Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic differential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. Directional cell motion in re...

Abstract. We investigate nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Given any general perturbation of magnitude δ, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of ln 1 δ . Our result can be interpreted as a rigourous mathematical ...

This article summarizes various aspects and results for some general formulations of the classical chemotaxis models also known as Keller-Segel models. It is intended as a survey of results for the most common formulation of this classical model for positive chemotactical movement and offers possible generalizations of these results to more universal models. Furthermore it collects open questio...

Chemotaxis, the orientedmovement of cells in response to ambient chemical gradients, is a prominent feature in the organization ofmany biological populations. Since the pioneer work of Keller and Segel [11] to propose mathematical models for chemotaxis, there has been great interest in modeling chemotaxis and in the mathematical analysis of systems like the Keller-Segel model. In this paper, mo...

Traveling wave (band) behavior driven by chemotaxis was observed experimentally by Adler and was modeled by Keller and Segel. For a quasilinear hyperbolic parabolic system that arises as a non-di®usive limit of the Keller Segel model with nonlinear kinetics, we establish the existence and nonlinear stability of traveling wave solutions with large amplitudes. The numerical simulations are perfor...

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on bounded domain Ω. is simplified version of chemotaxis model introduced by Keller and Segel in [14]. With aid priori L∞-estimates (for solutions system) built up this paper, we treat parabolic equation as linear equation, then use frequency function method localization technique to buil...

In the Langevin or Ornstein-Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein-Kramers or s...

Abstract We consider the singular limit problem for Cauchy of (Patlak–) Keller–Segel system parabolic-parabolic type. The is considered in uniformly local Lebesgue spaces and as relaxation parameter $$\tau $$ τ goes to infinity, solution equation converges a drift-diffusion strong topology. For proof, we foll...