Skip to main content
SpringerLink
Log in

New Schemes of Dynamic Preservation of Diversity: Remarks on Stability and Topology

  • Regular Article
  • Published:
Acta Biotheoretica Aims and scope Submit manuscript

Abstract

We address the biological dynamics problem of the persistence of several species in conditions of non-existence of an equilibrium, including an example of stabilization by predation and the very controversial “competitive exclusion” (which depends on the precise definition of persistence). We give normal forms for various examples of such (essentially dynamical) persistence and comments on the involved topology, which implies the presence of exceptional heteroclinic connections binding equilibria on the boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Armstrong RA, McGehee R (1980) Competitive exclusion. Am Nat 115:151–170

    Article  Google Scholar 

  • Brauner F, Castillo-Chavez C (2012) Mathematical models in population biology and epidemiology. Springer, Berlin

    Book  Google Scholar 

  • Hirsch M, Smale S, Devaney RL (2013) Differential equations, dynamical systems, and an introduction to chaos, 3rd edn. Elsevier/Academic Press, Amsterdam

    Google Scholar 

  • Hofbauer J, Sigmund K (1988) The theory of evolution and dynamical systems. Mathematical aspects of selection. Translated from the German. London Mathematical Society Student Texts, 7. Cambridge University Press, Cambridge

  • Lherminier P, Sanchez-Palencia E (2015) Some remarks and examples on transient processes and attractors in biological evolution. In: Electronic jour diff equations conference, vol 22, pp 63–77

  • McGehee R, Armstrong RA (1977) Some mathematical problems concerning the ecological principle of competitive exclusion. J Differ Equ 23:30–53

    Article  Google Scholar 

  • Paine RT (1966) Complexity and species diversity. Am Nat 100(910):65–75

    Article  Google Scholar 

  • Sanchez-Palencia E, Françoise JP (2019) Topological remarks and new examples of persistence of diversity in biological dynamics. Discret Contin Dyn Syst Ser S 12(6):1775–1789

    Google Scholar 

  • Sanchez-Palencia E, Lherminier PH (2019) Paradoxes of vulnerability to predator in biological dynamics and mediate versus immediate causality. Discret Contin Dyn Syst Ser S, to appear

Download references

Author information

Authors and Affiliations

  1. UPMC Univ Paris 06 and CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, Sorbonne Université, 75005, Paris, France

    Evariste Sanchez-Palencia

  2. Sorbonne-Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), 75005, Paris, France

    Jean-Pierre Françoise

Authors
  1. Evariste Sanchez-Palencia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Pierre Françoise
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Pierre Françoise.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sanchez-Palencia, E., Françoise, JP. New Schemes of Dynamic Preservation of Diversity: Remarks on Stability and Topology. Acta Biotheor 68, 157–169 (2020). https://doi.org/10.1007/s10441-019-09353-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10441-019-09353-z

Keywords

Search

Navigation