A species of fish was added to a lake. The population size $P(t)$ of this species can be modeled by the following function, where $t$ is the number of years from the time the species was added to the lake. \[ P(t)=\frac{800}{1+6 e^{-0.15 t}} \] Find the initial population size of the species and the population size after 9 years. Round your answers to the nearest whole number as necessary.

Step 1 ：The initial population size can be found by substituting \(t=0\) into the function \(P(t)\).

Step 2 ：The population size after 9 years can be found by substituting \(t=9\) into the function \(P(t)\).

Step 3 ：Using these substitutions, we find that the initial population size is \(P(0) = 114\) and the population size after 9 years is \(P(9) = 313\).

Step 4 ：Final Answer: The initial population size of the species is \(\boxed{114}\) and the population size after 9 years is \(\boxed{313}\).