We consider the birth-and-death process that was defined in Quiz 6:

Let a, b > 0 be real constants. Consider a population of amoeba where at every point in

time each amoeba either: does not change state, or dies, or divides into three children

(i.e. one division corresponds to +2 amoebas). We denote the number of amoeba at

time t 0 by Mt

. The initial number of amoeba M0 is a fixed number in N. Suppose

that amoebas divide or die independently of each other, and that births and deaths

follow these rules:

(i) P(Mt+h = k + 2 | Mt = k) = kbh + o(h)

(ii) P(Mt+h = k 1 | Mt = k) = kah + o(h)

(iii) P(Mt+h = k | Mt = k) = 1 k(a + b)h + o(h)

(iv) P(Mt+h = l | Mt = k) = o(h) for all l / {k 1, k, k + 2}.

Define pk(t) = P(Mt = k).

(a) For t > 0 and small h > 0, give a formula for pk(t + h) in terms of other pm(t) for

suitable m N.

(b) Use Part (a) to obtain a formula for p

0

k

(t) in terms of the same pm(t).

(c) Let E(t) := E(Mt). Use Part (b) to prove E0

(t) = (2b a)E(t).

(d) (Bonus) Use Part (c) to give E(t).