A linear birth and death process under the influence of another process

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Abstract
Let {X 1 (t), X 2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X 2(t), t ≧ 0} may influence the growth of the process {X 1(t), t ≧ 0}, while the process X 2 (·) itself grows without any influence whatsoever of the first process. The process X 2 (·) is taken to be a simple linear birth and death process with λ 2 and µ 2 as its birth and death rates respectively. The process X 1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X 2 (·) in the following manner: λ (t) = λ 1 (θ + X 2 (t)); µ(t) = µ 1 (θ + X 2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X 1 (·), first conditionally given a realization of the process {X 2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X 2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X 1 (t) always exists as t→∞, except only when both λ 1 > µ 1 and λ 2 > µ 2. Also, certain problems concerning moments of the process, regression of X 1 (t) on X 2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.