Example 6.4 (A Linear Growth Model with Immigration) A model in which n = n, n

Example 6.4 (A Linear Growth Model with Immigration) A model in which

μn = nμ, n  1

λn = nλ + θ, n  0 is called a linear growth process with immigration. Such processes occur naturally in the study of biological reproduction and population growth. Each individual in the population is assumed to give birth at an exponential rate λ; in addition, there is an exponential rate of increase θ of the population due to an external source such as immigration. Hence, the total birth rate where there are n persons in the system is nλ + θ. Deaths are assumed to occur at an exponential rate μ for each member of the population, so μn = nμ.

Let X(t) denote the population size at time t. Suppose that X(0) = i and let M(t) = E[X(t)]

We will determine M(t) by deriving and then solving a differential equation that it satisfies.

We start by deriving an equation for M(t + h) by conditioning on X(t). This yields M(t + h) = E[X(t + h)]

= E[E[X(t + h)|X(t)]]