I am reading ahead in some lecture notes that give the Beverton-Holt model with zero harvest:
$r_t=\dfrac{ar_{t-1}}{b+r_{t-1}}$.
It states that under a standard assumption of $a>b>0$, the 'intrinsic growth rate' is $r=\frac{a}{b}-1$. I am wondering what this means and how to show this?
I can show the carrying capacity is $a-b$ but struggling with understanding what the intrinsic growth rate could be? I have tried doing the ratio of $\frac{r_t}{r_{t-1}}$ and also tried differentiating.
I am not familiar with either the model nor the parameters you're talking about, but it would seem to me that you should obtain an analytic solution for $r_t$ in order to ascertain its properties. To that end, the Wikipedia article on the Beverton-Holt Model suggests that the nonlinear model you are looking at can be converted to a linear inhomogeneous form by a change of variables, ley's say, $f=1/r$. Then you can show that
$$f_{t+1}=\frac{b}{a}f_t+\frac{1}{a}$$
This can be converted to a homogeneous form by indexing $t+1\mapsto t+2$ and subtracting to eliminate to inhomogeneous term $1/a$. You then obtain
$$f_t=\bigg(1+\frac{b}{a}\bigg)f_{t-1}-\frac{b}{a}f_{t-2}$$
wuth the initial conditions $f_0=1/r_0$ and $f_1=b/a\cdot f_0+1/a$. This is a standard Fibonacci-type sequence. I have commented frequently on the solution to these types of equations, giving detailed analytic solutions. See, for example, Generalized Fibonacci Sequence. Can you take it from here? I have verified this solution numerically (comparing the sequence and the Fibonacci forms) with random values of $a$ and $b$. I'll also note that, mathematically at least, you are not confined to $a>b>0$.