Global Bifurcation Diagrams of Steady States of Systems of PDEs via Rigorous Numerics: a 3-Component Reaction-Diffusion System
- 341 Downloads
In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol. 53(4):617–641, 2006] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comput. 79(271):1565–1584, 2010], introduces new analytic estimates, a new gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs.
KeywordsEquilibria of systems of PDEs Bifurcation diagram Contraction mapping Rigorous numerics
Mathematics Subject Classification (2010)65N15 37M20 35K55
- 13.Keller, H.B.: Lectures on Numerical Methods in Bifurcation Problems. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 79. Tata, Bombay (1987). With notes by A.K. Nandakumaran and M. Ramaswamy Google Scholar