Advertisement

Pattern Formation for a Reaction Diffusion System with Constant and Cross Diffusion

  • Verónica Anaya
  • Mostafa Bendahmane
  • Michel Langlais
  • Mauricio Sepúlveda
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

In this work, we study a finite volume scheme for a reaction diffusion system with constant and cross diffusion modeling the spread of an epidemic disease within a host population structured with three subclasses of individuals (SIR-model). The mobility in each class is assumed to be influenced by the gradient of other classes. We establish the existence of a solution to the finite volume scheme and show convergence to a weak solution. The convergence proof is based on deriving a series of a priori estimates and using a general L p compactness criterion.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. Anaya, M. Bendahmane, M. Langlais, M. Sepúlveda, A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion, Preprint 2013–12, DIM, U. de ConcepciónGoogle Scholar
  2. 2.
    V. Anaya, M. Bendahmane, M. Sepúlveda, Numerical analysis for HP food chain system with nonlocal and cross diffusion, Preprint 2011–11, DIM, U. de ConcepciónGoogle Scholar
  3. 3.
    E. DiBenedetto, Degenerate Parabolic Equations. Universitext (Springer, New York, 1993)CrossRefzbMATHGoogle Scholar
  4. 4.
    G. Galiano, M.L. Garzón, A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. Rev. R. Acad. Cienc. Exactas Fís. Nat. Serie A. Mat. 95(2), 281–295 (2001)zbMATHGoogle Scholar
  5. 5.
    G. Galiano, M.L. Garzón, A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model. Numer. Math. 93(4), 655–673 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Y. Li, C. Zhao, Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discr. Contin. Dyn. Syst. 12(2), 185–192 (2005)zbMATHGoogle Scholar
  7. 7.
    Y. Lou, S. Martinez, Evolution of cross-diffusion and self-diffusion. J. Biol. Dyn. 3(4), 410–429 (2009)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Y. Wang, The global existence of solutions for cross-diffusion system. Acta Math. Appl. Sin. Engl. Ser. 21(3), 519–528 (2005)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Verónica Anaya
    • 1
  • Mostafa Bendahmane
    • 2
  • Michel Langlais
    • 2
  • Mauricio Sepúlveda
    • 3
  1. 1.Departamento de Matemática and GIMNAPUniversidad del Bío-BíoConcepciónChile
  2. 2.Institut de Mathématiques de Bordeaux UMR CNRS 5251Université Victor Segalen Bordeaux 2BordeauxFrance
  3. 3.DIM and CI2 MA, Universidad de ConcepciónEsteban Iturra s/n, Barrio UniversitarioConcepciónChile

Personalised recommendations