# Logistic Growth: Mean Field Equation

I was told a while ago that for the logistics growth process: $$x \underset{k_2}{\stackrel{k_1}{\rightleftharpoons}}x+x$$the mean field equation for the population $n$ is given by: $$\frac{d\bar n}{dt}=k_1\bar n-\frac{k_2}{V} \bar n^2$$ where $V$ is the volume. There are two things that I don't understand about this equation - both related to the second term on the RHS:

1. The presence of the volume $V$.
2. The fact we have $\bar n^2$ rather then $\bar n(\bar n-1)$ which would seem more appropriate for two particles coagulating.

Please can someone explain the reasons for these.

• Why would a general equation for population growth refer to only two particles ? – StephenG Mar 12 '18 at 9:51
• @StephenG sorry it doesn't. By $x \underset{k_2}{\stackrel{k_1}{\rightleftharpoons}}x+x$ I mean that any particle can split into two and any two particles can join to form one. E.g. particle 8 may defined into particle 9 and 10 and then 10 go onto divide, etc. I am not just considering a two state system. – Quantum spaghettification Mar 12 '18 at 11:25