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

Abstract

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.

Numerical High-Field Limits in Two-Stream Kinetic Models and 1D Aggregation Equations

“…Section 3 is devoted to a reminder of the strategy of [21] to derive WB scheme when the potential (∂ x S for (1.1) or E for (VPFP)) is assumed to be given. In §4, we develop our new strategy to build a WB and AP numerical scheme for the coupled models (see also [5,9,14]), the AP property of which is explained in §5. The main drawback of this strategy may be its CPU cost as it involves an iterative resolution of a nonlinear algebraic system (see Appendix A).…”

Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network

“…Recently an interest in these mathematical models evolving on networks is arising, due to their applications in the study of biological phenomena and traffic flows, both in parabolic cases [1,5,22] and in hyperbolic ones [9,6,27,12,2].…”

“…Finally, in [2] the authors introduce a numerical scheme to approximate the solutions to the problem (2.1); in that paper transmission conditions are set for the Riemann invariants of the hyperbolic part of the system, w ± i = 1 2 (u i ± v i ), and are equivalent to our ones for some choices of the transmission coefficients.…”

Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

“…In [14,29] coupling conditions for scalar hyperbolic equations on networks are discussed and investigated. [1,20,41] treat the wave equation and general nonlinear hyperbolic systems are considered in [9,3,2,15,6]. We finally note, that, for example, for hyperbolic systems on networks there are still many unsolved problems, like finding suitable coupling conditions without restricting to subsonic situations.…”