Geodesic
On the probability of the extinction of branching process with interaction of particles
Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 192-197 
A. V. Kalinkin. On the probability of the extinction of branching process with interaction of particles. Teoriâ veroâtnostej i ee primeneniâ, Tome 27 (1982) no. 1, pp. 192-197. http://geodesic.mathdoc.fr/item/TVP_1982_27_1_a21/
@article{TVP_1982_27_1_a21,
     author = {A. V. Kalinkin},
     title = {On the probability of the extinction of branching process with interaction of particles},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {192--197},
     year = {1982},
     volume = {27},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1982_27_1_a21/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.