I'm studying Morkov chains and I have a question about immigration process. Let's say I have two groups $X$ and $Y$ each individual of these groups give birth with the same rate $b$ and members of $X$ can leave their group and join to $Y$ with rate $l$. I define the transition with following table:
I use birth death process for the transition probability for $X$ and get $X^{'}_{i}(t)=b(i-1)X_{i-1}(t)+l(i+1)X_{i+1}(t)-(b+l)iX_{i}(t)$. I think it is true but I am not sure about the second transition: $Y^{'}_{j}(t)=b(j-1)Y_{j-1}(t)+l(j-1)Y_{j-1}(t)-(b+l)jY_{j}(t)$ Could you please help me to find out my mistake?