Direct Nonlinear Regression

If you are a mötorhead compu-stat monster, or just have access to good software (like Mathematica), you may regress directly, fitting the data,

$$\left\{(t_i, Y_i) \right\}_{i=1}{N},$$

to the solution (4),

$$Y = \frac{K Y_0 e^{rt}}{K + Y_0 \left(e^{rt} - 1\right) },$$

to determine $r$ and $K$. You can either put your initial observation for $Y_0$ in, or let it be a regression parameter and check it against your actual observed value to provide some measure of validation.