PIF2024 - Difusión no-conservativa con nolocalidad espacio-temporal en grafos-PID2023-149473NB-I00

Complex systems permeate biological life, society, and modern infrastructures. This basic research project foresees a significant advance in the knowledge of complex systems, developing a mathematical theory that will extend the knowledge of non-conservative diffusion to allow for non-locality in space and time, which will cover a very wide range of social, physical, biological, and technological processes occurring in the real world. This development is based on the elaboration of logistic reaction-diffusion models on graphs representing NC dynamics but reaching stationary equilibrium states. These models will be generalised to account for non-local effects in time and space. Spatial non-locality will be introduced by generalising the adjacency operator on graphs to the transformed d-path adjacency operators. Such transformations will be applied to account for the decay of spatial non-locality that takes place with the separation between vertices of the graph. On the other hand, temporal non-locality will be implemented by replacing the standard derivatives by Caputo time fractional derivatives in the NC diffusive models. Finally, the results will be integrated by studying the geometries induced by these non-local NC diffusive dynamics. For this purpose, the most probable trajectories of the diffusive particles will be studied through the geometry of the graphs. The models developed will be applied to the study of real-world problems such as chemical synapses in neural syste