So, namely i have $5$ models. And as we know the Malthusian model has form $$\frac{dP}{dt} = kt$$ And the solution of the ODE above is $$P(t) = ce^{kt}$$. And the formula for Relative Growth Rate (RGR) from Wikipedia is $$RGR = \frac{\ln(P(t_2)) - \ln(P(t_1))}{t_2-t_1}$$ Where $t_2$ and $t_1$ represent the year of the population. E.g. $t_2=2021$ and $t_1=2000$. We know we need $2$ points to determine the solution of Malthusian model, and the informations $t_2$ and $t_1$ have contribution on this.
And as i said before, i have 5 models means i use different $t_2$ and $t_1$ as their initial conditions. And i realized that the smaller the RGR value, the more accurate it is (at least on the nearest interval, suppose $t_2=2021$ and $t_1=2000$, the nearest interval is between $1980$ and $2025$). It is evident from the plot data I have drawn (I will upload if it's necessary). Can anyone confirm this thought, or is this just a coincidence?
Thanks in advance!
Edit:
In this edit section, i'd like to include the calculation of RGR and compare the results with the predicted data plot and the real (actual) data.
Here is my calculation of RGR on several models:
And the data plots:
$RGR=0.0195738$ :
$RGR=0.016577$
$RGR=0.0159$
For convenient reason, i only give 3 of them. And as you can see, the smaller RGR gives me better prediction on given interval (Might not accurate on another interval). Anyone can explain about this if this is just a conincidence or not? Thanks
The wikipedia formula for RGR is actually the calculation of the constant $k$ when you know two points on the exponential curve.
The fact that the fit happens to look best when you use any two particular points to find $k$ is a coincidence. The right way to find the "best" $k$ is to fit a linear regression line to the logarithmic plot of the data. Then $k$ is the slope of that line.
That's what Excel does when you ask it for an "exponential trendline". It's what you should do in this example.