Bulletin of Mathematical Biology

, Volume 75, Issue 5, pp 709–724 | Cite as

Cross-Feeding Dynamics Described by a Series Expansion of the Replicator Equation

Original Article


Understanding how ecosystems evolve and how they respond to external perturbations is critical if we are to predict the effects of human intervention. A particular class of ecosystems whose dynamics are poorly understood are those in which the species are related via cross-feeding. In these ecosystems the metabolic output of one species is being used as a nutrient or energy source by another species. In this paper we derive a mathematical description of cross-feeding dynamics based on the replicator equation. We show that under certain assumptions about the system (e.g., high flow of nutrients and time scale separation), the governing equations reduce to a second-order series expansion of the replicator equation. By analysing the case of two and three species we derive conditions for co-existence and show under which parameter conditions one can expect an increase in mean fitness. Finally, we discuss how the model can be parameterised from experimental data.


Syntrophy Cross-feeding Population dynamics 


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Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Sahlgrenska Cancer CenterInstitute of MedicineGöteborgSweden
  2. 2.Mathematical SciencesUniversity of Gothenburg and Chalmers University of TechnologyGöteborgSweden

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