On some Markov models of certain interacting populations

✍ Scribed by D. Kannan

• Publisher: Springer
• Year: 1976
• Tongue: English
• Weight: 677 KB
• Volume: 38
• Category: Article
• ISSN: 1522-9602
• DOI: 10.1016/s0092-8240(76)80012-2

✦ Synopsis

We give a stochastic foundation to the Volterra prey-predator population in the fol[owing case. We take Volterra's predator equation 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 construction 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 parameters a and b of the predator, whenever the host process visits the particular state a* = a/b the predator takes rest (saturates) from its evolution. We find the probability of asymptotic saturating of the predator.