In a study to develop the growth behavior for protozoa colonization in a particular lake, an experiment was conducted in which 15 sponges were placed in a lake and 3 sponges at a time were gathered. Then the number of protozoa were counted at 1, 3, 6, 15, and 21 days. In this case the MacArthur-Wilson equation was used to describe the growth mechanism. The model is given by
\[
y=\beta_{1}\left(1-e^{-\beta_{2} t}\right)
\]
where
\(y=\) total protozoa on the sponge
\(\beta_{1}=\) species equilibrium constant
\(\beta_{2}=\) parameter that measures how quickly growth rises
\(t=\) time, number of days
The data (collected by the Department of Biology at Virginia Tech) are as follows:

a. Estimate \(\beta_{1}\) and \(\beta_{2}\) using nonlinear regression. Supply your own starting values of the parameters.
b. Give estimated standard errors of the parameter estimates.

Transcribed Image Text:

Observation Day y (Total Protozoa) 11 1234567RADUAT 1 17 1 21 1 16 333 30 3 25 25 6 33 6 31 9 6 32 10 15 34 15 33 15 33 21 39 21 35 15 21 36