A population of mice is at its lowest in January $(t=0)$ and oscillates 18 above and below an average of 107 during the year. Find an equation for the population, $P$, in terms of the months since January, $t$.

Step 1 ：The population of mice oscillates above and below an average, which means it follows a sinusoidal pattern. The general equation for a sinusoidal function is \(y = A \sin(B(x - C)) + D\), where \(A\) is the amplitude (half the range of the function), \(B\) determines the period of the function, \(C\) is the horizontal shift, and \(D\) is the vertical shift (the average value of the function).

Step 2 ：In this case, the average population is 107, so \(D = 107\). The population oscillates 18 above and below this average, so the amplitude \(A = 18\). Since the population is at its lowest in January and we're measuring \(t\) in months since January, there's no horizontal shift, so \(C = 0\). The period of the function is one year, or 12 months, so \(B = \frac{2\pi}{12}\).

Step 3 ：So the equation for the population is \(P(t) = 18 \sin\left(\frac{2\pi}{12}t\right) + 107\).

Step 4 ：Final Answer: The equation for the population, \(P\), in terms of the months since January, \(t\), is \(\boxed{P(t) = 18 \sin\left(\frac{2\pi}{12}t\right) + 107}\).