Pattern formation and transition to chaos in a chemotaxis model of acute inflammation

We investigate a reaction-diffusion-chemotaxis system that describes the immune response during an inflammatory attack. The model is a modification of the system proposed in Penner, Ermentrout, and Swigon [SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 629-660]. We introduce a logistic term in the immune cell dynamics to reproduce the macrophages' activation, allowing us to describe the disease evolution from the early stages to the acute phase. We focus on the appearance of pattern solutions and their stability. We discover steady-state (Turing) and wave instabilities and classify the bifurcations deriving the corresponding amplitude equations. We study stationary radially symmetric solutions and show that they reproduce various inflammatory aggregates observed in the clinical practice. Moreover, the model supports oscillating-in-time spatial patterns, thus giving a theoretical explanation of the periodic appearance of inflammatory eruptions typical of recurrent erythema multiforme. A detailed numerical bifurcation analysis indicates that the inclusion of the logistic growth term is crucial for the occurrence of a sequence of bifurcations leading to spatio-temporal chaos. In the parameter space, there are large regions where the model system displays critical behavior.