An Approach Using Linear Regression

Divide equation (3) by $Y$ to get

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

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

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

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

$$\left\{ \left( \frac{(Y_{i+1}+Y_i)}{2} , \ 2\frac{Y_{i+1}-Y_i}{(Y_{i+1}+Y_i)(t_{i+1}-t_i)} \right)\right\}_{i=1}{N}$$

against the line

$$r - \frac{r}{K} Y$$

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