Question

Suppose that a one-celled organism can be in one of two states – either A or B. An individual in state A will change to state B at an exponential rate α; an individual in state B divides into two new individuals of type A at an exponential rate β. Define an appropriate CTMC for a population of such organisms and determine the appropriate parameters for this model.

Answer 1

P{x,y} {x - 1, y + 1} = βy/ αx + βy.

P{x,y} {x + 1, y - 1} = βx/ αx + βy.

Step-by-step explanation:

The full meaning for the acronym "CTMC" given in the question is known as the continuous-time Markov chain. The continuous-time Markov chain is an important tool or instrument which is used in the dealings with stochastic process.

Hence, Xa(t) is the number of organisms in state A, Xb(t) is the number of organisms in state B. Therefore, we can say that the continuous-time Markov chain(CTMC) is;

{ Xa(t), Xb(t)}.

Thus, b(x,y) = αx + βy.

Therefore, we will have;

P{x,y} {x - 1, y + 1} = βy/ αx + βy.

P{x,y} {x + 1, y - 1} = βx/ αx + βy.

Note that, if we are to use the generation of matrix to get differential equation as below;

I(x)α(ix - 1, iy + 1) (JxJy)(t) + iyβ

P (ix + 2, iy -1) (Jx,Jy) (t) - (ixα + iyβ) P(ix,iy)(JxJy) (t).

t is greater or equals to zero for ix,iy, Jx and Jy.