The size of a fish population follows a birth-death process. Suppose that the birth rate when there

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The size of a fish population follows a birth-death process. Suppose that the birth rate when there are no fish is some positive constant \(\lambda_{0}\) (i.e., migration to empty water is possible); otherwise the birth rate is proportional to the number of fish present. The death rate when there is only one fish is some positive constant \(\mu_{1}\); otherwise the death rate is proportional to the population size minus one. Find the limiting distribution of the population size.

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