The different types of stochasticity

The dynamics of each population is a mix of deterministic components (i.e. those that we can understand, predict and measure) and stochastic components (i.e. those that we are not able to understand and/or predict and/or measure). Sometimes, the deterministic components of the dynamics prevail, such as in the Seychelles warbler and the wandering albatross (see Fig 4). The first species, as a consequence of an active policy of conservation, has grown in a roughly logistic way stabilizing at the carrying capacity around which it fluctuates with a small stochastic component. The second species, instead, has suffered a high adult mortality (resulting in negative growth rate) because of poisoned baits attached to long-lines of commercial fishing boats. The consequence has been an exponential decrease of its numbers around which there have been small stochastic variations.

Figure 4: Fluctuations in the abundances of a wandering albatross (Diomedea exulans) population in southern Georgia, USA, and of a Seychelles warbler (Acrocephalus sechellensis) population at Cousin Island, Seychelles.

Deterministic factors can sometimes cause fluctuations and irregular oscillations that at first glance may seem completely stochastic. This is the so-called phenomenon of deterministic chaos which is linked to a strong density dependence (overcompensation), like the one that characterizes for example the Ricker model. However, in populations with no age structure, such as those of univoltine insects, rarely are intrinsic rates of demographic increase so large in reality as to cause large chaotic fluctuations. In populations with a rather long life expectation, hence with an age structure, there can be chaotic fluctuations for lower intrinsic growth rates, which in most cases are, nonetheless, considerably higher than the real ones. When irregular fluctuations of plant and animal population numbers are observed, it is fairly certain that in most cases these are linked with truly stochastic factors and more rarely with deterministic chaos.

As for the mechanisms generating these random fluctuations, we can distinguish among three main types of stochasticity (Lande and Saether, 2003). The first source of observed population vagaries is, so to say, trivial and will not be considered in the following: it is the random measurement error. Censuses, counts or estimates of a population size (for example, by means of capture-mark-recapture methods) are never perfect and therefore fluctuations of numbers can simply be caused by these sampling errors. The second type of source is the so-called demographic stochasticity which depends on random events operating at the level of a single individual mortality and reproduction. Like genetic drift it acts with particular strength in small populations. The third type is environmental stochasticity which depends on random events operating at the level of the environment in which populations live. This kind of stochasticity operates effectively and in a comparable way in both small and large populations.

To better understand what we mean by demographic stochasticity, it should be noted that, even in populations with no age or size structure, individuals are all equal only in the average. In particular, with regard to mortality, in a given amount of time each individual can either die or survive. If then in year $t$ a population is for example composed of 5 individuals, in year $t + 1$ there may be 0 or 1 or 2 or 3 or 4 or 5 surviving individuals. At the individual level, the mortality rate must be treated as the probability that an individual dies in the time unit. Therefore if, in the above example, the population had a survival rate from year to year amounting to 40%, there would be a probability of $0.4^{5}\simeq0.01$ that all 5 individuals survive until year $t + 1$ and a probability of $0.6^{5}\simeq0.077$ that all die. As one can remark, the extinction probability of the population, even in the course of a single year, is not at all negligible. A similar reasoning can take place for reproduction, because every sexually mature individual can only produce a whole number of offspring (0 or 1 or 2, etc.). For example, when we say that the population has a birth rate of 2.5 daughters per mother per year, this value is the average resulting from a probability distribution where some mothers produce 0 daughters, some one daughter, some two daughters, some three daughters and so on. In large populations, it is reasonable to use the law of large numbers and interpret probabilities as replaceable by proportions. If in the example under consideration the population consisted in year $t$ not of 5 but of 1000 individuals, the chances that all survive or die would be infinitesimal and we would not commit big mistakes saying that in year $t + 1$ there will be approximately 400 surviving individuals. Similarly, we would not commit big mistakes by saying that the newborn (assuming a sex ratio 1: 1) will be approximately 2,500. However, in small populations demographic stochasticity operates quite effectively, so that even with rather high rates of survival and reproduction the extinction risk cannot be overlooked. Fig. 5 illustrates the effect of demographic stochasticity on different populations of the bighorn sheep (Ovis canadensis). Only the populations that initially have more than 100 individuals have survived for longer than 50 years. Fig. 6 shows the actual decline and final extinction of small populations from four different species. One can note that usually the decline is not sudden, rather relatively gradual.

Figure 5: Relationship between the initial size $N$ and the percentage of surviving populations through time for the bighorn sheep (Ovis canadensis, inset). Data from 120 populations in south-western US (Berger, 1992).

Figure 6: Decline and extinction of four small populations: the golden plover (Pluvialis apricaria) in Scotland (Parr, 1992), the African wild dog (Lycaon pictus) in Serengeti Park, Tanzania (Burrows et al., 1995), two populations of sparrow (Passer domesticus) in Helgeland, Norway (Lande and Saether, 2003), and the California condor (Gymnogyps californianus Dennis et al., 1991).

Unlike demographic stochasticity, environmental stochasticity does not stem from differences between two individuals of the same population, but from temporal variations of the surrounding environment which affect the demographic parameters of all individuals of the population. It is therefore due to seasonal changes of factors external to the population, such as weather (temperature, precipitation, etc.), food availability, the presence of diseases, predation by other species, and so on. For this reason, as already stated, environmental stochasticity does produce effects on both large and small populations, because it influences more or less in the same way the survival and reproduction abilities of each of the organisms that belong to the population. Fig. 7 displays an example of this type of stochasticity. The reproductive success of flamingos of a South African park is correlated, albeit weakly, with rainfall because years with low precipitation are characterized by more frequent reproduction failures, while years with larger precipitation generally allow a good number of chicks to reach fledging.

Figure 7: The influence of rainfall on reproductive success in populations of two species of flamingos in the Etosha National Park, South Africa (Simmons et al., 1996). Vertical bars indicate the total annual rainfall recorded in the park. Reproductive success is indicated by circles: open circles indicate reproduction failure (eggs are laid but no chick will be able to fledge), filled circles indicate reproductive events of small, medium or large success depending on the circle size.