The transition probabilities of a bounded bivariate pure death process

Billiard [1] derived an expression for the transition probabilities of a bounded pure death process describing the competition between two species for survival. This expression involves some recursively defined constants. In this paper we derive a simplified nonrecursive expression for the transition probabilities of any bounded bivariate pure death process. We apply our result to Billard's competition model, to a stochastic prey-predator model based on the original Lotka-Volterra equations, and to the Weiss pre-predator model. Finally, we consider two bivariate generalizations of the simple stochastic epidemic model.