Bulletin of Mathematical Biology

, Volume 75, Issue 5, pp 709–724

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

Original Article

DOI: 10.1007/s11538-013-9828-3

Cite this article as:
Lundh, T. & Gerlee, P. Bull Math Biol (2013) 75: 709. doi:10.1007/s11538-013-9828-3


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 

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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