  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.
