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 $4 \mathrm{yr}$, initial population of 500 . Find an exponential function of the form $P(t)=P_{0} n^{\frac{t}{T}}$ that models the situation.

Step 1 ：Given that the initial population is 500, the population doubles every 4 years, and the general form of the function is \(P(t)=P_{0} n^{\frac{t}{T}}\).

Step 2 ：Substitute the given values into the general form to get the specific function that models this situation. Here, \(P_{0}\) is 500, \(n\) is 2 (since the population doubles), and \(T\) is 4 years.

Step 3 ：Substituting these values into the equation gives us the function \(P(t)=500 \cdot 2^{\frac{t}{4}}\).

Step 4 ：\(\boxed{P(t)=500 \cdot 2^{\frac{t}{4}}}\) is the exponential function that models the situation.