Steady-state bifurcation and pattern formation in an immune chemotaxis system with volume-filling effects

<title>Abstract</title> <p> This paper studies a three-component reaction-diffusion-advection immune chemotaxis model incorporating volume-filling effects, which accounts for spatial crowding and density limitation of immune cells. The primary focus is on the existence, direction, and stability of nonconstant steady-state solutions arising from steady-state bifurcation. We first derive the critical values of chemotactic sensitivity that lead to the instability of the constant positive steady state. Then the existence of nonconstant positive steady-state solutions bifurcating from the constant equilibrium by the Crandall-Rabinowitz bifurcation theorem is established. Furthermore, the direction and local stability of the bifurcation branches are investigated by analyzing the quadratic coefficients in the asymptotic expansion. It is proved that the minimal bifurcation point determines the stability of the bifurcated solutions, while other branches remain unstable. Numerical simulations are presented to verify the theoretical conclusions and exhibit the spatial patterns generated by chemotaxis and volume-filling mechanism. This work extends the classical immune chemotaxis model by incorporating the volume-filling mechanism and reveals how cellular crowding affects the stability and pattern formation in immune dynamics, enriching the mathematical theory of immune chemotaxis models. <bold>MSC</bold> : 35B40, 35B65, 35K57, 35Q92, 92C17. </p>