Positivity

pp 1–10 | Cite as

Criterion of positivity for semilinear problems with applications in biology

Article

Abstract

The goal of this article is to provide an useful criterion of positivity and well-posedness for a wide range of infinite dimensional semilinear abstract Cauchy problems. This criterion is based on some weak assumptions on the non-linear part of the semilinear problem and on the existence of a strongly continuous semigroup generated by the differential operator. To illustrate a large variety of applications, we exhibit the feasibility of this criterion through three examples in mathematical biology: epidemiology, predator-prey interactions and oncology.

Keywords

Positivity Well-posedness Dynamic systems Semilinear problems Population dynamics 

Mathematics Subject Classification

35A01 35B09 35Q92 92D25 

References

  1. 1.
    Alaa, N., Fatmi, I., Roche, J.-R., Tounsi, A.: Mathematical analysis for a model of nickel-iron alloy electrodeposition on rotating disk electrode: parabolic case. Int. J. Math. Stat. 2, 30–49 (2008)MathSciNetMATHGoogle Scholar
  2. 2.
    Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H.P., Moustakas, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-parameter semigroups of positive operators, Lecture Notes in Mathematics, vol. 1184. Springer (1986)Google Scholar
  3. 3.
    Chakrabarty, S., Hanson, F.B.: Distributed parameters deterministic model for treatment of brain tumors using galerkin finite element method. Math. Biosci. 219(2), 129–141 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Kermack, W.O., G, M.A.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 219, 700–721 (1927)CrossRefMATHGoogle Scholar
  5. 5.
    Magal, P., Ruan, S.: Structured Population Models in Biology and Epidemiology, Vol. 1936 of Lecture Notes in Mathematics/Mathematical Biosciences Subseries, Springer (2008)Google Scholar
  6. 6.
    Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  7. 7.
    Murray, J.D.: Mathematical Biology I, An introduction. Interdisciplinary Applied Mathematics, 3rd edn, vol. 17. Springer, Berlin (2002)Google Scholar
  8. 8.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)MATHGoogle Scholar
  9. 9.
    Perasso, A., Razafison, U.: Infection load structured si model with exponential velocity and external source of contamination, In: World Congress on Engineering, pp. 263–267 (2013)Google Scholar
  10. 10.
    Perasso, A., Razafison, U.: Asymptotic behavior and numerical simulations for an infection load-structured epidemiological model: application to the transmission of prion pathologies. SIAM J. Appl. Math. 74(5), 1571–1597 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Perasso, A., Richard, Q.: Implication of age-structure on the dynamics of Lotka Volterra equations, to appear in differential and integral equationsGoogle Scholar
  12. 12.
    Pierre, M.: Global existence in reaction-diffusion systems with control of mass: a survey. Milan J. Math. 78(2), 417–455 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Smith, H.L., Waltman, P.: The Theory of the Chemostat.: dynamics of microbial competition. Cambridge Studies in Mathematical Biology, vol. 13. Cambridge University Press, Cambridge (1995)Google Scholar
  14. 14.
    Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B Biol. Sci. 237(641), 37–72 (1952)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Institut de Mathmatiques de Marseille UMR CNRS 7373Université Aix-MarseilleMarseille Cedex 13France
  2. 2.Chrono-environnement UMR CNRS 6249Université Bourgogne Franche-ComtéBesançonFrance

Personalised recommendations