Pattern formation of a volume-filling chemotaxis model with a bistable source

In this paper, the pattern formation of a volume-filling chemotaxis model with bistable source terms was studied. First, it was shown that self-diffusion does not induce Turing patterns, but chemotaxis-driven instability occurs. Then, the asymptotic behavior of the chemotaxis model was analyzed by weakly nonlinear analysis with the method of multiple scales. When the chemotaxis coefficient exceeded a threshold value and there was a single unstable mode, the supercritical and subcritical bifurcation of the model was discussed. The amplitude equations and the asymptotic expressions of the patterns were obtained. When the chemotaxis coefficient was large enough, the two-mode competition behavior of the model with two unstable modes was analyzed, and the corresponding amplitude equations and the asymptotic expressions of the patterns were obtained. Finally, numerical simulations were provided to further illuminate the above analytical results.