Steady-State Analysis of a Competitive Age-Structured Population System with Two Inputs

Age-structured population models are an intuitive way to model competing bacteria populations in bioreactors, and are of interest for biotechnology processes, wastewater treatment, or epidemics. Such multi-population models with competition terms result in coupled partial differential equations with integral terms and non-local boundary conditions. They represent the population density of each species at a specific time and age. In this work, a model to represent two intra-and interspecific competing populations in a bioreactor is introduced. It has two system inputs, namely the dilution rate with nutrient solution and a recycling rate which introduces biomass in a steady-state from a second bioreactor. Adding a recycling rate to the multi-population models allows for influencing not only the entire biomass in the bioreactor but also the age distribution of the bacteria. In order to use population models to develop improved control concepts for such a cascaded bioreactor experiment, an extensive steady-state analysis is carried out. There exists an infinite number of steady-states of the system in dependence of the initial condition. Each choice of that initial condition lead to uniquely determined steady-state profiles, inputs and outputs. In a next step, a stabilization around these steady-states is necessary.