Risk analysis for populations subject to environmental stochasticity only

For populations whose size is not too small (say $N_c \ll N$) we can neglect demographic stochasticity and consider the environmental one only. Suppose then that individual fitnesses are all equal, but time-dependent

$$w_{i,t}=\overline {{w_t}} .$$

Since the population numbers are not too small, we can approximate $N$ as a real, not an integer variable and thus write

$${N_{t + 1}} = \sum\limits_{i = 1}^{{N_t}} {{w_{i,t}}} = {\overline{w _t}}{N_t} = {\lambda _t}{N_t} .$$

The finite growth rate $\lambda_t$ can differ from year to year due to changes in the external environment (climate, physical and chemical conditions in soil or water, the presence of predators, etc.). These changes affect the average fertility and/or survival of the organisms and thus indirectly affect the rate of population growth. The model we will use to describe environmental stochasticity will always be of this type

$$\lambda _t = \Lambda(N_t)\delta _t$$

namely

$$N_{t+1} = \delta _t \Lambda(N_t)N_t$$ (3.3)

where $\delta_t$ is a random factor. In other words, we assume that environmental variability influences the rate of demographic growth in a multiplicative way. $\delta_t$ is called environmental multiplicative noise and is a number $\geq 0$, so as to guarantee that the model always provides non-negative abundances $N_t$. As we assume that all years are mutually independent and that there is no environmental trend, $\delta_t$ is a random variable that is always drawn from the same probability distribution. For example, many data suggest that $\delta_t$ is often distributed as a lognormal - i.e. $\log (\delta _t)$ is distributed as a normal variable. It should be noted that the deterministic model

$$N_{t+1} = \Lambda(N_t)N_t$$

can be obtained from model 8 by setting the multiplication factor $\delta_t$ to 1. It is convenient to assume that $\delta _t= \exp (\varepsilon _t)$ with $\varepsilon _t$ “white noise”. A stochastic process is called white when it has the following properties

$$\left[\varepsilon_t\right] =$$ 0

$$\left[\varepsilon_t^2\right] = \left[\varepsilon_t\right] = \sigma _{\varepsilon}^2$$

$$\left[\varepsilon _t \varepsilon _{t-\tau}\right] = \tau\neq 0.$$

The last property implies no correlation between the different years. With these assumptions, the median of the random variable $\delta_t$ (the value corresponding to the 50-th percentile) is equal to 1. The deterministic model can therefore be thought of as the median model of the situation with environmental stochasticity. As stated before, it is reasonable to assume that $\varepsilon _t$ be normal and thus $\delta_t$ lognormal. In this case, the graph of the probability density of the multiplicative noise is as shown in Fig. 11. It should be strongly remarked that while the random variable $\varepsilon$, which is normally distributed, has the property that mean, mode and median coincide, this is not the case for the variable $\delta$ = $\exp (\varepsilon)$. Only the median is preserved after exponential transformation, or

   Median of $$\delta = \exp($$Median of $$\varepsilon) = \exp(0) = 1.$$

The mode of $\delta$, instead, is smaller than 1 and the mean larger than 1. In particular, one can prove that

   E$$[ \delta ] = \exp \left(\frac{{\sigma _\varepsilon ^2}}{2}\right)$$

where $\sigma^2_{\varepsilon}$ is the variance of $\varepsilon$. Notice that the variance of the environmental logarithmic noise ( $\sigma^2_{\varepsilon}$) should not be confused with the variance of $\overline{w_t}$ that we previously indicated by $\sigma _e^2$.

Figure 11: Lognormal probability density. The relevant parameters are, respectively, the mean $\mu _{\epsilon }$ and standard deviation $\sigma _{\epsilon }$ of the underlying normal random variable. In this graph $\mu _{\epsilon } = 0$ and $\sigma _{\epsilon } = 1$. The dotted, dash-dotted and solid segments, respectively, demarcate the mode, the median and the mean of the distribution. Since $\mu _{\epsilon } = 0$, the median of the lognormal is equal to 1.