Space Structures of Some Migrating Populations

  • Piero de Mottoni
Conference paper
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 54)

Abstract

The most primitive mathematical models of ecology describe populations as space-homogeneous entities; or, equivalently, space-averages rather than the spatial structure of the populations are taken into account. In recent years, several efforts have been made to overcome such a simplified representation: in particular by trying to provide a mathematical description of segregation effects, by which, in the long run, different populations occupy different sub-areas. Among the resulting mathematical models, we shall focus here on those involving quasilinear (or semilinear) parabolic systems, i.e., systems of partial differential equations describing the time course of communities subject both to kinetic interaction and to linear (respectively, nonlinear) diffusion.

Keywords

Migration Considera Tion 

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References

  1. Fife, P.C. (1979) The Mathematics of Reacting and Diffusing Systems. Lecture Notes in Biomathematics, Springer, Berlin-Heidelberg-New York.Google Scholar
  2. Hadeler, K.-P., an der Heiden, U., and Rothe, F. (1974) Nonhomogeneous spatial distri butions of populations, Journ. Math. Biol. 1: 165-176.Google Scholar
  3. Kawasaki, K. and Teramoto, E. Spatial pattern formation of prey-predator populations, Journ. Math. Biol. 8: 33 - 46.Google Scholar
  4. Krasnosel'skii, M.A. (1964) Positive Solutions of Operator Equations, Noordhoff, Groningen (transl. form the Russian edition, Moscow, 1962 ).Google Scholar
  5. Leung, A. (1980) Equilibria and stabilities for competing species equations with Dirichlet boundary data, Journ. Math. Anal. Appl. 73: 200264-218.Google Scholar
  6. Levin, S.A. (1979) Non uniform steady state solutions to reaction-diffusion equations: applications to ecological pattern formation. In: Pattern Formation by Dynamical Systems and Pattern Recognition, H. Haken ed., Springer Berlin-Heidelberg-New York: 210225.Google Scholar
  7. Lions, P.L. (1982) On the existence of positive solutions of semilinear elliptic equations, SIAM Review 24 (4): 441 - 467.MathSciNetMATHCrossRefGoogle Scholar
  8. Matano, H. and Mimura, M (1983). Preprint.University of Hiroshima.Google Scholar
  9. Mimura, M. (1981) Stationary pattern of some density-dependent diffusion systems with competitive dynamics, Hiroshima Math. Journ. 11: 621-635.Google Scholar
  10. Mimura, M (1983). Private communication. de Mottoni, P., Schiaffino, A., and Tesei, A. ( 1983 ). On stable space-dependent statio nary solutions of a competition system with diffusion. Zeitschrift für Analysis und ihre Anwendungen (Leipzig), to appear.Google Scholar
  11. Rothe, F. (1982). A priori estimates, global existence and asymptotic behaviour for weakly coupled systems of reaction-diffusion equations. Habilitationsschrift, Universi tät Tübingen.Google Scholar
  12. Sattinger, D.H. (1971-2) Monotone methods in nonlinear elliptic and parabolic equatio ns, Indiana University Math. Journ. 21: 979 – 1000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Piero de Mottoni
    • 1
  1. 1.Istituto di Matematica ApplicataUniversità dell’AquilaPoggio di Roio (L’Aquila)Italy

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