Population fluctuations

The numerical strength of plant and animal populations often displays a more or less large variability which contrasts with the simple deterministic models of population dynamics studied in basic courses of ecology. The cause of this variability is in many cases unknown, thus fluctuations are classified as random or, technically speaking, stochastic. It is anyway possible to distinguish between different types of stochasticity and provide a description for each type by using appropriate quantitative models that are not deterministic. These models are very useful in conservation ecology because among other things they allow predictions on the probability of a population future trends and hence the assessment of its risk of extinction.

Figures 1, 3 and 4 contain several examples of the fluctuations in numbers of animal populations (in these specific cases, birds and mammals). For each species, the corresponding graph shows the evolution of the numbers of individuals in different sites. It is important to compare the characteristics of species with short vs. long life expectancy. Consider, for example, the three populations of two birds of the genus Parus (tits) compared with the mute swan Cygnus olor. The blue tit and great tit have lifetimes of about 2 years and age at first reproduction of one year, while the swan has a lifespan of about 10 years and an age at first reproduction of about 4 years. The numbers of blue tits (panel A of Fig. 1) display fluctuations much faster and more pronounced than those of the swan (Panel B), whose population has a combination of long-term (about 30 years) and short-term (a few years) fluctuations. In fact, populations with a larger number of age classes are able to better absorb variations of the external conditions (for example, food availability and weather vagaries) because often it is only the younger classes that actually suffer from this variability.

Figure 1: Fluctuations in population abundances of the tit (panel A) and the mute swan (Panel B). For tits, the population data from Ghent (Belgium) are reported on the left axis, while the population data from Oxford (England) are reported on the right axis. The swan population is also English, living in the River Thames.

As the size $N$ of a population is always a non-negative variable, its variability over time can be effectively described by the coefficient of variation ($CV$), namely the ratio of the standard deviation of the series of abundances $\sigma _{N}$ and the average size of the population $\bar {N}$:

$$CV=\dfrac{\sigma _{N}}{\overline{N}}.$$

An alternative measure of variability is the standard deviation of the logarithm of size $N$, i.e. $\sigma_{\log N}$. For $CV$ values less than 30% the two variability indices are substantially equivalent. Fig. 2 reports the estimated coefficients of variation of different bird populations. You can note that $CV$ tends to decrease with the increase of the age at first reproduction, thus more generally confirming what we already stated for the fluctuations of tits and the mute swan.

Figure 2: Coefficient of variation ($CV$) of the fluctuations in numbers of different bird populations. The $CV$ is computed from the last 20 years only. Parameter $\alpha$ indicates the age at first reproduction of mothers.

Not always can the differences between average life expectations be sufficient to explain the different characteristics of population fluctuations. Consider, in this regard, the cases of chamois and Soay sheep (or primitive mouflon) shown in Fig. 3. While the chamois displays fluctuations with periodicity of about a decade, the Soay sheep has faster fluctuations with periodicity between two and three years. However, the two species have life expectations at birth that are roughly equal (10-12 years). In mammals, then, other factors seem to play a role in determining the stochasticity of population dynamics.

Figure 3: Fluctuations in the abundances of a chamois (Rupicapra rupicapra) population in the Swiss National Park and of a Soay sheep (Ovis aries) population in the island of Hirta, Scotland.