Stability of solutions of chemotaxis equations in reinforced random walks

Abstract

In this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation. The system arises in chemotaxis, a process whereby living organisms respond to chemical substance by moving toward higher, or lower, concentrations of the chemical substance, or by aggregating or dispersing. We prove that stationary solutions of the system are asymptotically stable.

Numerical resolution of a reinforced random walk model arising in haptotaxis

“…This kind of equations, containing haptotaxis terms, arise for example in modeling cáncer process, as angiogenesis, see for instance [1,12]. These problems also present mathematical challenges whereby several authors have been interested, as the literature shows, see for example [9,13,16], and references therein.…”

“…From the mathematical point of view, in [9], it is proved that any stationary state (u*, v*) of 1,2 is asymptotically stable provided:…”

“…This paper is involved with the numerical resolutionofa particular case of system 1,2, considered in [9] as an example. To precise, we shall consider the following parabolic-ODE system posed in a bounded domain íl c R 2 ,…”

“…The method here employed follows the ideas of [4]. In Section 3, we present some numerical results which ¡Ilústrate the stabilization of solutions when (9), and therefore (4), is satisfied. We also show the results concerning two cases for which (9) is not satisfied (neither (4)).…”

Mathematical analysis and stability of a chemotaxis model with logistic term

“…Friedman and Tello [4] study the models proposed in Levine and Sleeman [9] and Othmer and Stevens [12] under suitable conditions in chemotactic coefficient and production terms.…”