Self-sustained dynamics by modeling competing PHA-producers and non-PHA-producers bacteria population for a limited resource: local and homoclinic bifurcation analysis

We propose a mathematical model for two species competing for a limited resource associated to polyhydroxyalkanoate (PHA) production, which possesses regime of stable states: stable equilibrium, and periodic and aperiodic oscillations. Such regimes of stable oscillations are absent in the model without taking into account PHA production but is known to exist in experimental model associated to the production of PHA. It explains the capacity of the system to sustain itself at the lowest value of resource. Thus, the proposed system provides a simpler four-dimensional model containing monod functions with such behaviours. Using analytical tools and numerical bifurcation analysis, we describe parameter regions and bifurcation structures leading to the existence and the coexistence between stable, unstable equilibrium and limit cycle. These explain critical parameter sensitivity impact on the process. Considering the effects on the proposed system under a frequent alternation of the input resource, we investigate how the increase of the length of the feast period in the Feast-Famine conditions, increases the PHA-production or decreases the lowest value of the resource at the equilibrium.