Bifurcations in Nagumo Equations on Graphs and Fiedler Vectors

Reaction-diffusion equations serve as a basic framework for numerous dynamic phenomena like pattern formation and travelling waves. Spatially discrete analogues of Nagumo reaction-diffusion equation on lattices and graphs provide insights how these phenomena are strongly influenced by the discrete and continuous spatial structures. Specifically, Nagumo equations on graphs represent rich high dimensional problems which have an exponential number of stationary solutions in the case when the reaction dominates the diffusion. In contrast, for sufficiently strong diffusion there are only three constant stationary solutions. We show that the emergence of the spatially heterogeneous solutions is closely connected to the second eigenvalue of the Laplacian matrix of a graph, the algebraic connectivity. For graphs with simple algebraic connectivity, the exact type of bifurcation of these solutions is implied by the properties of the corresponding eigenvector, the so-called Fiedler vector.