Abstract
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.
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Kannan, D. On some Markov models of certain interacting populations. Bltn Mathcal Biology 38, 723–738 (1976). https://doi.org/10.1007/BF02458646
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DOI: https://doi.org/10.1007/BF02458646