Mean-field equation for a stochastic many-particle model of quorum-sensing microbial populations

We prove a mean-field equation for the dynamics of quorum-sensing microbial populations. In the stochastic many-particle process, individuals of a population produce public good molecules to different degrees. Individual production is metabolically costly such that non-producers replicate faster than producers. In addition, individuals sense the average production level in the well-mixed population and adjust their production in response ("quorum sensing"). Here we prove that the temporal evolution of such quorum-sensing populations converges to a macroscopic mean-field equation for increasing population sizes. To prove convergence, we introduce an auxiliary stochastic mean-field process that mimics the dynamics of the mean-field equation and that samples independently the individual's production degrees between consecutive update steps. This way, the law of large numbers is separated from the propagation of errors due to correlations. Our developed method of an auxiliary stochastic mean-field process may help to prove mean-field equations for other stochastic many-particle processes.