Modelling growth of bacteria The logistic growth is in the form
 where the constant $r$ defines the growth rate and $K$ is the carrying capacity 
$$
\frac{dP}{dt} = rP\left(1 - \frac{P}{K} \right)
$$
and Monod function is defined as 
$$
\mu = \mu_\max \frac{S}{K_s + S}
$$
What’s the difference between a logistic function and a Monod type function? They both saturate at some level. Mathematically what is the difference between the two functions?  
For example, in this article how is it decided that it is better to use Monod function than logistic growth as both can model the bacterial growth limited resource allocation. 
$$
\frac{dB_s}{dt} = G_s B_s - k_pPB_s - k_i IB_s - \mu B_s
$$
Here $B$ refers to Bacteria and $G_s$ is modelled as $\lambda_s = \Lambda_s \frac{R}{k+R}$
Where $R$ represents resource and $\Lambda _s$ denote maximum bacterial growth rate in the absence of resource limitation.   
But in this article the bacterial growth is modelled as a logistic term 
$$
\frac{dP}{dt} = rP\left(1 - \frac{P}{K_p} \right) - \frac{\gamma_{M_A}f(P,M_A^*)}{1 + k_dD M_A^*}M_A^*P - \gamma_N NP
$$
Is there a basis as to decide which form should be used? 
 A: 1. A logistic evolution $$P'(t)=rP(t)\left(1-\frac{P(t)}K\right)$$ predicts that, for every initial condition $0<P_0<K$, $P(t)$ grows from $P_0$ to $$P_\infty=K$$
2. On the other hand, experiences on bacterial growth led Monod to model the joint evolution of the concentrations of nutrient $S(t)$ and of bacteria $N(t)$ by the differential system $$N'(t)=\mu_\max N(t)\frac{S(t)}{K_s+S(t)}\qquad cS'(t)=-N'(t)$$ where $c>0$ is the growth yield. When $c$ is assumed constant, one gets $$N(t)=N_0+cS_0-cS(t)$$ hence $$N'(t)=\mu_\max N(t)\frac{cS_0+N_0-N(t)}{cK_s+cS_0+N_0-N(t)}$$ which predicts that $N(t)$ grows from $N(0)=N_0$ to $$N_\infty=N_0+cS_0$$
3. Thus, in the logistic model, the limiting quantity $P_\infty$ does not depend on its initial value $P_0$ but depends on the carrying capacity $K$, while, in Monod's model, the limiting quantity $N_\infty$ depends on its initial value $N_0$ but not on the analogue $K_s$ of the carrying capacity. This may help to discriminate these two classes of models.
A: The first equation is a differential equation for P.  It's general solution is $P(t)= \frac{CKe^{rt}}{1+ Ce^{rt}}$ where "C" is any constant.  If we take S to be $Ce^{rt}$, that becomes $P(t)= \frac{KS}{1+ S}$ which is of the "monod" form.
