Angela Stevens

Aggregation, blowup, and collapse: the abc's of taxis in reinforced random walks

In many biological systems, movement of an organism occurs in response to a diffusible or otherwise transported signal, and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by Patlak [Bull. of Math. Biophys., 15 (1953), pp. 311--338]. However, other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages. In these systems, one example of which is the myxobacteria, the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. Davis [Probab. Theory Related Fields, 84 (1990), pp. 203--229] has studied this question for a certain class of random walks, and here we extend this analysis to the continuum limit of such walks. We first derive several general classes of partial differential equations that depend on how the movement rules are...

The Derivation of Chemotaxis Equations as Limit Dynamics of Moderately Interacting Stochastic Many-Particle Systems

The chemotaxis equations are a well-known system of partial differential equations describing aggregation phenomena in biology. In this paper they are rigorously derived from an interacting stochastic many-particle system, where the interaction between the particles is rescaled in a moderate way as population size tends to infinity. The novelty of this result is that in all previous applications of this kind of limiting procedure, the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case. New techniques which deal with this difficulty are presented.

VARIATIONAL PRINCIPLES FOR PROPAGATION SPEEDS IN INHOMOGENEOUS MEDIA

An important problem in reactive flows is how to estimate the speed of front propagation, especially when inhomogeneities are present. Here we prove a variational characterization of the front speed for reaction-diffusion-advectionequations in periodically varying heterogeneous media. This formulation makes it possible to calculate sharp estimates for the speed explicitly. The method can be applied to any problem obeying a maximum principle. Three examples will be analyzed in detail: a shear flow problem, a problem with rapidly oscillating coefficients, and a discretized diffusion problem. In all cases the effects of the inhomogeneous medium on the speed are discussed in comparison to the homogeneous problem. For the shear flow problem, enhancement of the speed results.

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.

A STOCHASTIC CELLULAR AUTOMATON MODELING GLIDING AND AGGREGATION OF MYXOBACTERIA

The myxobacteria are ubiquitous soil bacteria which aggregate under starvation conditions and build fruiting bodies to survive. Until recently the mechanisms of their social gliding, aggregation, and fruiting body formation have not been well understood. In this paper a stochastic cellular automaton model is presented to describe and provide an understanding of how the bacteria manage to build higher organized structures. In the automaton myxobacteria move on a square grid with periodic boundary conditions. They respond to the four nearest neighbors of their frontal cell poles, mainly through two factors: slime and a diffusing chemoattractant; both are produced by the bacteria themselves. Simulations show the interdependence of the different mechanisms which finally cause aggregation. A simplified version of this model is formally approximated by a system of partial differential equations, a chemotaxis system. A related publication [SIAM J. Appl. Math., 61 (2000), pp. 183--212] deals with the first rigoro...

Global solutions of nonlinear transport equations for chemosensitive movement

A widespread phenomenon in moving microorganisms and cells is their ability to reorient themselves depending on changes of concentrations of certain chemical signals. In this paper we discuss kinetic models for chemosensitive movement, which also takes into account evaluations of gradient fields of chemical stimuli which subsequently influence the motion of the respective microbiological species. The basic type of model was discussed by Alt [J. Math. Biol., 9 (1980), pp. 147--177], [J. Reine Angew. Math., 322 (1981), pp. 15--41] and by Othmer, Dunbar, and Alt [J. Math. Biol., 26 (1988), pp. 263--298]. Chalub et al. rigorously proved that, in three dimensions, these kinds of kinetic models lead to the classical Keller--Segel model as its drift-diffusion limit when the equation for the chemo-attractant is of elliptic type [Monatsh. Math.}, 142 (2004), pp. 123--141], [On the Derivation of Drift-Diffusion Model for Chemotaxis from Kinetic Equations, ANUM preprint 14/02, Vienna Technical University, 2002]. In ...

Mass-Selection in Alignment Models with Non-Deterministic Effects

In this paper we consider a kinetic model for alignment of cells or filaments with probabilistic turning. For this equation existence of solutions is known, see [6]. To understand its qualitative behavior, especially with respect to the selection of orientations and mass distributions for long times, the model is approximated by a diffusion equation in the limit of small deviations of the interactions between the cell bundles. For this new equation existence of steady states is shown. In contrast to the kinetic equation discussed in [6] with deterministic turning, where local stability of two opposite orientations was shown but no selection of mass could be observed, for the new approximating problem with probabilistic turning additionally mass selection takes place. In the limit of small diffusion, steady states can only be constructed, if the aligning masses are either equal or the total mass is concentrated in one direction. By numerical simulations we tested stability of these steady states and for situations with 4 symmetrically placed smooth distributions of alignment. Convergence of the numerical code was proved. The simulations suggest, that only the 2- and the 1-peak steady states can be stable, whereas the 4 peak steady state is always unstable. We conjecture that the noise in the system is responsible for this final selection of masses. There exist other steady states with an arbitrary number of aligned bundles of cells or filaments, but we suspect that, as numerically shown for the 4 peak case, these multi-peak states are all unstable.

Topological Flow Structures in a Mathematical Model for Rotation-Mediated Cell Aggregation

In this paper we applyvector field topology methods to amathematical model for the fluid dynamics of reaggregation processes in tissue engineering. The experimental background are dispersed embryonic retinal cells, which reaggregate in a rotation culture on a gyratory shaker, according to defined rotation and culture conditions. Under optimal conditions, finally complex 3D spheres result. In order to optimize high throughput drug testing systems of biological cell and tissue models, a major aim is to understand the role which the fluid dynamics plays in this process. To allow for a mathematical analysis, an experimental model system was set up, using micro-beads instead of spheres within the culture dish. The qualitative behavior of this artificial model was monitored in time by using a camera. For this experimental setup amathematical model for the bead-fluid dynamics was derived, analyzed and simulated. The simulations showed that the beads form distinctive clusters in a layer close to the bottom of the petri dish. To analyze these patterns further, we perform a topological analysis of thevelocity field within this layer of the fluid. We find that traditional two-dimensional visualization techniques like path lines, streak lines and currenttime-dependent topology approaches are not able to answer the posed questions, for example they do not allow to find the location of clusters. We discuss the problems of these techniques and develop a new approach that measures thedensity of advected particles in the flow to find the moving point of particleaggregation. Using thedensity field the path of the movingaggregation point is extracted.

Qualitative Behavior of a Keller–Segel Model with Non-Diffusive Memory

In this paper a one-dimensional Keller–Segel model with a logarithmic chemotactic-sensitivity and a non-diffusing chemical is classified with respect to its long time behavior. The strength of production of the non-diffusive chemical has a strong influence on the qualitative behavior of the system concerning existence of global solutions or Dirac-mass formation. Further, the initial data play a crucial role.

Sorting Phenomena in a Mathematical Model For Two Mutually Attracting/Repelling Species

Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this paper we extend the analysis to a system for two species interacting with each other according to different inner- and intra-species attractions. Under suitable conditions on this self- and crosswise attraction an interesting effect can be observed, namely phase separation into neighbouring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is simply connected. Preliminary results indicate the existence of phase separation, i.e. spatial sorting of the different species. A detailed analysis in one spatial dimension follows. The existence and shape of segregated stationary solutions is shown via the Krein-Rutman theorem. Moreover, for small repulsion/nonlinear diffusion, also uniqueness of these stationary states is proved.