Journal of Mathematical Biology

, Volume 27, Issue 1, pp 65–80 | Cite as

Local vs. non-local interactions in population dynamics

  • J. Furter
  • M. Grinfeld


In this work we examine two models of single-species dynamics which incorporate non-local effects. The emphasis is on the ability of these models to generate stable patterns. Global behavior of the bifurcating branches is also investigated.

Key words

Reaction-diffusion equations Non-local effects Stable patterns 


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  1. 1.
    Ball, J. M.: On the asymptotic behavior of generalized processes with applications to nonlinear evolution equations. J. Diff. Equations 27, 224–265 (1978)Google Scholar
  2. 2.
    Casten, R., Holland, C.: Instability results for reaction-diffusion equations with Neumann boundary conditions. J. Differ. Equations 27, 266–273 (1978)Google Scholar
  3. 3.
    Chafee, N.: The electric balast resistor: homogeneous and nonhomogeneous equilibria. In: de Mottoni, P., Salvatori, L. (eds.) Nonlinear differential equations pp. 97–727. New York: Academic Press 1981Google Scholar
  4. 4.
    Cohen, D. S., Murray, J. D.: A generalized diffusion model for growth and dispersal in a population. J. Math. Biol. 12, 237–249 (1981)Google Scholar
  5. 5.
    Crandall, M., Rabinowitz, P.: Mathematical theory of bifurcation. In: Bardos, C., Bessis, D. (eds.) Bifurcation phenomena in mathematical physics and related topics. Dordrecht: Reidel 1980Google Scholar
  6. 6.
    Gierer, A., Meinhardt, H.: A theory of biological pattern formation, Kybernetik 12, 30–39 (1972)Google Scholar
  7. 7.
    Golubitsky, M., Schaeffer, D.: Singularities and groups in bifurcation theory. Berlin Heidelberg New York: Springer 1984Google Scholar
  8. 8.
    Hale, J. K., Sakamoto, K.: Shadow systems and attractors in reaction-diffusion equations. LCDS/CCS, Preprint # 87-28. Brown University, Providence 1987Google Scholar
  9. 9.
    Henry, D.: Geometric theory of semilinear parabolic equations (Lect. Notes Math., vol. 840) Berlin Heidelberg New York: Springer 1980Google Scholar
  10. 10.
    Matano, H.: Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. RIMS, Kyoto Univ. 15, 401–454 (1979)Google Scholar
  11. 11.
    Nishiura, Y.: Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13, 555–593 (1982)Google Scholar
  12. 12.
    Okubo, A.: Diffusion and ecological problems. Mathematical models. Berlin Heidelberg New York: Springer 1980Google Scholar
  13. 13.
    Protter, M., Weinberger, H.: Maximum principles in differential equations. Englewood Cliffs, N.J.: Prentice Hall 1967Google Scholar
  14. 14.
    Takagi, I.: Point-condensation solutions for a reaction-diffusion system. Preprint 1984Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • J. Furter
    • 1
  • M. Grinfeld
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityRiccartonScotland, UK

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