This paper aims to investigate the time fractional Keller-Segel system
with a small parameter. After the fractional order traveling wave
transformation, the heteroclinic orbit to the degenerate time fractional
Keller-Segel system is demonstrated through the method of constructing a
suitable invariant region. Moreover, the persistence of traveling waves
in the system with a small parameter can be further illustrated. The
results are mainly reliance on the application of geometric singular
perturbation theory and Fredholm theorem, which are fundamental
theoretical frameworks for dealing with problems of complexity and high
dimensionality. Eventually, the asymptotic behavior is depicted by the
asymptotic theory to illustrate the rate of decay for traveling waves.