A bifurcation analysis of a differential equations model for mutualism

Wendy Gruner Graves, Bruce Peckham, John Pastor

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25 Scopus citations


We develop from basic principles a two-species differential equations model which exhibits mutualistic population interactions. The model is similar in spirit to a commonly cited model [Dean, A.M., Am. Nat. 121(3), 409-417 (1983)], but corrects problems due to singularities in that model. In addition, we investigate our model in more depth by varying the intrinsic growth rate for each of the species and analyzing the resulting bifurcations in system behavior. We are especially interested in transitions between facultative and obligate mutualism. The model reduces to the familiar Lotka-Volterra model locally, but is more realistic for large populations in the case where mutualist interaction is strong. In particular, our model supports population thresholds necessary for survival in certain cases, but does this without allowing unbounded population growth. Experimental implications are discussed for a lichen population.

Original languageEnglish (US)
Pages (from-to)1851-1872
Number of pages22
JournalBulletin of mathematical biology
Issue number8
StatePublished - Nov 2006


  • Bifurcation analysis
  • Differential equations
  • Lotka-Volterra
  • Mathematical model
  • Mutualism


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