0000000000703844
AUTHOR
Valeria Giunta
Eckhaus and zigzag instability in a chemotaxis model of multiple sclerosis
We present a theoretical and numerical study of the bifurcations of the stationary patterns supported by a chemotactic model of Multiple Sclerosis (MS). We derive the normal forms of the dynamics which allows to predict the appearance and stabilization of the emerging branches describing the concentric patterns typical of Balo's sclerosis, a very aggressive variant of MS. Spatial modulation of the Turing-type structures through a zigzag instability is also addressed. The nonlinear stage of the Eckhaus and zigzag instability is investigated numerically: defect-mediated wavenumber adjustments are recovered and the time of occurrence of phase-slips is studied as the system parameters are varie…
Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model
<p style='text-indent:20px;'>We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the exis…
Axisymmetric solutions for a chemotaxis model of Multiple Sclerosis
In this paper we study radially symmetric solutions for our recently proposed reaction–diffusion–chemotaxis model of Multiple Sclerosis. Through a weakly nonlinear expansion we classify the bifurcation at the onset and derive the amplitude equations ruling the formation of concentric demyelinating patterns which reproduce the concentric layers observed in Balò sclerosis and in the early phase of Multiple Sclerosis. We present numerical simulations which illustrate and fit the analytical results.
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 an…