Basic Mechanisms Driving Complex Spike Dynamics in a Chemotaxis Model with Logistic Growth

We investigate the Keller--Segel (KS) model with logistic self-production terms that exhibits complex spatio-temporal dynamics of spikes. These dynamics are driven by merging of spikes on one hand, and spike insertion on the other. In this paper we analyze the basic mechanisms that initiate and sustain these events. We identify two distinguished regimes. In the first regime, a single interior spike drifts toward a boundary. This instability is responsible for spike merging. The same regime further exhibits spike insertion; we identify a fold-point bifurcation which is a precursor to the spike insertion event. In the second regime, we show that it is possible to stabilize a single interior spike, and we compute analytically a critical threshold which is responsible for spike stabilization. In particular, our calculation characterizes a stable spike in the KS model with logistic growth; this is in contrast to the classical KS model, where the interior spike is known to be unstable.