Environmental stochasticity in Malthusian population

In Malthusian populations $\Lambda(N_t) = \lambda$ constant so that

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

Note that $\lambda$ is actually the median rate of growth (in fact it can be obtained by setting $\varepsilon = 0$, i.e. $\delta = 1$). The corresponding deterministic model is

$$N_{t+1} = \lambda N_t$$

and then - if there were no stochasticity - the population would tend to become extinct if $\lambda <1$. If environmental stochasticity is introduced, the growth rate varies from one year to the next. Therefore, given an initial condition with abundance equal to $N_0$, the time evolution of the population size is

$$N_t = \lambda_{t-1}\lambda_{t-2}...\lambda_0 \cdot N_0 = \lambda^t \exp(\varepsilon _0+\varepsilon _1+...+\varepsilon _{t-1})\cdot N_0 .$$

To understand the long-run trend of abundance we consider the dynamics of $N_t$ in logarithmic scale, namely

$$\log N_t = t \log\lambda + \log N_0 + \psi_t = rt + \log N_0 + \psi_t$$ (3.4)

where $\psi_t =\sum\limits_{i = 0}^{t - 1} {{\varepsilon_i}}$ is an additive noise given by the sum of the $\varepsilon _t$ while $r=\log\lambda$. It is easy to deduce the properties of $\psi_t$, because

$$\left[\psi_t\right] = \sum\limits_{i = 0}^{t - 1} {\text{E}\left[ {{\varepsilon _i}} \right]}=0$$ (3.5)

$$[\psi_t] = \sum\limits_{i = 0}^{t - 1} {\text{Var}\left[ {{\varepsilon _i}} \right]} = t \text{Var}\left[\varepsilon _i\right] = t \sigma _{\varepsilon}^2.$$ (3.6)

The relationship 11 stems from the random variables $\varepsilon _t$ being statistically independent of each other (white noise property).

It is worthwhile to remark that, while the mean value of $\psi_t$ is null, its variance increases linearly with time. In conclusion, by considering abundances in logarithmic scale ( $Z_t = \log\left(N_t\right)$) one gets

$$Z_t = t\log\lambda + Z_0 + \psi_t = rt + Z_0 + \psi_t$$

$$\left[Z_t\right] = t\log\lambda + Z_0 = rt + Z_0$$

$$\left[Z_t\right] = \left[\psi_t\right] = t\sigma _{\varepsilon}^2 .$$

In particular, if $\varepsilon _t$ is normally distributed, $\psi_t$ is normal too, because it is the sum of normal and independent variables. Therefore, the logarithmic abundance is distributed as a Gaussian and its average increases (or decreases) linearly with time, while its standard deviation increases with the root of time (see Fig. 12). We thus conclude that a population subject to environmental stochasticity tends to grow on average if $r = \log\lambda > 0$ (i.e., if $\lambda>1$ ) just like in the deterministic model. However the variance of the logarithm of abundance also tends to grow over time. So, even with $\lambda>1$, for increasing times, there can occur very small population sizes with high probability (again see Fig. 12).

Figure 12: Distributions, at various times ( $t = 0, 20, 100, 200$ and $400$ years), of the natural logarithm of a hypothetical population size obtained by simulation from the initial condition $\log (N_0) = 5$. The assumption is Malthusian growth in the presence of environmental stochasticity only (no demographic stochasticity). Parameter values are $r = 0.05 yr^{-1}$ and $\sigma _{\epsilon }^2 =0.05$. Redrawn afterLande and Saether (2003).

It must be strongly remarked that $\lambda$ is the median, not the mean, value of $\lambda_t$. To establish whether a population is growing or declining on the average, one might be tempted to calculate $\lambda _t = \frac {N_{t+1}}{N_t}$ from data and then find the mean of the $\lambda_t$ and see if it is larger or smaller than 1. This procedure is however incorrect. In fact, we have already seen that the lognormal distribution of multiplicative noise $\delta_t$ (or equivalently the normal distribution of $\varepsilon _t$ ) implies

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

Therefore a population can be declining on average ($\lambda <1$) even if the mean value of the finite rates of increase is larger than 1. The likelihood of this event is larger if environmental stochasticity is higher (larger $\sigma _{\varepsilon}^2$) because the multiplication factor $\exp (\frac {{\sigma _\varepsilon ^2}}{2})$ can be much greater than unity. If we applied the incorrect estimate, we might falsely envisage a rosy future for a population by estimating its growth rate as being greater than 1, whereas in reality a correct estimate would provide a value less than 1. Note that we can instead correctly use the average of the logarithmic growth-rates. Indeed

   E$$\left [\log\lambda _t\right] =$$E$$\left [\log (\lambda\delta _t)\right] =\log\lambda +$$E$$\left [\log\delta _t\right] = r +$$E$$\left [\varepsilon _t\right] = r.$$

To illustrate this concept in an effective manner by means of an example, Lewontin and Cohen (1969) present the hypothetical case of a population with annual reproduction whose growth-rate is influenced by environmental conditions. In particular, they assume that usually the growth rate is $\lambda = 1.1$, however there are some critical years in which $\lambda = 0.3$. These critical years occur with low probability, let's say on average once every ten years. Thus the average of the $\lambda_t$ would amount to $1.1 \times \frac {9}{{10}} + 0.3 \times \frac {1}{{10}} = 1.02$. One might falsely conclude that the expected value of the abundance of the population would be growing by an average of 2% annually. Instead, by calculating the average of the logarithmic rates of increase one gets $\log 1.1 \times \frac {9}{{10}} + \log 0.3 \times \frac {1}{{10}} = -0.0346$ which implies that the instantaneous growth rate $r$ is negative. The correct estimate of $\lambda$ is therefore $\exp(-0.0346) = 0.966 < 1$, which leads to the conclusion that the population is doomed to extinction in the long term. Fig. 13 shows, on a logarithmic scale, the abundances obtained via 100 simulations for a population whose growth rate is randomly drawn each year and is just 1.1 with probability 90% and 0.3 in the remaining 10% of the cases. As one can see from the time evolution of the average abundance, the population's fate is extinction. The bottom panels of the same figure display the histograms of abundance frequencies in subsequent generations (respectively $t = 10, 100, 750, 2000$). From these histograms it is apparent that in almost all simulations abundance is practically vanishing after 2,000 generations.

Figure 13: Possible time evolutions of a Malthusian population subject to environmental stochasticity only. The top panel shows in logarithmic scale the time trend of population abundances for 100 repeated simulations starting from the same initial conditions ($N_0 = 100$), and their mean value (thick line). The bottom boxes show the histograms of frequencies of these abundances at four different times.

It is thus of utmost importance to correctly estimate $\lambda$ from available data (e.g. from periodical population census, as those of exercises 1.15, 1.17 e 1.19 proposed in Casagrandi et al., 2002). The proper way to address the problem is to not directly estimate $\lambda$, rather $r=\log\lambda$. To this purpose it is sufficient to use the standard linear regression. As suggested by Eq. 9, one must simply make a regression of $\log\left(N_t\right)$ against time $t$. The slope of the regression line provides a correct estimate $\widehat r$ of $r$. If $\widehat r$ is positive we can infer that the population tends to grow on average. Of course, the greater the number of data on which we base the estimate of $r$, the greater the confidence in this statement.