  Genetically engineered bacteria to increase the tumbling frequency of the
run-and-tumble motion for the higher local bacterial density form visible
stripe pattern composed of successive high and low density regions on an agar
plate. We propose a model that includes a simplified regulatory dynamics of the
tumbling frequency in individual cells to clarify the role of finite response
time. We show that the time-delay due to the response dynamics results in the
instability in a homogeneous steady state allowing a pattern formation. For
further understanding, we propose a simplified two-state model that allows us
to describe the response time dependence of the instability analytically. We
show that the instability occurs at long wave length as long as the response
time is comparable with the tumbling timescale and the non-linearity of the
response function to the change of the density is high enough. The minimum
system size to see the instability grows with the response time $\tau$,
proportional to $\sqrt{\tau}$ in the large delay limit.
