How to interpret this population growth model? I was working on this equation $$\frac{d P}{d t}=\frac{v P}{K+P}-d P$$
where $P$ is population density and $v,d,K$ is constant, the stedy-state of this equation is $P_1=0$ or $P_2=\dfrac{v}{d}-K$.
If we let $\frac{d P}{d t}=f(P)$, and take derivative wrt $P$, we get $\frac{df}{dP}=\frac{v}{P+K}-\frac{v P}{(P+K)^{2}}-d$, plug the stedy-state, we have $\dfrac{df(P_1)}{dP}=\dfrac{v}{K}-d$ and $\dfrac{df(P_2)}{dP}=\dfrac{Kd^2}{v}-d$, what is the biological meaning of the two steay-state if one of them is stable in terms of the three constant.
And how to interpret what this equation saying, I mean, I don't know the meaning of the three constant and the derivation of this equation. 
 A: If we linearize the ODE in the vicinity of the origin, we have
$$
\frac{d P}{ dt} = v\, \frac{P/K}{1+P/K} - d P \underset{(P\to 0)}{\simeq} \left(\frac{v}{K} - d\right) P .
$$
The coefficient $r = v/K - d$ is the growth rate at origin, where $v/K$ may be viewed as a birth rate and $d$ as a mortality rate. Interpreting the coefficient $d>0$ as a mortality rate is common in various dynamical systems (such as the SIR epidemiology model, for instance). 
One notes that $dP/dt$ vanishes at the origin $P=0$ and at $P=rK/d$, which are the steady states. Thus, by analogy with Verhulst's logistic growth model, we may rename the steady state $\kappa = rK/d$ as the carrying capacity. The dynamical system rewrites as
$$
\frac{d P}{ dt} = r P\, \frac{1 - P/\kappa}{1+ P/K} \, .
$$
One recognizes a modified logistic growth model, where the point of maximum growth is $P = \frac{\kappa}{1 + (1 + \kappa/K)^{1/2}}$. The corresponding instantaneous rates $dP/dt$ are represented below with respect to $P$ for several values of $K$ (and $r=1$, $\kappa = 1$). One notes that the curves are non-symmetric with respect to the maximum. Qualitatively, this model is similar to the $\theta$-logistic model.

The parameters $r$, $\kappa$ defined here are related to the original parameters $v$, $K$, $d$ through
$$
\begin{aligned}
r &= v/K - d \\
\kappa &= v/d - K
\end{aligned}
\qquad\text{i.e.}
\qquad
\begin{aligned}
v &= (\kappa + K)\, K\tfrac{r}{\kappa} \\
d &= K \tfrac{r}{\kappa}
\end{aligned}
$$

EDIT: Thanks to @Paichu for the comment mentioning Michaelis-Menten kinetics. Thus, the first term ${vP}/(K+P)$ of the growth rate is of Michaelis-Menten type (maximal growth rate $v$, half saturation $K$), and the second term $-dP$ is an additional mortality term. Similar terms are found in the Jacob-Monod predator-prey model.
