Estimating the risk of extinction

In the model we are considering, unlike the one we used to study demographic stochasticity, extinction is never possible in a theoretical sense because the variable $N_t$ can never vanish in finite time if the initial condition $N_0$ is positive. However, we know that below the critical threshold $N_c$ environmental stochasticity loses its importance and at the same time demographic stochasticity is no longer negligible and begins operating. In the same way, in cases where depensation or genetic deterioration plays a role, there exist abundance thresholds below which processes at the scale of each individual can strongly influence a population fate. It then becomes relevant to find the probability that $N_t$ falls below a critical size $N_c$ thus dragging the population into a so-called “vortex” of demographic stochasticity . To calculate this probability it is just sufficient to consider the logarithmic threshold $Z_c = \log\left(N_c\right)$ and calculate the size of the tail to the left of $Z_c$ for the probability distribution of $Z_t = \log\left(N_t\right)$. We know that, if $\varepsilon _t$ is normally distributed, $Z_t$ is normal too and has average equal to $rt + Z_0$ and variance equal to $t\sigma _{\varepsilon}^2$. If we then knew $r$ and $\sigma _{\varepsilon}^2$ we could easily estimate the probability for the population to fall below $Z_c$ at time $t$ (see Fig. 14).

Instead of the theoretical values ​​$r$ and $\sigma _{\varepsilon}^2$, which are generally unknown, we can use their estimates. We already learnt how to calculate $\widehat r$, but we have to understand how to estimate the environmental variance $\sigma _{\varepsilon}^2$. This is less elementary. If we have $m$ available data (all in consecutive years) one can proceed as follows (otherwise things are a bit more complicated). We know that

$$N_{t+1} = \lambda N_t \exp (\varepsilon _t)$$

and hence

$$\log\left(\frac{N_{t+1}}{N_t}\right) = \log\lambda + \varepsilon _t = r + \varepsilon _t.$$

If in the previous relationship we replace $r$ with $\widehat r$, we can get an estimate of the environmental noise as

$${\hat \varepsilon _t} = \log\left(\frac{N_{t+1}}{N_t}\right) - \widehat r$$

and consequently estimate the environmental variance $\sigma _{\varepsilon}^2$ as

$$\widehat \sigma _\varepsilon ^2 = \frac{1}{{m - 2}}\sum\limits_{t = 1}^{m - 1} {\widehat \varepsilon _t^2}$$

where the formula for non-biased variance estimation is used.

Figure: Hypothetical distributions of the logarithm of the abundances for three different instants of time $t = 0, 1$and$$ 10 in a population subject to environmental stochasticity only. The area of ​​the grey region corresponds to the probability for the population to drop below the critical threshold at time $t = 10$.