A modeling approach of the chemostat

Population dynamics and in particular microbial population dynamics, though intrinsically discrete and random, are conventionally represented as deterministic differential equations systems. In these type of models, populations are represented by continuous population sizes or densities usually with deterministic dynamics. Over the last decades, alternate individual-based models have been proposed where population is explicitly represented as a set of individuals. These may include stochastic dynamics or stochastic rules. With reference to the last class of models we can also associate pure jump processes where the population is described as a discrete population size with stochastic discrete event evolutions. In the first class of models the population dynamics and its representation may be viewed respectively as deterministic and continuous, in the second class they may be viewed respectively as stochastic and discrete. In this present work, we present a modeling approach that bridges the two representations. This link can be mathematically described as a functional law of large numbers in high population size asymptotics. These results suggest new strategies of modeling and simulation. We illustrate this approach on the modeling of the chemostat.

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