Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach

In this paper, we are concerned with the existence of traveling pulse solutions of one-dimensional generalized Keller-Segel system with nonlinear chemical gradients and small cell diffusion by using the dynamical systems approach, especially based on geometric singular perturbation theory and Poincare-Bendixson theorem. To show the existence of traveling pulse solutions, we first analyze the dynamics of the system by geometric singular perturbation theory. And then we seek an invariant region for the associated traveling wave equation. Finally, we apply Poincare-Bendixson theorem to analyze the flow on this invariant region to obtain the existence of traveling pulse solutions in this bounded invariant region. As applications, we present two examples to illustrate our main results. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the existence of traveling pulse solutions in one-dimensional generalized Keller-Segel system by using the dynamical systems approach, specifically based on geometric singular perturbation theory and Poincare-Bendixson theorem. The dynamics of the system is analyzed using geometric singular perturbation theory, an invariant region for the associated traveling wave equation is sought, and then Poincare-Bendixson theorem is applied to show the existence of traveling pulse solutions. The main results are illustrated with two examples presented as applications.