Stability and Bifurcation Structure of a Discrete-Time Chemostat Model with Nutrient Recycling

This work aims to study the dynamic behavior of a discrete-time chemostat model involving the effect of nutrient recycling, which is a common biological phenomenon usually left out in conventional models. Based on a continuous time model involving the interrelationship between bacterial population and nutrient concentration, a new system with nutrient recycling term is obtained by introducing the process of nutrient recycling from biomass. This model is subsequently transformed to a discrete form using the forward Euler approximation scheme. Existence and stability of all possible equilibria are examined using the characteristic equation. Necessary conditions for stability and bifurcations, including transcritical and flip types, have been derived. It turns out that the considered model cannot experience Neimark–Sacker bifurcation. The results indicate that the recycling parameter plays an important role in changing stability regions and can generate complex dynamics, including oscillations and chaos. Numerical simulations are used to verify the analytical results and to show transitions between different types of system behavior. The results imply that the relationship between nutrient input, bacterial proliferation, and the recycling process has an essential influence on the dynamics and improves the comprehension of discrete ecological models.