Demographic stochasticity in density-dependent populations

As the landscape is currently very much fragmented, it may well occur that the habitat of and available resources for wildlife populations are greatly reduced. Therefore, intraspecific competition can operate even in small populations for which it is mandatory to include demographic stochasticity in order to conduct a correct risk evaluation. The theory just illustrated can be adapted to the case where the rate $\nu$ and $\mu$ are not constant (as in the Malthusian growth model) but are dependent on the number $N$ of organisms. For example, one can consider the case in which the dependence on density is of the logistic type, namely

$$\nu _N = \nu _0 - aN$$

$$\mu _N = \mu _0 + bN$$

with $\nu _0$, $\mu _0$, $a$ and $b$ being positive constants. The per capita instantaneous growth rate is thus given by

$$r_N = \nu _0 - \mu _0 - (a + b)N = r_0 - cN.$$

It is possible to prove (Iannelli and Pugliese, 2014) that, if there is density dependence, extinction is always certain, which means that in any case it turns out that

$${\lim _{t \to \infty }}{p_0}(t) = 1 .$$

The case is in a sense analogous to the Malthusian one with stationary populations. In fact, in the deterministic logistic model the growth rate vanishes in correspondence to the carrying capacity, which is the equilibrium toward which the deterministic logistic growth tends. In the stochastic logistic model, we can still introduce a carrying capacity $K=r_0/c$, but the time evolutions of the population, after possibly oscillating randomly around $K$, will go to zero sooner or later with probability one. However, if the carrying capacity is large, $p_0 (t)$ tends to 1 very slowly, as shown in Fig. 10. Indeed, this figure clearly demonstrates the so-called phenomenon of the elbow curve. For carrying capacities larger than about 40 individuals, the time development of $p_0 (t)$ is divided into two phases: the first phase is a rapid increase over time of the extinction probability, while in the second phase the extinction probability increases so slowly that even after hundreds of generations it turns out to be much smaller than the theoretical asymptotic value, which is equal to 1. The carrying capacity of each habitat fragment thus plays a very important role in determining the actual risk of extinction.

Figure: Time evolutions of extinction probability $p_0 (t)$ in density-dependent logistic populations, obtained via simulation (with $r_0 =$ 0.03 time-unit$^{-1}$ , see text) for different values ​​of the carrying capacity $K$.