Spatio-temporal oscillations in the Keller-Segel system with logistic growth

The Keller-Segel system with the logistic growth term is discussed from the spatio-temporal-oscillation point of view. This system exhibits two different types of spatio-temporal oscillations in certain distinct parameter regimes. In this paper, we study the difference between the two types of spatio-temporal oscillations. In particular, the characteristic properties of the behaviors become clear in a limiting system when a certain parameter value tends to zero. Moreover, we demonstrate that the onset of one of the spatio-temporal oscillatory patterns is an infinite-dimensional relaxation oscillation that consists of slow and fast dynamics.