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An Approach Using Linear Regression

Divide equation (3) by $Y$ to get

\frac1Y \frac{dY}{dt} = r -\frac{r}{K} Y.
\end{displaymath} (5)

Approximate $Y^{-1} \dot{Y}$ by the average rate of change between observations divided by the average population between observations,

\left(\frac1Y \frac{dY}{dt} \right)_i \approx
2\frac{Y_{i+1}-Y_i}{(Y_{i+1}+Y_i)(t_{i+1}-t_i)} .

This gives a relationship between the right-hand-side of (5) and your data. To be consistent $Y$ in the left-hand-side should also be approximated using the average $\frac12(Y_{i+1}+Y_i)$. Now, to get an estimate for parameters $r$ and $K$ one should regress the data

\left\{ \left( \frac{(Y_{i+1}+Y_i)}{2} , \

against the line

r - \frac{r}{K} Y

to determine $r$ and $\frac{r}{K}$.

James Powell