Waterfowl Dynamics A population of B ducks lives on two ponds, one large, one small. Let N(t) be the number of birds on the small pond. You may assume that there are B - N(t) birds on the large pond. Let the probability of a departure from the small pond be proportional to the departure rate , to the interval, and to the number of birds on the small pond, Pr[N (t + At) = n - 1 \\ N(t) = n] = ranAt + o(At) . Similarly, assume that the probability of an arrival onto the small pond is proportional to the arrival rate ra, to the interval, and to the number of birds off the small pond, Pr[N (t + At) = n + 1 \\ N(t) = n] = ra(B - n) At + o(At) . (a) Derive a system of differential equations for Pn, the probability of n birds on the small pond. (b) Derive a partial differential equation for the probability generating function