Stability of steady-state solutions of a class of Keller–Segel type models with linear sensitivity and nonlinear consumption rate of chemical stimuli

This paper is devoted to the study of a class of Keller–Segel type models with Dirichlet boundary conditions and zero-flux boundary conditions on a one-dimensional bounded interval. We show the existence of non-trivial steady state solutions of these models by using sub-super solutions method and standard monotone iteration scheme method. Furthermore, we also show that the steady-state solution of these models is nonlinearly asymptotically stable by using the inverse derivative technique if the initial perturbation is sufficiently small.