. For some of the presented new models we give sufficient conditions for the existence of Lyapunov functionals. These new results are related to those of Wolansky (Scent and sensitivity: equilibria and stability of chemotactic systems in the absence of conflicts, preprint, 1998; Eur. J. Appl. Math. 13:641–661, 2002). Furthermore, a linear stability analysis is performed for uniform steady states, and results for the corresponding steady state problems are established. These include existence and nonexistence results for non-constant steady state solutions in some special cases.

A Constructive Approach to Traveling Waves in Chemotaxis

Abstract In this paper we study the existence of one-dimensional and multidimensional traveling wave solutions for general chemotaxis or so-called Keller-Segel models without reproduction of the chemotactic species. We present a constructive approach to give modelers a choice of chemotactic sensitivity functionals, production, and degradation terms for the chemical signal at hand. The main aim is to understand the type of functionals and the interplay between them that are needed for the traveling wave and pulse patterns to occur.

The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results

Abstract. Looking at the nonsymmetric case of a reaction-diffusion model known as the Keller-Segel model, we summarize known facts concerning (global in time) existence and prove new blowup results for solutions of this system of two strongly coupled parabolic partial differential equations. We show in Section 4, Theorem 4, that if the solution blows up under a condition on the initial data, blowup takes place at the boundary of a smooth domain

A positivity-preserving finite element method for chemotaxis problems in 3D

\Omega\subset{\mathbf {R}}^2

Uniqueness and symmetry of equilibria in a chemotaxis model

. Using variational techniques we prove in Section 5 the existence of nontrivial stationary solutions in a special case of the system.

Do some chemotaxis-growth models possess Lyapunov functionals?

We present an implicit finite element method for a class of chemotaxis models in three spatial dimensions. The proposed algorithm is designed to maintain mass conservation and to guarantee positivity of the cell density. To enforce the discrete maximum principle, the standard Galerkin discretization is constrained using a local extremum diminishing flux limiter. To demonstrate the efficiency and robustness of this approach, we solve blow-up problems in a 3D chemostat domain. To give a flavor of more complex and realistic chemotactic applications, we investigate the pattern dynamics and aggregating behavior of the bacteria Escherichia coli and Salmonella typhimurium. The obtained numerical results are in good qualitative agreement with theoretical studies and experimental data reported in the literature.

Die trigonometrischen Funktionen

Abstract We consider in a disc of a class of parameter-dependent, nonlocal elliptic boundary value problems that describes the steady states of some chemotaxis systems. If the appearing parameter is less than an explicit critical value, we establish several uniqueness results for solutions that are invariant under a group of rotations. Furthermore, we discuss the associated consequences for the time asymptotic behavior of the solutions to the corresponding time dependent chemotaxis systems. Our results also provide optimal constants in some Moser–Trudinger type inequalities.

Einstieg und grafische Darstellungen von Messdaten

Abstract In spirit of a result by W. Alt from 1980 we give some sufficient criteria that guarantee the existence of Lyapunov functionals for parabolic cross-diffusion models including chemotaxis-growth models with non-diffusive chemotactic signals (resp. with non-diffusive memory).

Rechnen mit Ungleichungen

Neben der Einfuhrung der trigonometrischen Funktionen und den mit ihnen verbundenen Rechengesetze wird ein Exkurs zur Zahl π gegeben. Das Kapitel endet mit der Darstellung der trigonometrischen Funktionen mithilfe der Exponentialfunktion und der komplexen Zahl \(\mathrm{i}\).

Weitere Anmerkungen zur Fehlerrechnung

In diesem einleitenden Kapitel werden verschiedene Moglichkeiten vorgestellt, wie man bei einem Experiment oder einer Befragung erhobene Stichprobendaten grafisch darstellen kann. Diese Moglichkeiten umfassen Flachen- und Kreisdiagramme sowie die Darstellung der Daten mittels eines Boxplots. Des Weiteren werden unterschiedliche Mittelwertbegriffe sowie die Begriffe der Stichprobenvarianz, der Standardabweichung und der des α-Qunatils einer Datenreihe eingefuhrt.