Demographic phenomena in populations threatened by extinction

Sections 3 and 4 of Chapter 1 have evidenced the biodiversity loss that affects our planet. It is therefore important to study in greater detail the phenomena that increase the vulnerability and extinction risk of animal and plant populations subject to increasing human pressure. To this purpose, it is necessary to both classify the different mechanisms of impact and increase the basic notions of population ecology that allow us to analyse the consequences of these impacts. In this chapter we will study only the simplest phenomena, i.e. those that operate at the level of individual populations (sets of organisms of the same species belonging to a given ecosystem). We will rely on the population dynamics basic concepts, which allow us to understand the fundamentals of demography. In particular, the basics that will be used in the sequel are the concepts of Malthusian population and of density dependent demography, as well as the distinction between populations with reproduction concentrated in time and populations that distribute progeny production over time without a specific reproductive season. To this regard remember that if we let $t$ be the time and $N$ the population size (i.e. the number of conspecific organisms in a given ecosystem) or its density (the number of organisms per unit area or volume), the main demographic models we introduced (without considering populations structured according to age, size or stage) are as follows:

Malthusian demography

As for populations with concentrated reproduction ($t$ = discrete time)

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

where the parameter $\lambda$ is constant and is called the finite rate of demographic increase; as for populations with continuous reproduction ($t$ = continuous time), instead, we can state

$$\frac{{dN}}{{dt}} = \dot N = rN$$

where $r$ is also constant and is called the per capita instantaneous rate of increase. It is the difference between the instantaneous birth rate $\nu$ and the instantaneous death rate $\mu$, which are also supposed to be constant.

Density-dependent demography

As for populations with concentrated reproduction

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

The finite rate of increase $\Lambda$ is not constant, but depends on the size or density of the population (which is equal to the size of the population divided by the area or volume occupied by the same population). The best-known models are those of Beverton and Holt, summarized by the equation

$${N_{t + 1}} = \frac{{\lambda {N_t}{\rm { }}}}{{1{\rm { + }}\alpha {N_t}}},$$

and that of Ricker given by the equation

$${N_{t + 1}} = \lambda {N_t}\exp ( - \beta {N_t}),$$

where $\lambda$, $\alpha$ and $\beta$ are positive constants.

Instead, for populations in which reproduction is continuous in time the general model is

$$\dot N = R(N)N$$

in which the per capita instantaneous rate of increase $R$ is not constant, but depends on the population density $N$. The most used model is the logistic one given by

$$\dot N = rN\left( {1 - \frac{N}{K}} \right)$$

where $K$ is the carrying capacity.

The main factors that lead populations to extinction or at least increase the extinction risk can be roughly classified in the following manner:

• inverse density dependence (the so-called Allee effect or depensation) caused by intraspecific interactions other than competition, such as for example sociality;
• genetic deterioration due to e.g. genetic drift or inbreeding in small populations;
• stochasticity due to either the important action of random events in small populations (demographic stochasticity) or the variability of external conditions that affect demographic parameters (environmental stochasticity).

All of these phenomena do not have in general a large impact on very abundant populations, but they become fundamental in all those species that are represented by small numbers either for natural reasons (for example because they are high in the food chain, e.g. top predators) or because they have been severely depleted due to human pressure. The simple models we have previously illustrated are sometimes inadequate to properly describe these phenomena. It is therefore necessary to introduce new modelling tools with different functional forms. In particular, compared to the traditional viewpoint, we will be obliged to introduce a probabilistic description of some phenomena and consider organisms as discrete entities, thus giving up the traditional approximation that utilizes continuous variables to describe animal and plant abundance.