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Models with several species and trophic levels

  • Mimmo Iannelli
  • Andrea Pugliese
Chapter
  • 1.8k Downloads
Part of the UNITEXT book series (UNITEXT, volume 79)

Abstract

The interspecific competition and predation models for two species interaction, presented in Chaps. 6 and 7, can be used as building blocks for designing more general mechanisms and ecosystems. Moving beyond the two species framework, we are led to consider non-planar systems and we meet systems that are mathematically more complex, possibly requiring specific treatment and analysis, and provide new insights into ecological dynamics. Even though we will not aim to a complete description of the World in a Wall, this chapter is devoted to a few cases of three species dynamics that illustrate some complexities of multiple species interaction and introduce to the use of some more advanced mathematical technique.

Keywords

Periodic Solution Hopf Bifurcation Unstable Manifold Stable Manifold Global Attractor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Benicà, E., Huisman, J., Heerkloss, R., Johnk, K.D., Branco, P., Van Nes, E.H., Scheffer, M., Ellner, S.P.: Chaos in a long-term experiment with a plankton community. Nature 451, 822–826 (2008)CrossRefGoogle Scholar
  2. 2.
    Candaten, M., Rinaldi, S.: Peak-to-peak dynamics: A critical survey. Int. Jour. of Bifurcation and Chaos 10, 1805–1819 (2000)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cushing, J.M., Costantino, R.F., Dennis, B., Desharnais, R.A., Henson, S.M.: Chaos in Ecology: Experimental Nonlinear Dynamics. Theoretical Ecology Series, Academic Press, San Diego (2003)Google Scholar
  4. 4.
    Hairston, N.G., Smith, F.E., Slobodkin, L.B.: Community structure, population control, and competition. Am. Nat. 44, 421–425 (1960)CrossRefGoogle Scholar
  5. 5.
    Hastings, A., Powell, T.: Chaos in three-species food chain. Ecology 72, 896–903 (1991)CrossRefGoogle Scholar
  6. 6.
    Hutson, V., Vickers, G.T.: A criterion for permanent coexistence of species, with an application to a two-prey one-predator system, Mathematical Biosciences 63, 253–269 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Kuznetsov, Y.A., Rinaldi, S.: Remarks on food chain dynamics. Mathematical Biosciences 134, 1–33 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    May, R., Leonard, W.J.: Nonlinear aspects of competition between three species. SIAM J. on Appl. Math. 29, 243–253 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    McCann, K., Yodzis, P.: Bifurcation structure of a three species food chain model. Theoretical Population Biology 48, 93–125 (1993)CrossRefGoogle Scholar
  10. 10.
    Oksanen, L., Fretwell, D.S., Arruda, J., Niemela, P.: Exploitation ecosystems in gradients of primary productivity. Am. Nat. 118, 240–261 (1981)CrossRefGoogle Scholar
  11. 11.
    Paine, R.T.: Food web complexity and species diversity. Amer. Nat. 100 65–75 (1966)CrossRefGoogle Scholar
  12. 12.
    Perry, J.N., Smith, R.H., Woiwod, I.P., Morse, D.R.: Chaos in Real Data: the Analysis of Nonlinear Dynamics from Short Ecological Time Series, Kluwer the Netherlands (2000)CrossRefGoogle Scholar
  13. 13.
    Smith, H.L., Thieme, H.R.: Dynamical Systems and Population Persistence. Graduate Studies in Mathematicas, 118, American Mathematical Society (2011)Google Scholar
  14. 14.
    Takeuchi, Y.: Global dynamical properties of Lotka-Volterra systems. World Scientific (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Vance, R.R.: Predation and resource-partitioning in one predator-two prey model communities, Amer. Nat. 112 798–813 (1978)CrossRefGoogle Scholar
  16. 16.
    Volterra, V.: Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. della R. Accademia dei Lincei, ser. VI, vol II, 31–113 (1926)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Mimmo Iannelli
    • 1
  • Andrea Pugliese
    • 1
  1. 1.Department of MathematicsUniversity of TrentoItaly

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