Existence Of Steady-state Distribution Research Articles

The multitype branching random walk, II

Limit theorems for the multitype branching random walk as n → ∞ are given ( n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of Kesten and Stigum are given.

On some Markov models of certain interacting populations.

We give a stochastic foundation to the Volterra prey-predator population in the following case. We take Volterra's predator equations and let a free host birth and death process support the evolution of the predator population. The purpose of this article is to present a rigorous population sample path construction of this interacted predator process and study the properties of this interacted process. The constructions yields a strong Markov process. The existence of steady-state distribution for the interacted predator process means the existence of equilibrium population level. We find a necessary and sufficient condition for the existence of a steady-state distribution. Next we see that if the host process possesses a steady-state distribution, so does the interacted predator process and this distribution satisfies a difference equation. For special choices of the auto death and interaction parametersa andb of the predator, whenever the host process visits the particular statea *=a/b the predator takes rest (saturates) from its evolution. We find the probability of asymptotic saturating of the predator.

Steady state theory of hot atom reactions

A simple steady state theory of hot atom reactions is developed based on the Boltzmann equation. The solution to this equation is approximated by a local Maxwell distribution involving the temperature of the hot atoms and steady state distributions are calculated by determining steady values of the hot atom temperature. General considerations imply the existence of steady state distributions with temperatures just below the ambient temperature and a simple model calculation indicates that very reactive systems should have steady or quasisteady states at temperatures much higher than ambient. An application of the theory to a photochemical reaction is shown to be compatible with more extensive computer calculations. If a hot atom distribution becomes steady, it is shown that the loss of hot atoms can be described by a pseudo-first-order rate equation with a rate constant differing from the equilibrium rate constant only by the appearance of the steady state temperature.