So I'm starting out learning how to parameterize ODE models with demographic data. I'm having a slight problem with parameterizing birth rates due to some various ways I have seen?
For simplicity sake, let's assume a differential equation below.
$$ \frac{dN}{dt} = bN(t)$$
$N(t)$ is the total population at time $t$ and $b$ is the per capita birth rate.
To parameterize this, let's say that I had some real-life data that says there is an initial population of $500$ and one year later there were $100$ new births. I have seen people calculate the birth rate, $b$, as new births/initial population so in this case $100/500$ so the birth rate is $0.20$ births/year. This is one approach I found.
But, this is an approach that I thought was the best way. I would calculate the birth rate by knowing that the solution to the above equation is
$$N(t) = N_0\exp(bt)$$
And replacing $N(t)$ with $600(500\text{ initial} + 100\text{ births})$ and $N_0$ with $500$.
Then taking the natural log,
$$\ln\left(\frac{N(t)}{N_0}\right) = bt$$
We're assuming $t$ is $1$ (for one year).
I get that $b$ is $0.1823216$, which while close to the $0.20$ birth/year above, it's not the same.
So I'm wondering what the correct approach is.