Scaling methods and approximative equations for homogeneous reaction-diffusion systems and applications to epidemics

J. Müller

Research output: Contribution to journalArticlepeer-review

Abstract

We consider a reaction-diffusion equation that is homogeneous of degree one. This homogeneity is a symmetry. The dynamics is factorized into trivial evolution due to symmetry and nontrivial behavior by a projection to an appropriate hypermanifold. The resulting evolution equations are rather complex. We examine the bifurcation behavior of a stationary point of the projected system. For these purposes we develop techniques for dimension reduction similar to the Ginzburg-Landau (GL) approximation, the modulation equations. Since we are not in the classical GL situation, the remaining approximative equations have a quadratic nonlinearity and the amplitude does not scale with ∈ but with ∈2 where ∈2 denotes the bifurcation parameter. Moreover, the symmetry requires that not only one but two equations are necessary to describe the behavior of the system. We investigate traveling fronts for the modulation equations. This result is used to analyze an epidemic model.

Original languageEnglish
Pages (from-to)149-168
Number of pages20
JournalJournal of Nonlinear Science
Volume9
Issue number2
DOIs
StatePublished - 1999
Externally publishedYes

Keywords

  • Epidemic models
  • Modulation equations
  • Phase equation
  • Symmetry

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