Periodic coexistence of four species competing for three essential resources

Bingtuan Li, Hal Smith

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.

Original languageEnglish (US)
Pages (from-to)115-135
Number of pages21
JournalMathematical Biosciences
Volume184
Issue number2
DOIs
StatePublished - Aug 2003

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Competing Species
Coexistence
coexistence
Resources
new species
resource
Hopf bifurcation
Bifurcation (mathematics)
Asymptotic stability
Periodic Solution
bifurcation
matrix
Computer simulation
Asymptotically Stable
Symmetric matrix
Hopf Bifurcation
Asymptotic Stability
Standard Model
Exceed
Bifurcation

Keywords

  • Bifurcation
  • Coexistence
  • Periodic orbit
  • Resource competition

ASJC Scopus subject areas

  • Agricultural and Biological Sciences(all)
  • Ecology, Evolution, Behavior and Systematics

Cite this

Periodic coexistence of four species competing for three essential resources. / Li, Bingtuan; Smith, Hal.

In: Mathematical Biosciences, Vol. 184, No. 2, 08.2003, p. 115-135.

Research output: Contribution to journalArticle

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