Stability analysis of switching hyperbolic systems: the example of SMB chromatography

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 behavior of solutions and to discuss the stability and the accuracy of our approximation.1991 Mathematics Subject Classification. 65M06, 35L50, 92B05, 92C17, 92C42.. IntroductionThe movement of bacteria, cells or other microorganisms under the effect of a chemical stimulus, represented by a chemoattractant, has been widely studied in mathematics in the last two decades, see [21,23,26], and numerous models involving partial differential equations have been proposed. The basic unknowns in these chemotactic models are the density of individuals and the concentrations of some chemical attractants. One of the most considered models is the Patlak-Keller-Segel system [19], where the evolution of the density of cells is described by a parabolic equation, and the concentration of a chemoattractant is generally given by a parabolic or elliptic equation, depending on the different regimes to be described and on authors' choices. The behavior of this system is quite well known now: in the one-dimensional case, the solution is always global in time, while in two and more dimensions the solutions exist globally in time or blow up according to the size of the initial data. However, a drawback of this model is that the diffusion leads to a fast dissipation or an explosive behavior, and prevents us to observe intermediate organized structures, like aggregation patterns.By contrast, models based on hyperbolic/kinetic equations for the evolution of the density of individuals, are characterized by a finite speed of propagation and have registered a growing consideration in the last few years [5-7, 15, 26]. In such models, the population is divided in compartments depending on the velocity of propagation of individuals, giving raise to kinetic type equations, either with continuous or discrete velocities.

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Stability analysis of switching hyperbolic systems: the example of SMB chromatography

Cited by 2 publications

References 15 publications

On the finite-time boundary dissipative for a class of hyperbolic systems. The networks example

On the finite-time boundary dissipative for a class of hyperbolic systems. The networks example

A hyperbolic model of chemotaxis on a network: a numerical study

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