Date Published: March 27, 2019
Publisher: Public Library of Science
Author(s): Dimitrios Voulgarelis, Ajoy Velayudhan, Frank Smith, Chengming Huang.
This paper considers a novel dynamical behaviour of two microbial populations, competing in a chemostat over a single substrate, that is only possible through the use of population balance equations (PBEs). PBEs are partial integrodifferential equations that represent a distribution of cells according to some internal state, mass in our case. Using these equations, realistic parameter values and the assumption that one population can deploy an emergency mechanism, where it can change the mean mass of division and hence divide faster, we arrive at two different steady states, one oscillatory and one non-oscillatory both of which seem to be stable. A steady state of either form is normally either unstable or only attainable through external control (cycling the dilution rate). In our case no external control is used. Finally, in the oscillatory case we attempt to explain how oscillations appear in the biomass without any explicit dependence on the division rate (the function that oscillates) through the approximation of fractional moments as a combination of integer moments. That allows an implicit dependence of the biomass on the number of cells which in turn is directly dependent on the division rate function.
In this paper we have taken a different angle of approach to the classical two populations with one substrate in a chemostat system using population balance equations instead of ODE. We wanted to investigate under which conditions can more complex models allow for a stable coexistence state that would arise solely from internal interactions. From our work it seems that one of the simplest cases where coexistence is stable is when an adaptive response to one of the populations is included. That response emerges from the ability of one of the populations to sense their biomass and adapt when it crosses some critical value by changing the mean mass of division. By formulating two models for this adaptive response competition, one semi-equation based and the other purely equation based whose dynamics are very similar, we showed that, with realistic growth parameters and dilution rate, two steady-states are possible depending on whether or not there is any delay in the sensing. The stability of these steady-states was explored through different means, namely stochastic simulations and parameters sweep. Finally, building on the work of Alexiadis et. al.  we were able to explain how oscillations appears in biomass equations.
Our work has been on the investigation of coexistence in a chemostat from a different modelling aspect and an effort to identify the path of least assumptions to achieve that. That led to the use of PBE models and the assumption of an adaptive response mechanism that only affects the mean mass of division of the cell population that deploys it. With that we were able to show that coexistence is possible and stable and that depending on the biological premises this can have different forms, an oscillatory and a non-oscillatory form. The stability is proven through stochastic simulations and parameter sweeps and we indeed observe that the steady states are reached for a wide range of noise intensities as well as a wide range of two important parameters, the dilution rate and response time.