Probability of Population Extinction

Definition

For a birth and death process with absorbing state $0$ (${\lambda }_0={\mu }_0=0$), the probability of population extinction is ${\lim }_{t\to \infty }{p}_0(t)$.

Theorem 6.2 (Allen). Let ${\mu }_0=0={\lambda }_0$ with initial population $X(0)=m\ge 1$.

(i) Infinite State Space. Suppose ${\mu }_i>0$ and ${\lambda }_i>0$ for all $i=1,2,\dots$

If ${\sum }_{i=1}^{\infty }\frac{{\mu }_1{\mu }_2\cdots {\mu }_i}{{\lambda }_1{\lambda }_2\cdots {\lambda }_i}=\infty ,$ then ${\lim }_{t\to \infty }{p}_0(t)=1$ (certain extinction).

If ${\sum }_{i=1}^{\infty }\frac{{\mu }_1{\mu }_2\cdots {\mu }_i}{{\lambda }_1{\lambda }_2\cdots {\lambda }_i}<\infty ,$ then

${\lim }_{t\to \infty }{p}_0(t)=\frac{{\sum }_{i=m}^{\infty }\frac{{\mu }_1{\mu }_2\cdots {\mu }_i}{{\lambda }_1{\lambda }_2\cdots {\lambda }_i}}{1+{\sum }_{i=1}^{\infty }\frac{{\mu }_1{\mu }_2\cdots {\mu }_i}{{\lambda }_1{\lambda }_2\cdots {\lambda }_i}}$

(ii) Finite State Space. If ${\lambda }_i=0$ for $i\ge N$, then ${\lim }_{t\to \infty }{p}_0(t)=1$ (extinction is certain).

Interpretation

When the death rates dominate the birth rates sufficiently (making the infinite sum diverge), extinction is guaranteed. When birth rates dominate more strongly, there is a positive probability that the population grows indefinitely (i.e., extinction is not certain).

Example: Simple Birth and Death Process

For ${\lambda }_i=\lambda i$ and ${\mu }_i=\mu i$:

${\sum }_{i=1}^{\infty }\frac{{\mu }_1\cdots {\mu }_i}{{\lambda }_1\cdots {\lambda }_i}={\sum }_{i=1}^{\infty }{\left(\frac{\mu }{\lambda }\right)}^i=\{\begin{array}{ll}\infty ,& \mu \ge \lambda \\ \frac{\mu /\lambda }{1-\mu /\lambda }<\infty ,& \mu <\lambda \end{array}$

• If $\mu \ge \lambda$: extinction is certain ($\lim p_0(t)=1$)
• If $\mu <\lambda$: $\lim p_0(t)=(\mu /\lambda {)}^m$ (not certain; positive probability of indefinite growth)

Related

• Expected Time to Extinction
• Birth and Death Process
• Extinction Probability
• Stochastic Logistic Growth Process