In a seasonal-growth model, a period function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for examples, be caused by seasonal changes in the availability of food.

(a) Find the solution of the seasonal-growth model

$ \frac {dP}{dt} = kP \cos(rt - \phi) P(0) = P_0 $

where $ k, r, $ and $ \phi $ are positive constants.

(b) By graphing the solution for several values of $ k, r, $ and $ \phi, $ explain how to the values of $ k, r, $ and $ \phi $ affect the solution. What can you say about lim $ _{t \to \infty} P(t)? $