Moving spike patterns in a chemotaxis model and oscillatory traveling waves in a traffic flow model

For the first project, we prove rigorously the existence of moving spike patterns in an attractive chemotaxis model with small diffusion coefficient for the chemical. In the zero diffusion limit, ϵ → 0, we prove that the nonmonotone traveling wave solutions of the system with ϵ > 0 converge to those of the system with ϵ = 0. Moreover, we show that the traveling wave solutions are linearly unstable. We perform numerical simulations and find existence of the moving spike patterns when ϵ > 0. We confirm that as ϵ → 0 the traveling wave solutions of the system with ϵ > 0 converge to the traveling wave solutions of the system with ϵ = 0.