Linear Growth Model with Immigration

Example

Consider a population model with the following rates:

${\mu }_n=n\mu ,n\ge 1$ ${\lambda }_n=n\lambda +\theta ,n\ge 0$

This is a birth and death process where:

• Each individual gives birth at rate $\lambda$ (total: $n\lambda$)
• There is an external immigration source at rate $\theta$
• Each individual dies at rate $\mu$ (total: $n\mu$)

Expected Population Size

Let $X(t)$ be the population size at time $t$ with $X(0)=i$, and $M(t)=\mathbb{E}[X(t)]$.

Given $X(t)$, the population at $t+h$ behaves as:

$X(t+h)=\{\begin{array}{ll}X(t)+1,& \text{w.p. }[\theta +X(t)\lambda ]h+o(h)\\ X(t)-1,& \text{w.p. }X(t)\mu h+o(h)\\ X(t),& \text{w.p. }1-[\theta +X(t)(\lambda +\mu )]h+o(h)\end{array}$

Taking expectations yields the differential equation:

$M^{\prime }(t)=(\lambda -\mu )M(t)+\theta$

Solution (when $\lambda \ne \mu$)

$M(t)=\frac{\theta }{\lambda -\mu }\left[e^{(\lambda -\mu )t}-1\right]+i{e}^{(\lambda -\mu )t}$

Solution (when $\lambda =\mu$)

$M(t)=\theta t+i$

Interpretation

When $\lambda >\mu$, the population grows exponentially. When $\lambda =\mu$, growth is linear (driven only by immigration). When $\lambda <\mu$, the population eventually stabilizes around $\theta /(\mu -\lambda )$.

Related

• Birth and Death Process
• Continuous-Time Markov Chain