Suppose autoinducer $A$ diffuses across the cell membrane, and that the local volume fraction of cells is $\rho$. Then,

$$(1-\rho)({dE\over dt}+k_EE) = \delta (A-E).$$ (9)

$${\frac {dA}{dt}} = -k_{RA}R A +k_PP+k_2L-k_AA -{\delta\over \rho}(A-E).$$ (10)

Question: Does this system exhibit quorum sensing?

Take everything but $A$ and $R$ to be in steady state:

$${\frac {dR}{dt}} = V_R\frac{P}{K_R+P} - k_RR +R_0.$$ (11)

$${\frac {dA}{dt}} = V_A\frac{P}{K_A+P} +A_0 -d(\rho)A.$$ (12)

where $P = {k_{RA} R A\over k_P}$ and $d(\rho) = k_A +{\delta\over \rho}({ k_E(1-\rho)\over \delta + k_E(1-\rho)})$.

Figure: Nullclines ${dR\over dt} = - k_RR + \frac{V_RRA}{K_R+RA} +R_0 =0$ and ${dA\over dt} = \frac{V_ARA}{K_A+RA} +A_0 -d(\rho)A =0$ with three different values of $\rho$, shown for parameter values $V_R = 2.0, V_A = 2.0, K_R = 1.0, K_A = 1.0, R_0 = 0.05, A_0 = 0.05, \delta = 0.2, k_E = 0.1, k_R = 0.7, k_A = 0.02$.