Analysis and numerical simulations of a reaction-diffusion model with fixed active bodies

We propose an interaction model for two substances (molecules) of concentrations f and g. The substance f is assumed to be involved in the growth process of g, and in contrast, the substance g inhibits the production of f. Our model is based on a reaction-diffusion system for the interaction of two considered substances (molecules). We analyze its well-posedness and perform a numerical scheme to simulate the behavior of the solutions. As an application, we investigate the role of Aβ-oligomers in the early stage of Alzheimer's disease propagation, more precisely for the impact of Aβ-oligomers in neuron impairment as studied in [M. Andrade-Restrepo et al., Modeling the spatial propagation of Aβoligomers in Alzheimer's disease, in CEMRACS 2018 - Numerical and Mathematical Modeling for Biological and Medical Applications: Deterministic, Probabilistic and Statistical Descriptions, Marseille, France, 2018, pp. 1-10].