Department of Mathematics
University of Oviedo
Oviedo, Spain
email: galiano@uniovi.es
Department of Mathematics
University of Santiago de Compostela
Santiago de Compostela, Spain
email: victor.gonzalez.tabernero@rai.usc.es
Gonzalo Galiano, Victor Gonzalez-Tabernero
Abstract:
The population model by Busenberg and Travis is a paradigmatic model in ecology
and tumor modeling because its ability to capture interesting phenomena
such as segregation of populations. Its singular mathematical structure enforces
the consideration of regularized problems to deduce properties as fundamental
as the existence of solutions.
In this article we perform a weakly nonlinear stability analysis of a general
class of regularized problems to study the convergence of the instability modes
in the limit of the regularization parameter. We demonstrate with some specific
examples that the pattern formation observed in the regularized problems,
with unbounded wave numbers, is not present in the limit problem because of the
amplitude decay of the oscillations.
We also check the results of the stability analysis with direct finite element
simulations of the problem.
Submitted December 8, 2020. Published June 21, 2021.
Math Subject Classifications: 35K55, 35B36, 92D25.
Key Words: Cross-diffusion; Turing instability; weakly nonlinear equation;
finite element method.
DOI: https://doi.org/10.58997/ejde.2021.55
Show me the PDF file (448 KB), TEX file for this article.
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Gonzalo Galiano Department of Mathematics University of Oviedo Oviedo, Spain email: galiano@uniovi.es |
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Víctor González-Tabernero Department of Mathematics University of Santiago de Compostela Santiago de Compostela, Spain email: victor.gonzalez.tabernero@rai.usc.es |
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