From pair-wise interactions to triplet dynamics

Abstract

The present thesis investigates the properties of a system of ordinary di_erential equations, that describes cross-feeding in two- and three-species systems of bacteria. The system is studied statistically and the probability of permanence | a stable state where no species is driven extinct | is computed under the assumption that the energy-uptake parameters of the system are either independent or organised in a hierarchy where any excreted metabolites carry less energy than previous nutrients. For a system of two species, we derive the probability of permanence analytically. For three-species systems, we di_erentiate between di_erent modes of coexistence with respect to boundary behaviour of the system. We are able to show that the a_ne _tness function described by Lundh & Gerlee (Lundh, T., Gerlee, P., Bull Math Biol, 75, 2013) is equivalent to the linear _tness function investigated by Bomze (Bomze, I. M., Biol Cybern, 48, 1983) and hence that the dynamics derived by Bomze holds for the cross-feeding paradigm of Lundh & Gerlee. For the question implicit in the title of the thesis, the pair-wise interactions of a three-species system are not enough to draw any deterministic conclusions on permanence of the triplet. We _nd, however, that the probability of permanence is close to 50% for systems with three coexistent pairs on the boundary and for so-called intransitive systems. Systems with two and one coexistent pairs on the boundary are more likely to exist for random interactions parameters, but are not as likely to be permanent. Keywords: Biomathematics, Population dynamics, Cross-feeding. v