Environmental stochasticity in density-dependent populations

So far we have studied the effects of environmental stochasticity in Malthusian populations. This is not unreasonable because, if our main aim is to estimate the extinction risk in not too big populations, we can, as a first approximation, neglect the phenomena of intraspecific competition. However, we have previously seen that intraspecific competition is not the only mechanism of density dependence. Sometimes there may be density dependence effects even in relatively small populations (such as for example in the case of depensation). Also, if the habitat suitable for a species has been greatly reduced, there can be competition even in populations with numbers close enough to the threshold of demographic stochasticity or genetic deterioration. Therefore, in some cases it may be reasonable to consider a model like the following

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

where $\Lambda(N_t)$ is the median growth rate, $\varepsilon _t$ is the white noise that incorporates environmental stochasticity and $\Lambda(N_t)$ is not a constant, but depends on density. First it is to be noted that

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

and thus, in order to estimate $\Lambda(N)$, one can make a regression (possibly non-linear) of $\log\lambda _t$ against $N_t$. The resulting regression curve can be taken as an estimate of $\log\Lambda(N)$ from which one can derive $\Lambda(N)$.

Figure 15: (A) Time evolution of the population of red deer (Cervus elaphus) in the Yellowstone Park, (B) Logarithmic growth rates vs. deer abundance (filled circles) and fitted linear regression line (in red). Reworked after Lande and Saether (2003).

The properties of the deterministic model depend on the functional form of $\Lambda(N)$. If only intraspecific competition is operating, $\Lambda(N)$ is a decreasing function of $N$ with $\Lambda(0)>1$. Typical examples are the Beverton-Holt and Ricker models already mentioned above. For example, Fig. 15A shows the time evolution of the red deer population in the Yellowstone Park, USA. Clearly growth is not Malthusian, but density dependent. We can test whether the Ricker model provides a good description of available data. To this end we note that the Ricker model can also be written as

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

and thus - apart from the environmental noise $\varepsilon _t$ - the logarithmic growth-rates are related to density $N_t$ via a linearly decreasing relationship. In fact Fig. 15B shows that as a first approximation a straight line interpolates the data quite well. Using the classic linear regression formulas we obtain the following estimates of the demographic parameters: $\lambda = 1.597$, $\beta = 4.135 \times 10^{-5}$.

At present it is difficult, if not impossible, to theoretically analyse the properties of stochastic models with density dependence. In almost all cases we must resort to extensive computer simulation. Usually a fixed time horizon (for example, 100 years) is chosen and a number of simulations (for example 500) is carried out from given initial conditions. One can then calculate various indices, such as for example:

• the mean, standard deviation and median of the population abundance as a function of time;
• the extinction probability as a function of time, estimated as the percentage of replicates that have fallen below the critical threshold $N_c$ within time $t$;
• the mean and median extinction times, estimated as the mean and median of the times at which the population size reaches the critical threshold (of course only for replicates that become extinct).

Figure 16: Simulations obtained through the stochastic Ricker model, with the following parameters: $\lambda = 4$, $\beta =$ 0.001, $\sigma _{\epsilon } =$ 0.35 . The deterministic equilibrium is $1,386$ individuals. (A) Time course of three simulations starting from $500$ initial individuals. (B) Time evolution of the average (dotted yellow line), the median (thin black line) and standard deviation (thick pink line) of $500$ simulations. (C) Time evolution of the extinction probability as a function of time, calculated as the fraction of simulations that fall below the threshold of 200 individuals.

As an example, consider a population whose fluctuations are well described by a stochastic Ricker model, that is

$$N_{t+1}=\lambda {N_t}\exp \left( { - \beta {N_t}} \right)\exp ({\varepsilon _t})$$ (3.7)

with $\lambda>1$. Fig. 16A shows three simulations with a stochastic Ricker model in which $\lambda = 4$ and $\beta = 0.001$. The equilibrium of the deterministic model is equal to $\frac {{\log \lambda }}{\beta } = 1,386$ individuals. Even though the deterministic model is characterized by a stable equilibrium (with damped oscillations because the slope of the curve at equilibrium is $1 - \log\lambda = -0.386$), the environmental variance is so high as to produce permanent large fluctuations around this value. However, carrying out 500 simulations and reporting the time evolution of the mean, median and standard deviation of $N_t$, we see that the process is basically stationary, because each of these indicators tends to a constant value (Fig. 16B). Among other things, the mean tends to the deterministic equilibrium value. However, one should not be tempted to generalize this outcome, because this is a peculiarity of the Ricker model that does not apply to other models with density dependence (for istance, it does not apply to the Beverton-Holt model). In addition, one can note that the median abundance is always below the mean.

To calculate the extinction risk we must first set a threshold value as critical population size. Suppose, for example, that $N_c = 200$ . The trend of the extinction probability is reported in Fig. 16C which shows that the risk of extinction within 100 years is approximately equal to 40%. For that 40% of simulations that become extinct, the average time to extinction is 53.7 years and the median time is 57 years. Of course, things would radically change if we considered a different critical threshold. If the threshold were to be reduced to 50 individuals the probability of extinction within 100 years would decrease down to 0.6%. Since one does not actually know very well the critical thresholds for the different species, it is always recommendable that extinction risk studies include a sensitivity analysis for varying extinction thresholds $N_c$.