  In this paper, we consider a generalized model of $2\times 2$ Keller-Segel
system with nonlinear chemical gradient and small cell diffusion. The existence
of the traveling pulses for such equations is established by the methods of
geometric singular perturbation (GSP in short) and trapping regions from
dynamical systems theory. By the technique of GSP, we show that the necessary
condition for the existence of traveling pulses is that their limiting profiles
with vanishing diffusion can only consist of the slow flows on the critical
manifold of the corresponding algebraic-differential system. We also consider
the linear instability of these pulses by the spectral analysis of the
linearized operators.
