Existence and stability of traveling waves in a Keller-Segel system with nonlinear stimulation

Abstract In this article, we reformulate the Keller-Segel model by incorporating a nonlinear cellular stimulation term. We then investigate its dynamical behavior by reducing the Keller-Segel model to a dynamical system on a two-dimensional invariant flow. Our analysis confirms the existence of a pulse in bacteria density and a kink front in chemical density. Furthermore, we derive precise conditions for an oscillatory tail and monotonic decay in the front. Two illustrative examples are presented to demonstrate our findings, and the range of wave speeds is determined numerically for each system. Finally, we examine the stability of the pulse by examining the spectrum of the differential operator associated with the governing partial differential equation.