Abstract
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.
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Notes
Alternatively we could fix \( \lambda =1\), which is more common, but we prefer to fix the diffusion parameter because of the direct natural connection between the value of \(\lambda \) and the algebraic connectivity \(\lambda _2\).
We omit the case \( \lambda '(0) = 0 \) and \( \lambda ''(0) = 0 \), since the behavior of bifurcation depends on higher order terms for which we do not obtain representing formulas. Consequently, the case \(\left|a-\tfrac{1}{2}\right|= \delta \) is missing in the statement of Theorem 1.1, case 3.
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Acknowledgements
The authors gratefully appreciate discussions with Hermen Jan Hupkes, Tomáš Kaiser, Roman Nedela, and Antonín Slavík. The authors acknowledge the support of the project LO1506 of the Czech Ministry of Education, Youth and Sports under the program NPUI. Moreover, the first and second authors have been supported by the Czech Science Foundation, grants no. GA20-11164L and GA18-03253S, respectively.
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Stehlík, P., Švígler, V. & Volek, J. Bifurcations in Nagumo Equations on Graphs and Fiedler Vectors. J Dyn Diff Equat 35, 2397–2412 (2023). https://doi.org/10.1007/s10884-021-10101-6
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DOI: https://doi.org/10.1007/s10884-021-10101-6