Study of a virus–bacteria interaction model in a chemostat: application of geometrical singular perturbation theory

Abstract

This paper provides a mathematical analysis of a virus-marine bacteria interaction model. The model is a simplified case of the model published and used by Middelboe (Middelboe, M. 2000 Microb. Ecol. 40, 114-124). It takes account of the virus, the susceptible bacteria, the infected bacteria and the substrate in a chemostat. We show that the numerical values of the parameters given by Middelboe allow two different time scales to be considered. We then use the geometrical singular perturbation theory to study the model. We show that there are two invariant submanifolds of dimension two in the four-dimensional phase space and that these manifolds cross themselves on the boundary of the domain of biological relevance. We then perform a rescaling to understand the dynamics in the vicinity of the intersection of the manifolds. Our results are discussed in the marine ecological context.