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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.

This paper formulates two 3D stochastic differential equations (SDEs) of two microbial populations in a chemostat competing over a single substrate. The two models have two distinct noise sources. One is general noise whereas the other is dilution rate induced noise. Nonlinear Monod growth rates are assumed and the paper is mainly focused on the parameter values where coexistence is present deterministically. Nondimensionalising the equations around the point of intersection of the two growth rates leads to a large parameter which is the nondimensional substrate feed. This in turn is used to perform an asymptotic analysis leading to a reduced 2D system of equations describing the dynamics of the populations on and close to a line of steady states retrieved from the deterministic stability analysis. That reduced system allows the formulation of a spatially 2D Fokker-Planck equation which when solved numerically admits results similar to those from simulation of the SDEs. Contrary to previous suggestions, one particular population becomes dominant at large times. Finally, we briefly explore the case where death rates are added.

For certain values of the parameter space a close match between the equation-based and agent-based models is achieved. The need for division of labour between the two approaches is explored.

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