Numerical study of the volcano effect in chemotactic aggregation based on a kinetic transport equation with non-instantaneous tumbling

Abstract 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.