Steadily propagating waves of a chemotaxis model

This paper studies the existence, asymptotic decay rates, nonlinear stability, wave speed and chemical diffusion limits of traveling wave solutions to a chemotaxis model describing the initiation of angiogenesis and reinforced random walk. By transforming the chemotaxis system, via a Hopf-Cole transformation, into a system of conservation laws, the authors studied the traveling wave solutions of the transformed system in previous papers. One of the purposes of this paper is to transfer the results of the transformed system to the original Keller–Segel chemotaxis model. It turns out that only partial results of the transformed system have physical meaning when they are passed back to the original system. Thus the transformed system is not entirely equivalent to the original system. Particularly the chemical growth rate parameter appeared in the original system vanishes in the transformed system. Hence to understand the role of this parameter, one has to go back to the original system. Moreover, we establish some new results on zero chemical diffusion limits of traveling wave solutions. Numerical simulations of steadily propagating waves are shown.

► The chemotaxis model describes the initiation of angiogenesis and contains a singularity in the cross-diffusion term.
► We transform the model, via a smart Hopf-Cole transformation, into a system of conservation laws without singularity.
► The existence and nonlinear stability of traveling wave solutions are established.
► Numerical simulations of steadily propagating waves are illustrated and interpretation of biological implications is given.