A multivariate competition process (M.C.P.) is a stationary, continuous time Markov process whose state space is the lattice points of the positive orthant in $N$-dimensional space and whose transition probability matrix only allows jumps to certain nearest neighbors. As such it is the natural generalization of birth and death processes. In this paper we extend the results of Reuter  to obtain sufficient conditions for a M.C.P. to be regular, positive recurrent, absorbed with certainty, and to have finite mean absorption time. Some explicit examples are given and references to various applications indicated.
"Multivariate Competition Processes." Ann. Math. Statist. 35 (1) 350 - 361, March, 1964. https://doi.org/10.1214/aoms/1177703758