Extended fractional-polynomial generalizations of diffusion and Fisher–KPP equations on directed networks

Abstract

In a variety of practical applications, there is a need to investigate diffusion or reaction–diffusion processes on complex structures, including brain networks, that can be modeled as weighted undirected and directed graphs. As an instance, the celebrated Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) reaction–diffusion equation is becoming increasingly popular for use in graph frameworks by substituting the standard graph Laplacian operator for the continuous one to study the progression of neurodegenerative diseases such as Alzheimer’s disease (AD). In this work, we establish existence, uniqueness, and boundedness of solutions for generalized Fisher–KPP reaction–diffusion equations on undirected and directed networks with fractional polynomial (FP) terms. This type of model has possible applications for modeling spreading of diseases within neuronal fibers whose porous structure may cause particles to diffuse anomalously. In the case of pure diffusion, convergence of solutions and stability of equilibria are also analyzed. Moreover, different families of positively invariant sets for the proposed equations are derived. Finally, we conclude by investigating nonlinear diffusion on a directed one-dimensional lattice.