I would like to know what is the purpose of using the term $P\over (k+P)$ in the following. I found it when reading the article found here but it was commonly used in few other related articles .
Is this similar to a logistic function?
What is the advantage of using this instead of a logistic function?
If this function can saturate, which of the terms does it?
I would be grateful if someone can clarify these.
You are right! the $sX\frac{P}{k+P}$ term is precisely the one that saturates. As $P\to \infty$, this terms goes to $sX$, and so $\frac{dX}{dt}$ remains bounded.
Another way to look at it, is to solve for $X(P)$, yielding
$$\frac{dX}{dP}=\frac{a+sX\left(\frac{P}{k+P}\right)-dX}{rP-hPX} \, .$$
For a given $X(0)$, As $P\to \infty$, you get that $\frac{dX}{dP} \sim O(\frac{1}{P})$ and so the growth in $P$ hardly effects the marginal growth in $X$.
