## The Immigration - Death Model

In the immigration death model individuals join the population according to a Poisson process with parameterand live a life of length model by an exponential distribution with parameter

Because immigration is constant, there can be no extinction. Similarly the population cannot increase to infinity since if the population isthe number of deaths in any time periodis approximately equal toand this will exceed the constant level of immigration for x big enough.

The Kolmogorov equations give

for

The population may vary about some equilibrium level. Settingand letting represent the equilibrium level of the population being of sizewe obtain

andfor

If the number of immigrants is equal to the number of deaths thenor

Hence

This is a probability distribution soso that

Hence

The probability distribution of the population in the long run is a Poisson distribution with parameter