Question
Let a, b > 0 be real constants. Consider a population of amoeba where at every point in time each amoeba either: does not change
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).
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