  In quorum sensing (QS), bacteria exchange molecular signals to work together.
An analytically-tractable model is presented for characterizing QS signal
propagation within a population of bacteria and the number of responsive
cooperative bacteria (i.e., cooperators) in a two-dimensional (2D) environment.
Unlike prior works with a deterministic topology and a simplified molecular
propagation channel, this work considers continuous emission, diffusion,
degradation, and reception among randomly-distributed bacteria. Using
stochastic geometry, the 2D channel response and the corresponding probability
of cooperation at a bacterium are derived. Based on this probability, new
expressions are derived for the moment generating function and different orders
of moments of the number of cooperators. The analytical results agree with the
simulation results obtained by a particle-based method. In addition, the
Poisson and Gaussian distributions are compared to approximate the distribution
of the number of cooperators and the Poisson distribution provides the best
overall approximation. The derived channel response can be generally applied to
any molecular communication model where single or multiple transmitters
continuously release molecules into a 2D environment. The derived statistics of
the number of cooperators can be used to predict and control the QS process,
e.g., predicting and decreasing the likelihood of biofilm formation.
