A study on the positive nonconstant steady states of nonlocal chemotaxis systems

Abstract

In one spatial dimension, we perform global bifurcation analysis on a general nonlocal Keller-Segel chemotaxis model, showing that positive nonconstant steady states exist, if the chemotactic coefficient $\chi$ is larger than a bifurcation value $\overline{\chi}_1$, which is expressible in terms of the parameters and the nonlocal sampling radius in the model. We then show that the positive solutions of the nonlocal model converge at least in $C^2([0,l])\times C^2([0,l])$ to that of the corresponding ``local' model as the nonlocal sampling radius $\rho\rightarrow 0+$. Finally, we use Helly's compactness theorem to establish the profiles of these steady states, when the ratio of the chemotactic coefficient and the cell diffusion rate is large and the nonlocal sampling radius is small, exhibiting whether they are either spiky, of transition layer structure or just flat everywhere. Our results supply understandings on how the biological parameters affect pattern formation for the nonlocal model. In the limit of $\rho\rightarrow 0+$, our results agree with those of local models studied in Wang and Xu [29].