Survival probability \(P(Z_n>0)\) and time to extinction \(\tau\)

Lemma If \(\xi \ge 0\) with probability 1 and is not identical to 0, then \[\begin{equation} \label{10.1} P(\xi >0) \ge \frac{(E\xi)^2}{E\xi^2}. \end{equation}\]

Proof. For random variables \(X\ge 0\) and \(Y\ge 0\) on Hölder’s inequality reads \[ E(XY)\le (E(X^p))^{1/p}(E(Y^q))^{1/q}, \qquad \frac{1}{p}+\frac{1}{q}=1. \] By Holder inequality with \(p=q=2\) \[ E(\xi)=E (\xi I\{\xi>0\})\le \sqrt{E\xi^2EI^2\{\xi>0\}}=\sqrt{E\xi^2P(\xi>0)}, \] where \[ I\{\xi>0\}= \left\{ \begin{array}{ll} 0, & \mbox{if} \quad \xi=0,\\ 1, & \mbox{if}\quad \xi>0. \end{array} \right. \] Squaring both sides of this inequality, we arrive at (10.1). The lemma is proved.

Applying the lemma, for the branching process \(Z_n\) with offspring mean \(\mu\) and offspring variance \(\sigma^2\), we obtain \[ \mu^n=EZ_n\ge P(Z_n>0)\ge \frac{(EZ_n)^2}{EZ^2_n}\\ =\frac{\mu^{2n}}{\sigma^2\frac{\mu^{n-1}(\mu^n-1)}{\mu-1}+\mu^{2n}}=\frac{\mu^{n+1}(1-\mu)}{\sigma^2(1-\mu^n)+\mu^{n+1}(1-\mu)}. \]

Here we used the formulas for \(EZ_n\) and \(EZ^2_n\) obtained using the probability generating functions technique. These upper and lower bounds for the survival probability \(P(Z_n>0)\) assume that the process starts with one ancestor, i.e., \(Z_0=1\). The following theorem holds for any fixed number of ancestors \(Z_0=N\ge 1\).

Theorem 10.1 Consider a subcritical Galton-Watson process, starting with \(Z_0=N\ge1\) individuals. If the offspring (reproduction) variance \(\sigma^2 <\infty\), then \[\begin{equation} \label{10.2} \frac{N\mu^{n+1}(1-\mu^n)^{N-1}(1-\mu)}{\sigma^2(1-\mu^n)+\mu^{n+1}(1-\mu)}\le P(Z_n>0)\le N\mu^n. \end{equation}\]

Note that the lower bound of the survival probability in (\(\ref{10.2}\)) depends on both offspring mean \(\mu\) and offspring variance \(\sigma^2\).

Next we will discuss the expected value of the time to extinction \(\tau\), say, when the process begins with large number \(N\to \infty\) ancestors. Define the time to extinction of a subcritical process \(Z_n\) starting with \(N\) ancestors by \[ \tau= \min_n \{Z_n \ |\ X_0=N\}=0. \] Next theorem reveals the asymptotic behavior of the expectation of \(\tau\) as \(N\to \infty\).

Theorem 10.2 If \(\mu<1\) and \(E(Z_1(1+Z_1)<\infty\), then \[\begin{equation} \label{10.3} E(\tau\ |\ Z_0=N)\sim \frac{\ln N}{|\ln \mu|}, \qquad \mbox{as}\quad N\to \infty. \end{equation}\] It is interesting that the behavior of the expected time to extinction depends on the parameters \(\mu\) and \(N\) only, but not on \(\sigma^2\).

Real-world application

The following discussion is based on the paper Caswell, H., Fujiwara, M., Brault, S. (1999) “Declining survival probability threatens the North Atlantic right whale”. Proc. Natl. Acad. sci USA, 96(1999), 3308-3313.

ABSTRACT The North Atlantic northern right whale (Eubalaena glacialis) is considered the most endangered large whale species. Its population has recovered only slowly since the cessation of commercial whaling and numbers about 300 individuals. We applied mark-recapture statistics to a catalog of photographically identified individuals to obtain the first statistically rigorous estimates of survival probability for this population. The crude survival (chance of remaining alive) decreased from about 0.99 per year in 1980 to about 0.94 in 1994. We combined this survival trend with a reported decrease in reproductive rate into a branching process model to compute population growth rate and extinction probability. Population growth rate declined from about 1.053 in 1980 to about 0.976 in 1994. Under current conditions the population is doomed to extinction; an upper bound on the expected time to extinction is 191 years. The most effective way to improve the prospects of the population is to reduce mortality. The right whale is at risk from entanglement in fishing gear and from collisions with ships. Reducing this human-caused mortality is essential to the viability of this population.

This paper study the threats posed by a decline of survival probabilities for the North Atlantic right whale, within the framework of the following model. A female right whale may produce 0 or 1 females the following year. It is assumed that the death of a parent results in the death of a calf in the first year. Thus, a female at time \(n\) produces no o§spring if she dies before \(n+1\); one offspring (herself) if she survives without reproducing female offspring and two offspring (herself and her calf) if she survives and gives birth to a female calf. Generation length is then one year. Let \(p\) be the survival probability and \(g\) be the probability of begetting a female calf. The probabilities \(p_i\) of producing \(i\) offspring are then \[ p_0=(1-p)(1-g)+(1-p)g, \] \[ p_1=p(1-g), \qquad p_2=pg. \] The reproduction (offspring) generating function of the process becomes \[ 1-p+p(1-g)s+pgs^2, \] with mean \(\mu=p(1-g)+2pg=p(1+g)\). Caswell et al. (1999), using different sources, give the following estimates for \(p\), \(g\), and \(\mu\): \[ p=0.945,\quad g=0.051, \quad \mu=0.988, \] \[ p=0,945, \quad g=0.038, \quad \mu=0.976. \]

Applying formulas (\(\ref{10.2}\)) to the data, we obtain the following estimates from below of the number n of generations (years) which the population of whales (now having around 150 female members) can survive with probability higher than 0.99 and from above for the number of generations within which the population will die out with a probability greater than 0.99:

In particular this shows that, provided reproduction conditions remain the same, in the future, then under the worst scenario the whale population will die out within 400 years with a probability of more than 99 percent.

For the expected time to extinction of the North Atlantic Right whales, applying Theorem~10.2 we obtain \[ \mu=0.988, \qquad E(\tau\ | \ Z_0=150)\approx 415, \]

\[ \mu=0.976, \qquad E(\tau\ | \ Z_0=150)\approx 206, \] These numbers agree with the results given in Caswell et al. (1999), showing through direct calculations that \(E(\tau \ |\ Z_0 = 150) \approx 191\) if \(p = 0.94\) and \(g = 0.038\) and, therefore, \(\mu = 0.976\). Our method is however robust in the sense that it is built upon (10.3), so the result does not depend on the particular form of the reproduction distribution.

Figure 1 The probability distribution of extinction time for a population of 150 female whales with constant survival probability p=0.94 and inter-birth interval 5 years (Caswell et al. (1999))
Figure 1 The probability distribution of extinction time for a population of 150 female whales with constant survival probability \(p=0.94\) and inter-birth interval 5 years (Caswell et al. (1999))