Patterns in a generalized volume-filling chemotaxis model with cell proliferation

In this paper, we consider the following system [Equation presented here] which corresponds to the stationary system of a generalized volume-filling chemotaxis model with logistic source in a bounded domain in RN(N ≥ 1) with zero Neumann boundary conditions. Here the parameters D, χ, μ, ucare positive and α, β ∈ R, and ν denotes the outward unit normal vector of ∂Ω. With the priori positive lower- and upper-bound solutions derived by the Moser iteration technique and maximum principle, we apply the degree index theory in an annulus to show that if the chemotactic coefficient χ is suitably large, the system with α + β > 1 admits pattern solutions under certain conditions. Numerical simulations of the pattern formation are shown to illustrate the theoretical results and predict the interesting phenomenon for further studies.