Find an exponential function of the form $P(t)=P_{0} n^{\frac{t}{T}}$ that models the situation, and then find the equivalent exponential model of the form $P(t)=P_{0} e^{r t}$. Doubling time of $3 \mathrm{yr}$, initial population of 550 . Find an exponential function of the form $P(t)=P_{0} n^{\frac{t}{T}}$ that models the situation. The exponential function is $\mathrm{P}(\mathrm{t})=$ (Use integers or fractions for any numbers in the expression.)

Step 1 ：Given that the initial population is 550, the population doubles every 3 years, and the general form of the exponential function is \(P(t)=P_{0} n^{\frac{t}{T}}\).

Step 2 ：Substitute the given values into the general form to get the specific exponential function for this situation. Here, \(P_{0} = 550\), \(n = 2\) (since the population doubles), and \(T = 3\) years.

Step 3 ：Substituting these values into the equation gives us \(P(t)=550 \cdot 2^{\frac{t}{3}}\).

Step 4 ：\(\boxed{P(t)=550 \cdot 2^{\frac{t}{3}}}\) is the exponential function that models the situation.