Consider a branching process with the following reproduction rules:
- A living parent always generates $m$ children. The position/site of each child is fixed.
- With probability $p$ a child is born alive, with $1-p$ is is born dead.
- If a child is born alive AND its parent's neigbour is dead or immortal, the child is born immortal with probability $q$ and can never reproduce. With probability $1-q$ this child is a normal child and can reproduce.
- Middle children can never become immortal.
- Periodic boundary conditions on the position/sites.
Denote dead = 0, living = 1, immortal = 2.
Example:
Consider $m=3$, the first generation could for example be:
$(0\qquad 1\qquad 1)$
Now the second generation could be the following:
$(0\qquad 211\qquad 012)$
And so on. Note that the first 2 was possible because the left neigbor of the 2's parent was dead. The second 2 was possible because the right neigbor of that 2's parent was dead (periodic boundary condition). Here's a drawing of this process:
Can someone help me find the extinction probability and a generating function for this process? I would appreciate any information you can give me. Thanks!