On the probability of the extinction of branching process with interaction of particles

We consider Markov chains with state space 0, 1, 2, and transition probabilities
$$ p_{ij}(t)= \begin{cases} i(i-1)\dots(i-k+1)p_{j-i+k}t+o(t), &j\ge i-k,\ j\ne k\\ 1+i(i-1)\dots(i-k+1)p_kt+o(t), &j=i\\ o(t) &j<i-k \end{cases} $$
where $t\to 0$, $p_i\ge 0\ (i\ne k)$, $p_k<0$, $\displaystyle\sum_{i=1}^\infty p_i=0$, the number $k$ is fixed. Such chains may be considered as branching processes with interaction of particles. The probability of extinction of such chain is investigated.