Existence and stability of bifurcating solution of a chemotaxis model

Abstract

This paper focuses on a chemotaxis model with no-flux boundary conditions. We first discuss the stability of the unique positive equilibrium by treating the chemotaxis coefficient ξ \xi as the Hopf bifurcation and the steady state bifurcation parameter. Hereafter, we perform the existence and stability of the bifurcating solution, which bifurcated from the steady state bifurcation, by using the Crandall-Rabinowitz local bifurcation theory. It is noticed that few existing literatures give a discriminant to determine the stability of the bifurcating solution for the chemotaxis models. To this end, we will fill this gap, and an explicit formula will be presented. This technique can also be applied in other ecological models with chemotaxis.

Chain sequences and zeros of polynomials related to a perturbed RII type recurrence relation

“…In other words, a stable system will return to its equilibrium state after being perturbed. The exploration of stability analysis holds significant research interest across various domains, including difference equations [1], differential equations [6], fluid mechanics [22], etc. In this section, we make an attempt to establish connections between the methodologies developed in these fields and the framework of orthogonal polynomials.…”