Multivariable birth and death process

Question

I read about birth and death process in a paper. The paper then generalized the birth and death process as multivariable. The paper part is like in the picture : . Can anyone explain to me about this part? I totally don't understand. I understand about the one dimension birth and death process though.

Answer 1

$\newcommand\vec[1]{{\mathbf #1}}$ This is a general way to think about master equations: the rate of change of probability of the system being at a state $\vec k$ equals the rate at which the system transitions from some state $\vec l$ to the state $\vec k$ time the probability of the system being at that state $\vec l$ (summed over all such states $\vec l$) minus the probability of the system being already at the state $\vec k$ times the rate of the system transitioning out of $\vec k$ to some other state $\vec l$ (again summed over all such states $\vec l$):

$$\frac{dP(\vec k, t)}{dt} = \sum_{\vec l} T(\vec l\to \vec k)P(\vec l,t) - \sum_{\vec l} T(\vec k\to \vec l)P(\vec k,t), $$

where $T(\vec x\to\vec y)$ is the transition rate from the state $\vec x$ to the state $\vec y$.

In your case, the transitions are due to birth-death processes of $n$ different species. The $j$th birth process only allows the state $(k_1, \dots, k_j-1,\dots, k_n)\equiv \vec E_j^- \vec k$ to transition to the state $\vec k = (k_1, \dots,k_n)$. In your equation, the state dependent rate of this transition is given by $g_j$ which depends on the state that you transition from, that is $\vec E_j^- \vec k$.

Similarly, the $j$th death process only allows the transitions of the state $(k_1, \dots, k_j+1,\dots, k_n)\equiv \vec E_j^+ \vec k$ to the state $\vec k$. The rate of this transition in your equation is given by $r_j(\vec E_j^+ \vec k)$.

Transitions out of the state $\vec k$ are due to $j$th birth process at the rate $g_j(\vec k)$ and $j$th death process at the rate $r_j(\vec k)$ for all $j=1,\dots,n$.

In summary, for the first term, $\sum_{\vec l} T(\vec l\to \vec k)P(\vec l,t)$, transition only from the states $\vec l = \vec E_j^\pm \vec k$ (for all $j$s) are allowed and have the rates $g_j(\vec E_j^- \vec k)$ and $r_j(\vec E_j^+ \vec k)$. This gives you the first two terms of your equation. And for the second term, transitions only to the states $\vec l = \vec E_j^\pm \vec k$ (for all $j$) are allowed, and they have the rates $g_j(\vec k)$ and $r_j(\vec k)$. This gives you the last term in your equation.