### A class of kinetic models for chemotaxis with threshold to prevent overcrowding.

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We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than ${\left(8\pi \right)}^{2}$, whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known $8\pi $ problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel system...

In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the...

In this paper we derive a model describing the dynamics of HIV-1 infection in tissue culture where the infection spreads directly from infected cells to healthy cells trough cell-to-cell contact. We assume that the infection rate between healthy and infected cells is a saturating function of cell concentration. Our analysis shows that if the basic reproduction number does not exceed unity then infected cells are cleared and the disease dies out. Otherwise, the infection is persistent with the existence...

Vasculogenesis and angiogenesis are two different mechanisms for blood vessel formation. Angiogenesis occurs when new vessels sprout from pre-existing vasculature in response to external chemical stimuli. Vasculogenesis occurs via the reorganization of randomly distributed cells into a blood vessel network. Experimental models of vasculogenesis have suggested that the cells exert traction forces onto the extracellular matrix and that these forces may play an important role in the network forming...

This note contains some remarks on the paper of Y. Naito concerning the parabolic system of chemotaxis and published in this volume.

A model of tumor growth in a spatial environment is analyzed. The model includes proliferating and quiescent compartments of tumor cells indexed by successively mutated cell phenotypes of increasingly proliferative aggressiveness. The model incorporates spatial dependence due to both random motility and directed movement haptotaxis. The model structures tumor cells by both cell age and cell size. The model consists of a system of nonlinear partial differential equations for the compartments of...

Proteus mirabilis are bacteria that make strikingly regular spatial-temporal patterns on agar surfaces. In this paper we investigate a mathematical model that has been shown to display these structures when solved numerically. The model consists of an ordinary differential equation coupled with a partial differential equation involving a first-order hyperbolic aging term together with nonlinear degenerate diffusion. The system is shown to admit global weak solutions.

In this article we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the assumptions of radial symmetry, it is known that this system has at least two stable mechanisms of singularity formation (see e.g. M. P. Brenner et al. 1999, Nonlinearity 12, 1071-1098); one type is self-similar, and may be viewed as a trade-off between diffusion...

We consider a nonlinear parabolic system modelling chemotaxis ${u}_{t}=\nabla \xb7(\nabla u-u\nabla v)$, ${v}_{t}=\Delta v+u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

In this paper, we will consider blowup solutions to the so called Keller-Segel system and its simplified form. The Keller-Segel system was introduced to describe how cellular slime molds aggregate, owing to the motion of the cells toward a higher concentration of a chemical substance produced by themselves. We will describe a common conjecture in connection with blowup solutions to the Keller-Segel system, and some results for solutions to simplified versions of the Keller-Segel system, giving the...

A class of nonlinear viscous transport equations describing aggregation phenomena in biology is considered. General conditions on an interaction potential are obtained which lead either to the existence or to the nonexistence of global-in-time solutions.