Nonlinear Stability of Traveling Waves to a Hyperbolic-Parabolic System Modeling Chemotaxis

Abstract

We prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic-parabolic system modeling repulsive chemotaxis. In contrast to the previous related results, where various smallness conditions on wave strengths were imposed, we are able to prove the nonlinear stability of the traveling waves with arbitrary amplitudes under small perturbations in spite of partial diffusion in the model. Moreover, we perform numerical experiments to verify our theoretical results. Finally, the biological implications are discussed. Our results indicate that when the dissipative effect is not negligible, the cell density distribution approaches a smooth viscous shock profile asymptotically if the chemotaxis is repulsive.