Aggregation of chemotactic bacteria under a unimodal distribution of chemical
cues was investigated by Monte Carlo simulation and asymptotic analysis based
on a kinetic transport equation, which considers an internal adaptation
dynamics as well as a finite tumbling duration. It was found that there exist
two different regimes of the adaptation time, between which the effect of the
adaptation time on the aggregation behavior is reversed; that is, when the
adaptation time is as small as the running duration, the aggregation becomes
increasingly steeper as the adaptation time increases, while, when the
adaptation time is as large as the diffusion time of the population density,
the aggregation becomes more diffusive as the adaptation time increases.
Moreover, notably, the aggregation profile becomes bimodal (volcano) at the
large adaptation-time regime while it is always unimodal at the small
adaptation-time regime. The volcano effect occurs in such a way that the
population of tumbling cells considerably decreases in a diffusion layer which
is created near the peak of the external chemical cues due to the coupling of
diffusion and internal adaptation of the bacteria. Two different
continuum-limit models are derived by the asymptotic analysis according to the
scaling of the adaptation time; that is, at the small adaptation-time regime,
the Keller-Segel model while, at the large adaptation-time regime, an extension
of KS model, which involves both the internal variable and the tumbling
duration. Importantly, either of the models can accurately reproduce the MC
results at each adaptation-time regime, involving the volcano effect. Thus, we
conclude that the coupling of diffusion, adaptation, and finite tumbling
duration is crucial for the volcano effect