5.4 Effects of Immigration

Consider a simple birth and death process which is subject to random immigrants with immigration rate \(v\).
Hence, chance of an increase in population by one unite in interval of length \(\Delta t\), when the population at time \(t\) is \(n\), is given by \[\left (n\, \lambda \Delta t + v\, \Delta t\right )\] which the chance of a loss of one unit in time \(\Delta t\) is still \(\mu \, X(t)\, \Delta t\) \begin {align*} P_n(t + \Delta t) = & P_n(t) (1 -n\, \lambda \, \Delta t - v\, \Delta t)(1 - n\, \mu \, \Delta t) + P_{n - 1}(t)(\overline {n - 1}\, \lambda \, \Delta t + v\, \Delta t)(1 - \overline {n - 1}\, \mu \, \Delta t)\\ & + P_{n + 1}(t) (1 - \overline {n + 1}\, \lambda \, \Delta t - v\, \Delta t)(\overline {n + 1}\, \mu \, \Delta t)\, , \hspace {0.4cm} n\geq 0 \end {align*}

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